The concept of the Switch as layer 3 is that, you are converting your switch into a Router and the vlan interfaces as its route interfaces. All devices connected to each of the vlans configured must use the ip address of the vlan interface of the Switch connected to that vlan as their gateway. The vlan where you specified the default route 0.0.0.0/0 must be connected to the router which provides internet connection and there must be route configured on that router going to each of the subnets since subnets are now 1 hop away from the router. Also, NAT on the router must be configured to allow the translations of the ip address of each subnet. When configured that way, all packets from any subnet destined to internet will be forwarded to the router that provides internet and the router forwards it to the destination. Now it is important to make sure that from the internet-router all devices in different subnets must be reachable or else the reply packet from the internet will not reach the host on the subnet. If you are using a modem router for internet connection, you can still configure it statistically and used the vlan interface ip of the switch where the route to 0.0.0.0/0 is configured.
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