Theorem ecqusaddd 13826

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 1008	. . . . 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 6029	. . . . . 6 |- ( R /.s .~ ) = ( R /.s ( R ~QG I ) )
8	5, 7	eqtri 2252	. . . . 5 |- Q = ( R /.s ( R ~QG I ) )
9		ecqusaddd.b	. . . . 5 |- B = ( Base ` R )
10		eqid 2231	. . . . 5 |- ( +g ` R ) = ( +g ` R )
11		eqid 2231	. . . . 5 |- ( +g ` Q ) = ( +g ` Q )
12	8, 9, 10, 11	qusadd 13822	. . . 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 6740	. . . 4 |- [ A ] .~ = [ A ] ( R ~QG I )
15	6	eceq2i 6740	. . . 4 |- [ C ] .~ = [ C ] ( R ~QG I )
16	14, 15	oveq12i 6030	. . 3 |- ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = ( [ A ] ( R ~QG I ) ( +g ` Q ) [ C ] ( R ~QG I ) )
17	6	eceq2i 6740	. . 3 |- [ ( A ( +g ` R ) C ) ] .~ = [ ( A ( +g ` R ) C ) ] ( R ~QG I )
18	13, 16, 17	3eqtr4g 2289	. 2 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) = [ ( A ( +g ` R ) C ) ] .~ )
19	18	eqcomd 2237	1 |- ( ( ph /\ ( A e. B /\ C e. B ) ) -> [ ( A ( +g ` R ) C ) ] .~ = ( [ A ] .~ ( +g ` Q ) [ C ] .~ ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6018   [cec 6700   Basecbs 13083   +g cplusg 13161   /._s cqus 13384  NrmSGrpcnsg 13756   ~_QG cqg 13757

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 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 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-er 6702  df-ec 6704  df-qs 6708  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13086  df-slot 13087  df-base 13089  df-sets 13090  df-iress 13091  df-plusg 13174  df-mulr 13175  df-0g 13342  df-iimas 13386  df-qus 13387  df-mgm 13440  df-sgrp 13486  df-mnd 13501  df-grp 13587  df-minusg 13588  df-subg 13758  df-nsg 13759  df-eqg 13760

This theorem is referenced by:  ecqusaddcl  13827