Theorem ecqusaddd 13311

Description: Addition of equivalence classes in a quotient group. (Contributed by AV, 25-Feb-2025.)

Hypotheses

Ref	Expression
ecqusaddd.i	|- ( ph -> I e. (NrmSGrp ` R ) )
ecqusaddd.b	|- B = ( Base ` R )
ecqusaddd.g	|- .~ = ( R ~QG I )
ecqusaddd.q	|- Q = ( R /.s .~ )

Assertion

Ref	Expression
ecqusaddd	|- ( ( ph /\ ( A e. B /\ C e. B ) ) -> [ ( A ( +g ` R ) C ) ] .~ = ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) )

Proof of Theorem ecqusaddd

Step	Hyp	Ref	Expression
1		ecqusaddd.i	. . . . . 6 |- ( ph -> I e. (NrmSGrp ` R ) )
2	1	anim1i 340	. . . . 5 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( I e. (NrmSGrp ` R ) /\ ( A e. B /\ C e. B ) ) )
3		3anass 984	. . . . 5 |- ( ( I e. (NrmSGrp ` R ) /\ A e. B /\ C e. B ) <-> ( I e. (NrmSGrp ` R ) /\ ( A e. B /\ C e. B ) ) )
4	2, 3	sylibr 134	. . . 4 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( I e. (NrmSGrp ` R ) /\ A e. B /\ C e. B ) )
5		ecqusaddd.q	. . . . . 6 |- Q = ( R /.s .~ )
6		ecqusaddd.g	. . . . . . 7 |- .~ = ( R ~QG I )
7	6	oveq2i 5930	. . . . . 6 |- ( R /.s .~ ) = ( R /.s ( R ~QG I ) )
8	5, 7	eqtri 2214	. . . . 5 |- Q = ( R /.s ( R ~QG I ) )
9		ecqusaddd.b	. . . . 5 |- B = ( Base ` R )
10		eqid 2193	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		eqid 2193	. . . . 5 |- ( +g ` Q ) = ( +g ` Q )
12	8, 9, 10, 11	qusadd 13307	. . . 4 |- ( ( I e. (NrmSGrp ` R ) /\ A e. B /\ C e. B ) -> ( [ A ] ( R ~QG I ) ( +g ` Q ) [ C ] ( R ~QG I ) ) = [ ( A ( +g ` R ) C ) ] ( R ~QG I ) )
13	4, 12	syl 14	. . 3 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( [ A ] ( R ~QG I ) ( +g ` Q ) [ C ] ( R ~QG I ) ) = [ ( A ( +g ` R ) C ) ] ( R ~QG I ) )
14	6	eceq2i 6627	. . . 4 |- [ A ] .~ = [ A ] ( R ~QG I )
15	6	eceq2i 6627	. . . 4 |- [ C ] .~ = [ C ] ( R ~QG I )
16	14, 15	oveq12i 5931	. . 3 |- ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = ( [ A ] ( R ~QG I ) ( +g ` Q ) [ C ] ( R ~QG I ) )
17	6	eceq2i 6627	. . 3 |- [ ( A ( +g ` R ) C ) ] .~ = [ ( A ( +g ` R ) C ) ] ( R ~QG I )
18	13, 16, 17	3eqtr4g 2251	. 2 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = [ ( A ( +g ` R ) C ) ] .~ )
19	18	eqcomd 2199	1 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> [ ( A ( +g ` R ) C ) ] .~ = ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1364   e. wcel 2164   ` cfv 5255  (class class class)co 5919   [cec 6587   Basecbs 12621   +g cplusg 12698   /._s cqus 12886  NrmSGrpcnsg 13241   ~_QG cqg 13242

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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-tp 3627  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-er 6589  df-ec 6591  df-qs 6595  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-0g 12872  df-iimas 12888  df-qus 12889  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-minusg 13079  df-subg 13243  df-nsg 13244  df-eqg 13245

This theorem is referenced by:  ecqusaddcl  13312