Theorem gsumwrev 17796

Description: A sum in an opposite monoid is the regular sum of a reversed word. (Contributed by Stefan O'Rear, 27-Aug-2015.) (Proof shortened by Mario Carneiro, 28-Feb-2016.)

Hypotheses

Ref	Expression
gsumwrev.b	|- B = ( Base ` M )
gsumwrev.o	|- O = (oppg ` M )

Assertion

Ref	Expression
gsumwrev	|- ( ( M e. Mnd /\ W e. Word B ) -> ( O gsumg W ) = ( M gsumg (reverse ` W ) ) )

Proof of Theorem gsumwrev

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . 5 |- ( x = (/) -> ( O gsumg x ) = ( O gsumg (/) ) )
2		fveq2 6191	. . . . . . 7 |- ( x = (/) -> (reverse ` x ) = (reverse ` (/) ) )
3		rev0 13513	. . . . . . 7 |- (reverse ` (/) ) = (/)
4	2, 3	syl6eq 2672	. . . . . 6 |- ( x = (/) -> (reverse ` x ) = (/) )
5	4	oveq2d 6666	. . . . 5 |- ( x = (/) -> ( M gsumg (reverse ` x ) ) = ( M gsumg (/) ) )
6	1, 5	eqeq12d 2637	. . . 4 |- ( x = (/) -> ( ( O gsumg x ) = ( M gsumg (reverse ` x ) ) <-> ( O gsumg (/) ) = ( M gsumg (/) ) ) )
7	6	imbi2d 330	. . 3 |- ( x = (/) -> ( ( M e. Mnd -> ( O gsumg x ) = ( M gsumg (reverse ` x ) ) ) <-> ( M e. Mnd -> ( O gsumg (/) ) = ( M gsumg (/) ) ) ) )
8		oveq2 6658	. . . . 5 |- ( x = y -> ( O gsumg x ) = ( O gsumg y ) )
9		fveq2 6191	. . . . . 6 |- ( x = y -> (reverse ` x ) = (reverse ` y ) )
10	9	oveq2d 6666	. . . . 5 |- ( x = y -> ( M gsumg (reverse ` x ) ) = ( M gsumg (reverse ` y ) ) )
11	8, 10	eqeq12d 2637	. . . 4 |- ( x = y -> ( ( O gsumg x ) = ( M gsumg (reverse ` x ) ) <-> ( O gsumg y ) = ( M gsumg (reverse ` y ) ) ) )
12	11	imbi2d 330	. . 3 |- ( x = y -> ( ( M e. Mnd -> ( O gsumg x ) = ( M gsumg (reverse ` x ) ) ) <-> ( M e. Mnd -> ( O gsumg y ) = ( M gsumg (reverse ` y ) ) ) ) )
13		oveq2 6658	. . . . 5 |- ( x = ( y ++ <" z "> ) -> ( O gsumg x ) = ( O gsumg ( y ++ <" z "> ) ) )
14		fveq2 6191	. . . . . 6 |- ( x = ( y ++ <" z "> ) -> (reverse ` x ) = (reverse ` ( y ++ <" z "> ) ) )
15	14	oveq2d 6666	. . . . 5 |- ( x = ( y ++ <" z "> ) -> ( M gsumg (reverse ` x ) ) = ( M gsumg (reverse ` ( y ++ <" z "> ) ) ) )
16	13, 15	eqeq12d 2637	. . . 4 |- ( x = ( y ++ <" z "> ) -> ( ( O gsumg x ) = ( M gsumg (reverse ` x ) ) <-> ( O gsumg ( y ++ <" z "> ) ) = ( M gsumg (reverse ` ( y ++ <" z "> ) ) ) ) )
17	16	imbi2d 330	. . 3 |- ( x = ( y ++ <" z "> ) -> ( ( M e. Mnd -> ( O gsumg x ) = ( M gsumg (reverse ` x ) ) ) <-> ( M e. Mnd -> ( O gsumg ( y ++ <" z "> ) ) = ( M gsumg (reverse ` ( y ++ <" z "> ) ) ) ) ) )
18		oveq2 6658	. . . . 5 |- ( x = W -> ( O gsumg x ) = ( O gsumg W ) )
19		fveq2 6191	. . . . . 6 |- ( x = W -> (reverse ` x ) = (reverse ` W ) )
20	19	oveq2d 6666	. . . . 5 |- ( x = W -> ( M gsumg (reverse ` x ) ) = ( M gsumg (reverse ` W ) ) )
21	18, 20	eqeq12d 2637	. . . 4 |- ( x = W -> ( ( O gsumg x ) = ( M gsumg (reverse ` x ) ) <-> ( O gsumg W ) = ( M gsumg (reverse ` W ) ) ) )
22	21	imbi2d 330	. . 3 |- ( x = W -> ( ( M e. Mnd -> ( O gsumg x ) = ( M gsumg (reverse ` x ) ) ) <-> ( M e. Mnd -> ( O gsumg W ) = ( M gsumg (reverse ` W ) ) ) ) )
23		gsumwrev.o	. . . . . . 7 |- O = (oppg ` M )
24		eqid 2622	. . . . . . 7 |- ( 0g ` M ) = ( 0g ` M )
25	23, 24	oppgid 17786	. . . . . 6 |- ( 0g ` M ) = ( 0g ` O )
26	25	gsum0 17278	. . . . 5 |- ( O gsumg (/) ) = ( 0g ` M )
27	24	gsum0 17278	. . . . 5 |- ( M gsumg (/) ) = ( 0g ` M )
28	26, 27	eqtr4i 2647	. . . 4 |- ( O gsumg (/) ) = ( M gsumg (/) )
29	28	a1i 11	. . 3 |- ( M e. Mnd -> ( O gsumg (/) ) = ( M gsumg (/) ) )
30		oveq2 6658	. . . . . 6 |- ( ( O gsumg y ) = ( M gsumg (reverse ` y ) ) -> ( z ( +g ` M ) ( O gsumg y ) ) = ( z ( +g ` M ) ( M gsumg (reverse ` y ) ) ) )
31	23	oppgmnd 17784	. . . . . . . . . 10 |- ( M e. Mnd -> O e. Mnd )
32	31	adantr 481	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> O e. Mnd )
33		simprl 794	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> y e. Word B )
34		simprr 796	. . . . . . . . . 10 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> z e. B )
35	34	s1cld 13383	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> <" z "> e. Word B )
36		gsumwrev.b	. . . . . . . . . . 11 |- B = ( Base ` M )
37	23, 36	oppgbas 17781	. . . . . . . . . 10 |- B = ( Base ` O )
38		eqid 2622	. . . . . . . . . 10 |- ( +g ` O ) = ( +g ` O )
39	37, 38	gsumccat 17378	. . . . . . . . 9 |- ( ( O e. Mnd /\ y e. Word B /\ <" z "> e. Word B ) -> ( O gsumg ( y ++ <" z "> ) ) = ( ( O gsumg y ) ( +g ` O ) ( O gsumg <" z "> ) ) )
40	32, 33, 35, 39	syl3anc 1326	. . . . . . . 8 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( O gsumg ( y ++ <" z "> ) ) = ( ( O gsumg y ) ( +g ` O ) ( O gsumg <" z "> ) ) )
41	37	gsumws1 17376	. . . . . . . . . . 11 |- ( z e. B -> ( O gsumg <" z "> ) = z )
42	41	ad2antll 765	. . . . . . . . . 10 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( O gsumg <" z "> ) = z )
43	42	oveq2d 6666	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( ( O gsumg y ) ( +g ` O ) ( O gsumg <" z "> ) ) = ( ( O gsumg y ) ( +g ` O ) z ) )
44		eqid 2622	. . . . . . . . . 10 |- ( +g ` M ) = ( +g ` M )
45	44, 23, 38	oppgplus 17779	. . . . . . . . 9 |- ( ( O gsumg y ) ( +g ` O ) z ) = ( z ( +g ` M ) ( O gsumg y ) )
46	43, 45	syl6eq 2672	. . . . . . . 8 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( ( O gsumg y ) ( +g ` O ) ( O gsumg <" z "> ) ) = ( z ( +g ` M ) ( O gsumg y ) ) )
47	40, 46	eqtrd 2656	. . . . . . 7 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( O gsumg ( y ++ <" z "> ) ) = ( z ( +g ` M ) ( O gsumg y ) ) )
48		revccat 13515	. . . . . . . . . . 11 |- ( ( y e. Word B /\ <" z "> e. Word B ) -> (reverse ` ( y ++ <" z "> ) ) = ( (reverse ` <" z "> ) ++ (reverse ` y ) ) )
49	33, 35, 48	syl2anc 693	. . . . . . . . . 10 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> (reverse ` ( y ++ <" z "> ) ) = ( (reverse ` <" z "> ) ++ (reverse ` y ) ) )
50		revs1 13514	. . . . . . . . . . 11 |- (reverse ` <" z "> ) = <" z ">
51	50	oveq1i 6660	. . . . . . . . . 10 |- ( (reverse ` <" z "> ) ++ (reverse ` y ) ) = ( <" z "> ++ (reverse ` y ) )
52	49, 51	syl6eq 2672	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> (reverse ` ( y ++ <" z "> ) ) = ( <" z "> ++ (reverse ` y ) ) )
53	52	oveq2d 6666	. . . . . . . 8 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( M gsumg (reverse ` ( y ++ <" z "> ) ) ) = ( M gsumg ( <" z "> ++ (reverse ` y ) ) ) )
54		simpl 473	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> M e. Mnd )
55		revcl 13510	. . . . . . . . . 10 |- ( y e. Word B -> (reverse ` y ) e. Word B )
56	55	ad2antrl 764	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> (reverse ` y ) e. Word B )
57	36, 44	gsumccat 17378	. . . . . . . . 9 |- ( ( M e. Mnd /\ <" z "> e. Word B /\ (reverse ` y ) e. Word B ) -> ( M gsumg ( <" z "> ++ (reverse ` y ) ) ) = ( ( M gsumg <" z "> ) ( +g ` M ) ( M gsumg (reverse ` y ) ) ) )
58	54, 35, 56, 57	syl3anc 1326	. . . . . . . 8 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( M gsumg ( <" z "> ++ (reverse ` y ) ) ) = ( ( M gsumg <" z "> ) ( +g ` M ) ( M gsumg (reverse ` y ) ) ) )
59	36	gsumws1 17376	. . . . . . . . . 10 |- ( z e. B -> ( M gsumg <" z "> ) = z )
60	59	ad2antll 765	. . . . . . . . 9 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( M gsumg <" z "> ) = z )
61	60	oveq1d 6665	. . . . . . . 8 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( ( M gsumg <" z "> ) ( +g ` M ) ( M gsumg (reverse ` y ) ) ) = ( z ( +g ` M ) ( M gsumg (reverse ` y ) ) ) )
62	53, 58, 61	3eqtrd 2660	. . . . . . 7 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( M gsumg (reverse ` ( y ++ <" z "> ) ) ) = ( z ( +g ` M ) ( M gsumg (reverse ` y ) ) ) )
63	47, 62	eqeq12d 2637	. . . . . 6 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( ( O gsumg ( y ++ <" z "> ) ) = ( M gsumg (reverse ` ( y ++ <" z "> ) ) ) <-> ( z ( +g ` M ) ( O gsumg y ) ) = ( z ( +g ` M ) ( M gsumg (reverse ` y ) ) ) ) )
64	30, 63	syl5ibr 236	. . . . 5 |- ( ( M e. Mnd /\ ( y e. Word B /\ z e. B ) ) -> ( ( O gsumg y ) = ( M gsumg (reverse ` y ) ) -> ( O gsumg ( y ++ <" z "> ) ) = ( M gsumg (reverse ` ( y ++ <" z "> ) ) ) ) )
65	64	expcom 451	. . . 4 |- ( ( y e. Word B /\ z e. B ) -> ( M e. Mnd -> ( ( O gsumg y ) = ( M gsumg (reverse ` y ) ) -> ( O gsumg ( y ++ <" z "> ) ) = ( M gsumg (reverse ` ( y ++ <" z "> ) ) ) ) ) )
66	65	a2d 29	. . 3 |- ( ( y e. Word B /\ z e. B ) -> ( ( M e. Mnd -> ( O gsumg y ) = ( M gsumg (reverse ` y ) ) ) -> ( M e. Mnd -> ( O gsumg ( y ++ <" z "> ) ) = ( M gsumg (reverse ` ( y ++ <" z "> ) ) ) ) ) )
67	7, 12, 17, 22, 29, 66	wrdind 13476	. 2 |- ( W e. Word B -> ( M e. Mnd -> ( O gsumg W ) = ( M gsumg (reverse ` W ) ) ) )
68	67	impcom 446	1 |- ( ( M e. Mnd /\ W e. Word B ) -> ( O gsumg W ) = ( M gsumg (reverse ` W ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   (/)c0 3915   ` cfv 5888  (class class class)co 6650  Word cword 13291   ++ cconcat 13293   <"cs1 13294  reversecreverse 13297   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294  opp_gcoppg 17775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-tpos 7352  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-reverse 13305  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-oppg 17776

This theorem is referenced by:  symgtrinv  17892