Theorem ecqusaddd 13929

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 1009	. . . . 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 6052	. . . . . 6 |- ( R /.s .~ ) = ( R /.s ( R ~QG I ) )
8	5, 7	eqtri 2253	. . . . 5 |- Q = ( R /.s ( R ~QG I ) )
9		ecqusaddd.b	. . . . 5 |- B = ( Base ` R )
10		eqid 2232	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		eqid 2232	. . . . 5 |- ( +g ` Q ) = ( +g ` Q )
12	8, 9, 10, 11	qusadd 13925	. . . 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 6796	. . . 4 |- [ A ] .~ = [ A ] ( R ~QG I )
15	6	eceq2i 6796	. . . 4 |- [ C ] .~ = [ C ] ( R ~QG I )
16	14, 15	oveq12i 6053	. . 3 |- ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = ( [ A ] ( R ~QG I ) ( +g ` Q ) [ C ] ( R ~QG I ) )
17	6	eceq2i 6796	. . 3 |- [ ( A ( +g ` R ) C ) ] .~ = [ ( A ( +g ` R ) C ) ] ( R ~QG I )
18	13, 16, 17	3eqtr4g 2290	. 2 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = [ ( A ( +g ` R ) C ) ] .~ )
19	18	eqcomd 2238	1 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> [ ( A ( +g ` R ) C ) ] .~ = ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   ` cfv 5343  (class class class)co 6041   [cec 6756   Basecbs 13186   +g cplusg 13264   /._s cqus 13487  NrmSGrpcnsg 13859   ~_QG cqg 13860

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4218  ax-sep 4221  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-cnex 8206  ax-resscn 8207  ax-1cn 8208  ax-1re 8209  ax-icn 8210  ax-addcl 8211  ax-addrcl 8212  ax-mulcl 8213  ax-addcom 8215  ax-addass 8217  ax-i2m1 8220  ax-0lt1 8221  ax-0id 8223  ax-rnegex 8224  ax-pre-ltirr 8227  ax-pre-ltadd 8231

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-pw 3667  df-sn 3688  df-pr 3689  df-tp 3690  df-op 3691  df-uni 3908  df-int 3943  df-iun 3986  df-br 4103  df-opab 4165  df-mpt 4166  df-id 4405  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-f1 5348  df-fo 5349  df-f1o 5350  df-fv 5351  df-riota 5994  df-ov 6044  df-oprab 6045  df-mpo 6046  df-er 6758  df-ec 6760  df-qs 6764  df-pnf 8298  df-mnf 8299  df-ltxr 8301  df-inn 9226  df-2 9284  df-3 9285  df-ndx 13189  df-slot 13190  df-base 13192  df-sets 13193  df-iress 13194  df-plusg 13277  df-mulr 13278  df-0g 13445  df-iimas 13489  df-qus 13490  df-mgm 13543  df-sgrp 13589  df-mnd 13604  df-grp 13690  df-minusg 13691  df-subg 13861  df-nsg 13862  df-eqg 13863

This theorem is referenced by:  ecqusaddcl  13930