Metamath Proof Explorer

Theorem ringexp0nn

Description: Zero to the power of a positive integer is zero. (Contributed by metakunt, 5-May-2025)

Ref Expression
Hypotheses ringexp0nn.1 
|- ( ph -> R e. Ring )
ringexp0nn.2 
|- ( ph -> N e. NN )
ringexp0nn.3 
|- .^ = ( .g ` ( mulGrp ` R ) )
Assertion ringexp0nn 
|- ( ph -> ( N .^ ( 0g ` R ) ) = ( 0g ` R ) )

Proof

Step Hyp Ref Expression
1 ringexp0nn.1 
 |-  ( ph -> R e. Ring )
2 ringexp0nn.2 
 |-  ( ph -> N e. NN )
3 ringexp0nn.3 
 |-  .^ = ( .g ` ( mulGrp ` R ) )
4 2 ancli 
 |-  ( ph -> ( ph /\ N e. NN ) )
5 oveq1 
 |-  ( x = 1 -> ( x .^ ( 0g ` R ) ) = ( 1 .^ ( 0g ` R ) ) )
6 5 eqeq1d 
 |-  ( x = 1 -> ( ( x .^ ( 0g ` R ) ) = ( 0g ` R ) <-> ( 1 .^ ( 0g ` R ) ) = ( 0g ` R ) ) )
7 oveq1 
 |-  ( x = y -> ( x .^ ( 0g ` R ) ) = ( y .^ ( 0g ` R ) ) )
8 7 eqeq1d 
 |-  ( x = y -> ( ( x .^ ( 0g ` R ) ) = ( 0g ` R ) <-> ( y .^ ( 0g ` R ) ) = ( 0g ` R ) ) )
9 oveq1 
 |-  ( x = ( y + 1 ) -> ( x .^ ( 0g ` R ) ) = ( ( y + 1 ) .^ ( 0g ` R ) ) )
10 9 eqeq1d 
 |-  ( x = ( y + 1 ) -> ( ( x .^ ( 0g ` R ) ) = ( 0g ` R ) <-> ( ( y + 1 ) .^ ( 0g ` R ) ) = ( 0g ` R ) ) )
11 oveq1 
 |-  ( x = N -> ( x .^ ( 0g ` R ) ) = ( N .^ ( 0g ` R ) ) )
12 11 eqeq1d 
 |-  ( x = N -> ( ( x .^ ( 0g ` R ) ) = ( 0g ` R ) <-> ( N .^ ( 0g ` R ) ) = ( 0g ` R ) ) )
13 ringmnd 
 |-  ( R e. Ring -> R e. Mnd )
14 1 13 syl 
 |-  ( ph -> R e. Mnd )
15 eqid 
 |-  ( Base ` R ) = ( Base ` R )
16 eqid 
 |-  ( 0g ` R ) = ( 0g ` R )
17 15 16 mndidcl 
 |-  ( R e. Mnd -> ( 0g ` R ) e. ( Base ` R ) )
18 14 17 syl 
 |-  ( ph -> ( 0g ` R ) e. ( Base ` R ) )
19 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
20 19 15 mgpbas 
 |-  ( Base ` R ) = ( Base ` ( mulGrp ` R ) )
21 20 a1i 
 |-  ( ph -> ( Base ` R ) = ( Base ` ( mulGrp ` R ) ) )
22 18 21 eleqtrd 
 |-  ( ph -> ( 0g ` R ) e. ( Base ` ( mulGrp ` R ) ) )
23 eqid 
 |-  ( Base ` ( mulGrp ` R ) ) = ( Base ` ( mulGrp ` R ) )
24 23 3 mulg1 
 |-  ( ( 0g ` R ) e. ( Base ` ( mulGrp ` R ) ) -> ( 1 .^ ( 0g ` R ) ) = ( 0g ` R ) )
25 22 24 syl 
 |-  ( ph -> ( 1 .^ ( 0g ` R ) ) = ( 0g ` R ) )
26 simplr 
 |-  ( ( ( ph /\ y e. NN ) /\ ( y .^ ( 0g ` R ) ) = ( 0g ` R ) ) -> y e. NN )
27 22 ad2antrr 
 |-  ( ( ( ph /\ y e. NN ) /\ ( y .^ ( 0g ` R ) ) = ( 0g ` R ) ) -> ( 0g ` R ) e. ( Base ` ( mulGrp ` R ) ) )
28 eqid 
 |-  ( +g ` ( mulGrp ` R ) ) = ( +g ` ( mulGrp ` R ) )
29 23 3 28 mulgnnp1 
 |-  ( ( y e. NN /\ ( 0g ` R ) e. ( Base ` ( mulGrp ` R ) ) ) -> ( ( y + 1 ) .^ ( 0g ` R ) ) = ( ( y .^ ( 0g ` R ) ) ( +g ` ( mulGrp ` R ) ) ( 0g ` R ) ) )
30 26 27 29 syl2anc 
 |-  ( ( ( ph /\ y e. NN ) /\ ( y .^ ( 0g ` R ) ) = ( 0g ` R ) ) -> ( ( y + 1 ) .^ ( 0g ` R ) ) = ( ( y .^ ( 0g ` R ) ) ( +g ` ( mulGrp ` R ) ) ( 0g ` R ) ) )
31 simpr 
 |-  ( ( ( ph /\ y e. NN ) /\ ( y .^ ( 0g ` R ) ) = ( 0g ` R ) ) -> ( y .^ ( 0g ` R ) ) = ( 0g ` R ) )
32 31 oveq1d 
 |-  ( ( ( ph /\ y e. NN ) /\ ( y .^ ( 0g ` R ) ) = ( 0g ` R ) ) -> ( ( y .^ ( 0g ` R ) ) ( +g ` ( mulGrp ` R ) ) ( 0g ` R ) ) = ( ( 0g ` R ) ( +g ` ( mulGrp ` R ) ) ( 0g ` R ) ) )
33 eqid 
 |-  ( .r ` R ) = ( .r ` R )
34 19 33 mgpplusg 
 |-  ( .r ` R ) = ( +g ` ( mulGrp ` R ) )
35 34 eqcomi 
 |-  ( +g ` ( mulGrp ` R ) ) = ( .r ` R )
36 15 35 16 ringrz 
 |-  ( ( R e. Ring /\ ( 0g ` R ) e. ( Base ` R ) ) -> ( ( 0g ` R ) ( +g ` ( mulGrp ` R ) ) ( 0g ` R ) ) = ( 0g ` R ) )
37 1 18 36 syl2anc 
 |-  ( ph -> ( ( 0g ` R ) ( +g ` ( mulGrp ` R ) ) ( 0g ` R ) ) = ( 0g ` R ) )
38 37 adantr 
 |-  ( ( ph /\ y e. NN ) -> ( ( 0g ` R ) ( +g ` ( mulGrp ` R ) ) ( 0g ` R ) ) = ( 0g ` R ) )
39 38 adantr 
 |-  ( ( ( ph /\ y e. NN ) /\ ( y .^ ( 0g ` R ) ) = ( 0g ` R ) ) -> ( ( 0g ` R ) ( +g ` ( mulGrp ` R ) ) ( 0g ` R ) ) = ( 0g ` R ) )
40 32 39 eqtrd 
 |-  ( ( ( ph /\ y e. NN ) /\ ( y .^ ( 0g ` R ) ) = ( 0g ` R ) ) -> ( ( y .^ ( 0g ` R ) ) ( +g ` ( mulGrp ` R ) ) ( 0g ` R ) ) = ( 0g ` R ) )
41 30 40 eqtrd 
 |-  ( ( ( ph /\ y e. NN ) /\ ( y .^ ( 0g ` R ) ) = ( 0g ` R ) ) -> ( ( y + 1 ) .^ ( 0g ` R ) ) = ( 0g ` R ) )
42 6 8 10 12 25 41 nnindd 
 |-  ( ( ph /\ N e. NN ) -> ( N .^ ( 0g ` R ) ) = ( 0g ` R ) )
43 4 42 syl 
 |-  ( ph -> ( N .^ ( 0g ` R ) ) = ( 0g ` R ) )