Theorem ecqusaddd 13368

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 5933	. . . . . 6 |- ( R /.s .~ ) = ( R /.s ( R ~QG I ) )
8	5, 7	eqtri 2217	. . . . 5 |- Q = ( R /.s ( R ~QG I ) )
9		ecqusaddd.b	. . . . 5 |- B = ( Base ` R )
10		eqid 2196	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		eqid 2196	. . . . 5 |- ( +g ` Q ) = ( +g ` Q )
12	8, 9, 10, 11	qusadd 13364	. . . 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 6630	. . . 4 |- [ A ] .~ = [ A ] ( R ~QG I )
15	6	eceq2i 6630	. . . 4 |- [ C ] .~ = [ C ] ( R ~QG I )
16	14, 15	oveq12i 5934	. . 3 |- ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = ( [ A ] ( R ~QG I ) ( +g ` Q ) [ C ] ( R ~QG I ) )
17	6	eceq2i 6630	. . 3 |- [ ( A ( +g ` R ) C ) ] .~ = [ ( A ( +g ` R ) C ) ] ( R ~QG I )
18	13, 16, 17	3eqtr4g 2254	. 2 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = [ ( A ( +g ` R ) C ) ] .~ )
19	18	eqcomd 2202	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 2167   ` cfv 5258  (class class class)co 5922   [cec 6590   Basecbs 12678   +g cplusg 12755   /._s cqus 12943  NrmSGrpcnsg 13298   ~_QG cqg 13299

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-er 6592  df-ec 6594  df-qs 6598  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-iress 12686  df-plusg 12768  df-mulr 12769  df-0g 12929  df-iimas 12945  df-qus 12946  df-mgm 12999  df-sgrp 13045  df-mnd 13058  df-grp 13135  df-minusg 13136  df-subg 13300  df-nsg 13301  df-eqg 13302

This theorem is referenced by:  ecqusaddcl  13369