Theorem grpridd 13637

Description: The identity element of a group is a right identity. Deduction associated with grprid 13635. (Contributed by SN, 29-Jan-2025.)

Hypotheses

Ref	Expression
grpbn0.b	|- B = ( Base ` G )
grplid.p	|- .+ = ( +g ` G )
grplid.o	|- .0. = ( 0g ` G )
grplidd.g	|- ( ph -> G e. Grp )
grplidd.1	|- ( ph -> X e. B )

Assertion

Ref	Expression
grpridd	|- ( ph -> ( X .+ .0. ) = X )

Proof of Theorem grpridd

Step	Hyp	Ref	Expression
1		grplidd.g	. 2 |- ( ph -> G e. Grp )
2		grplidd.1	. 2 |- ( ph -> X e. B )
3		grpbn0.b	. . 3 |- B = ( Base ` G )
4		grplid.p	. . 3 |- .+ = ( +g ` G )
5		grplid.o	. . 3 |- .0. = ( 0g ` G )
6	3, 4, 5	grprid 13635	. 2 |- ( ( G e. Grp /\ X e. B ) -> ( X .+ .0. ) = X )
7	1, 2, 6	syl2anc 411	1 |- ( ph -> ( X .+ .0. ) = X )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2201   ` cfv 5325  (class class class)co 6020   Basecbs 13102   +g cplusg 13180   0gc0g 13359   Grpcgrp 13603

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4206  ax-pow 4263  ax-pr 4298  ax-un 4529  ax-cnex 8125  ax-resscn 8126  ax-1re 8128  ax-addrcl 8131

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3653  df-sn 3674  df-pr 3675  df-op 3677  df-uni 3893  df-int 3928  df-br 4088  df-opab 4150  df-mpt 4151  df-id 4389  df-xp 4730  df-rel 4731  df-cnv 4732  df-co 4733  df-dm 4734  df-rn 4735  df-res 4736  df-iota 5285  df-fun 5327  df-fn 5328  df-fv 5333  df-riota 5973  df-ov 6023  df-inn 9146  df-2 9204  df-ndx 13105  df-slot 13106  df-base 13108  df-plusg 13193  df-0g 13361  df-mgm 13459  df-sgrp 13505  df-mnd 13520  df-grp 13606

This theorem is referenced by:  rnglidlmcl  14515