Theorem ecqusaddd 13972

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 6063	. . . . . 6 |- ( R /.s .~ ) = ( R /.s ( R ~QG I ) )
8	5, 7	eqtri 2255	. . . . 5 |- Q = ( R /.s ( R ~QG I ) )
9		ecqusaddd.b	. . . . 5 |- B = ( Base ` R )
10		eqid 2234	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		eqid 2234	. . . . 5 |- ( +g ` Q ) = ( +g ` Q )
12	8, 9, 10, 11	qusadd 13968	. . . 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 6807	. . . 4 |- [ A ] .~ = [ A ] ( R ~QG I )
15	6	eceq2i 6807	. . . 4 |- [ C ] .~ = [ C ] ( R ~QG I )
16	14, 15	oveq12i 6064	. . 3 |- ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = ( [ A ] ( R ~QG I ) ( +g ` Q ) [ C ] ( R ~QG I ) )
17	6	eceq2i 6807	. . 3 |- [ ( A ( +g ` R ) C ) ] .~ = [ ( A ( +g ` R ) C ) ] ( R ~QG I )
18	13, 16, 17	3eqtr4g 2292	. 2 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = [ ( A ( +g ` R ) C ) ] .~ )
19	18	eqcomd 2240	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 2205   ` cfv 5354  (class class class)co 6052   [cec 6767   Basecbs 13229   +g cplusg 13307   /._s cqus 13530  NrmSGrpcnsg 13902   ~_QG cqg 13903

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-tp 3699  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-er 6769  df-ec 6771  df-qs 6775  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-mulr 13321  df-0g 13488  df-iimas 13532  df-qus 13533  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-subg 13904  df-nsg 13905  df-eqg 13906

This theorem is referenced by:  ecqusaddcl  13973