Theorem gsumwmhm 17382

Description: Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)

Hypothesis

Ref	Expression
gsumwmhm.b	|- B = ( Base ` M )

Assertion

Ref	Expression
gsumwmhm	|- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) )

Proof of Theorem gsumwmhm

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

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . 5 |- ( W = (/) -> ( M gsumg W ) = ( M gsumg (/) ) )
2		eqid 2622	. . . . . 6 |- ( 0g ` M ) = ( 0g ` M )
3	2	gsum0 17278	. . . . 5 |- ( M gsumg (/) ) = ( 0g ` M )
4	1, 3	syl6eq 2672	. . . 4 |- ( W = (/) -> ( M gsumg W ) = ( 0g ` M ) )
5	4	fveq2d 6195	. . 3 |- ( W = (/) -> ( H ` ( M gsumg W ) ) = ( H ` ( 0g ` M ) ) )
6		coeq2 5280	. . . . . 6 |- ( W = (/) -> ( H o. W ) = ( H o. (/) ) )
7		co02 5649	. . . . . 6 |- ( H o. (/) ) = (/)
8	6, 7	syl6eq 2672	. . . . 5 |- ( W = (/) -> ( H o. W ) = (/) )
9	8	oveq2d 6666	. . . 4 |- ( W = (/) -> ( N gsumg ( H o. W ) ) = ( N gsumg (/) ) )
10		eqid 2622	. . . . 5 |- ( 0g ` N ) = ( 0g ` N )
11	10	gsum0 17278	. . . 4 |- ( N gsumg (/) ) = ( 0g ` N )
12	9, 11	syl6eq 2672	. . 3 |- ( W = (/) -> ( N gsumg ( H o. W ) ) = ( 0g ` N ) )
13	5, 12	eqeq12d 2637	. 2 |- ( W = (/) -> ( ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) <-> ( H ` ( 0g ` M ) ) = ( 0g ` N ) ) )
14		mhmrcl1 17338	. . . . . 6 |- ( H e. ( M MndHom N ) -> M e. Mnd )
15	14	ad2antrr 762	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> M e. Mnd )
16		gsumwmhm.b	. . . . . . 7 |- B = ( Base ` M )
17		eqid 2622	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
18	16, 17	mndcl 17301	. . . . . 6 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
19	18	3expb 1266	. . . . 5 |- ( ( M e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` M ) y ) e. B )
20	15, 19	sylan 488	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` M ) y ) e. B )
21		wrdf 13310	. . . . . . 7 |- ( W e. Word B -> W : ( 0..^ ( # ` W ) ) --> B )
22	21	ad2antlr 763	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W : ( 0..^ ( # ` W ) ) --> B )
23		wrdfin 13323	. . . . . . . . . . . 12 |- ( W e. Word B -> W e. Fin )
24	23	adantl 482	. . . . . . . . . . 11 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> W e. Fin )
25		hashnncl 13157	. . . . . . . . . . 11 |- ( W e. Fin -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
26	24, 25	syl 17	. . . . . . . . . 10 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( ( # ` W ) e. NN <-> W =/= (/) ) )
27	26	biimpar 502	. . . . . . . . 9 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( # ` W ) e. NN )
28	27	nnzd 11481	. . . . . . . 8 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( # ` W ) e. ZZ )
29		fzoval 12471	. . . . . . . 8 |- ( ( # ` W ) e. ZZ -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
30	28, 29	syl 17	. . . . . . 7 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( 0..^ ( # ` W ) ) = ( 0 ... ( ( # ` W ) - 1 ) ) )
31	30	feq2d 6031	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( W : ( 0..^ ( # ` W ) ) --> B <-> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> B ) )
32	22, 31	mpbid 222	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W : ( 0 ... ( ( # ` W ) - 1 ) ) --> B )
33	32	ffvelrnda 6359	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) -> ( W ` x ) e. B )
34		nnm1nn0 11334	. . . . . 6 |- ( ( # ` W ) e. NN -> ( ( # ` W ) - 1 ) e. NN0 )
35	27, 34	syl 17	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. NN0 )
36		nn0uz 11722	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
37	35, 36	syl6eleq 2711	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( ( # ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
38		simpll 790	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> H e. ( M MndHom N ) )
39		eqid 2622	. . . . . . 7 |- ( +g ` N ) = ( +g ` N )
40	16, 17, 39	mhmlin 17342	. . . . . 6 |- ( ( H e. ( M MndHom N ) /\ x e. B /\ y e. B ) -> ( H ` ( x ( +g ` M ) y ) ) = ( ( H ` x ) ( +g ` N ) ( H ` y ) ) )
41	40	3expb 1266	. . . . 5 |- ( ( H e. ( M MndHom N ) /\ ( x e. B /\ y e. B ) ) -> ( H ` ( x ( +g ` M ) y ) ) = ( ( H ` x ) ( +g ` N ) ( H ` y ) ) )
42	38, 41	sylan 488	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ ( x e. B /\ y e. B ) ) -> ( H ` ( x ( +g ` M ) y ) ) = ( ( H ` x ) ( +g ` N ) ( H ` y ) ) )
43		ffn 6045	. . . . . . 7 |- ( W : ( 0 ... ( ( # ` W ) - 1 ) ) --> B -> W Fn ( 0 ... ( ( # ` W ) - 1 ) ) )
44	32, 43	syl 17	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W Fn ( 0 ... ( ( # ` W ) - 1 ) ) )
45		fvco2 6273	. . . . . 6 |- ( ( W Fn ( 0 ... ( ( # ` W ) - 1 ) ) /\ x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) -> ( ( H o. W ) ` x ) = ( H ` ( W ` x ) ) )
46	44, 45	sylan 488	. . . . 5 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) -> ( ( H o. W ) ` x ) = ( H ` ( W ` x ) ) )
47	46	eqcomd 2628	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( ( # ` W ) - 1 ) ) ) -> ( H ` ( W ` x ) ) = ( ( H o. W ) ` x ) )
48	20, 33, 37, 42, 47	seqhomo 12848	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H ` ( seq 0 ( ( +g ` M ) , W ) ` ( ( # ` W ) - 1 ) ) ) = ( seq 0 ( ( +g ` N ) , ( H o. W ) ) ` ( ( # ` W ) - 1 ) ) )
49	16, 17, 15, 37, 32	gsumval2 17280	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( M gsumg W ) = ( seq 0 ( ( +g ` M ) , W ) ` ( ( # ` W ) - 1 ) ) )
50	49	fveq2d 6195	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H ` ( M gsumg W ) ) = ( H ` ( seq 0 ( ( +g ` M ) , W ) ` ( ( # ` W ) - 1 ) ) ) )
51		eqid 2622	. . . 4 |- ( Base ` N ) = ( Base ` N )
52		mhmrcl2 17339	. . . . 5 |- ( H e. ( M MndHom N ) -> N e. Mnd )
53	52	ad2antrr 762	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> N e. Mnd )
54	16, 51	mhmf 17340	. . . . . 6 |- ( H e. ( M MndHom N ) -> H : B --> ( Base ` N ) )
55	54	ad2antrr 762	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> H : B --> ( Base ` N ) )
56		fco 6058	. . . . 5 |- ( ( H : B --> ( Base ` N ) /\ W : ( 0 ... ( ( # ` W ) - 1 ) ) --> B ) -> ( H o. W ) : ( 0 ... ( ( # ` W ) - 1 ) ) --> ( Base ` N ) )
57	55, 32, 56	syl2anc 693	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H o. W ) : ( 0 ... ( ( # ` W ) - 1 ) ) --> ( Base ` N ) )
58	51, 39, 53, 37, 57	gsumval2 17280	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( N gsumg ( H o. W ) ) = ( seq 0 ( ( +g ` N ) , ( H o. W ) ) ` ( ( # ` W ) - 1 ) ) )
59	48, 50, 58	3eqtr4d 2666	. 2 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) )
60	2, 10	mhm0 17343	. . 3 |- ( H e. ( M MndHom N ) -> ( H ` ( 0g ` M ) ) = ( 0g ` N ) )
61	60	adantr 481	. 2 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( H ` ( 0g ` M ) ) = ( 0g ` N ) )
62	13, 59, 61	pm2.61ne 2879	1 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   (/)c0 3915   o. ccom 5118   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   1c1 9937   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326  ..^cfzo 12465   seqcseq 12801   #chash 13117  Word cword 13291   Basecbs 15857   +g cplusg 15941   0gc0g 16100   gsum_g cgsu 16101   Mndcmnd 17294   MndHom cmhm 17333

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-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335

This theorem is referenced by:  frmdup3lem  17403  symgtrinv  17892  frgpup3lem  18190