Theorem gsumwmhm 13060

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 j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2193	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
2		eqid 2193	. . . . 5 |- ( 0g ` N ) = ( 0g ` N )
3	1, 2	mhm0 13030	. . . 4 |- ( H e. ( M MndHom N ) -> ( H ` ( 0g ` M ) ) = ( 0g ` N ) )
4	3	ad2antrr 488	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( H ` ( 0g ` M ) ) = ( 0g ` N ) )
5		oveq2 5918	. . . . . 6 |- ( W = (/) -> ( M gsumg W ) = ( M gsumg (/) ) )
6	5	adantl 277	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( M gsumg W ) = ( M gsumg (/) ) )
7		mhmrcl1 13025	. . . . . . 7 |- ( H e. ( M MndHom N ) -> M e. Mnd )
8	7	ad2antrr 488	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> M e. Mnd )
9	1	gsum0g 12969	. . . . . 6 |- ( M e. Mnd -> ( M gsumg (/) ) = ( 0g ` M ) )
10	8, 9	syl 14	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( M gsumg (/) ) = ( 0g ` M ) )
11	6, 10	eqtrd 2226	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( M gsumg W ) = ( 0g ` M ) )
12	11	fveq2d 5550	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( H ` ( M gsumg W ) ) = ( H ` ( 0g ` M ) ) )
13		coeq2 4814	. . . . . . 7 |- ( W = (/) -> ( H o. W ) = ( H o. (/) ) )
14		co02 5171	. . . . . . 7 |- ( H o. (/) ) = (/)
15	13, 14	eqtrdi 2242	. . . . . 6 |- ( W = (/) -> ( H o. W ) = (/) )
16	15	oveq2d 5926	. . . . 5 |- ( W = (/) -> ( N gsumg ( H o. W ) ) = ( N gsumg (/) ) )
17	16	adantl 277	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( N gsumg ( H o. W ) ) = ( N gsumg (/) ) )
18		mhmrcl2 13026	. . . . . 6 |- ( H e. ( M MndHom N ) -> N e. Mnd )
19	18	ad2antrr 488	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> N e. Mnd )
20	2	gsum0g 12969	. . . . 5 |- ( N e. Mnd -> ( N gsumg (/) ) = ( 0g ` N ) )
21	19, 20	syl 14	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( N gsumg (/) ) = ( 0g ` N ) )
22	17, 21	eqtrd 2226	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( N gsumg ( H o. W ) ) = ( 0g ` N ) )
23	4, 12, 22	3eqtr4d 2236	. 2 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) )
24	7	ad2antrr 488	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> M e. Mnd )
25		gsumwmhm.b	. . . . . . 7 |- B = ( Base ` M )
26		eqid 2193	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
27	25, 26	mndcl 12994	. . . . . 6 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
28	27	3expb 1206	. . . . 5 |- ( ( M e. Mnd /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` M ) y ) e. B )
29	24, 28	sylan 283	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ ( x e. B /\ y e. B ) ) -> ( x ( +g ` M ) y ) e. B )
30		wrdf 10910	. . . . . . 7 |- ( W e. Word B -> W : ( 0..^ (♯ ` W ) ) --> B )
31	30	ad2antlr 489	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W : ( 0..^ (♯ ` W ) ) --> B )
32		wrdfin 10923	. . . . . . . . . . . 12 |- ( W e. Word B -> W e. Fin )
33	32	adantl 277	. . . . . . . . . . 11 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> W e. Fin )
34		hashnncl 10856	. . . . . . . . . . 11 |- ( W e. Fin -> ( (♯ ` W ) e. NN <-> W =/= (/) ) )
35	33, 34	syl 14	. . . . . . . . . 10 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( (♯ ` W ) e. NN <-> W =/= (/) ) )
36	35	biimpar 297	. . . . . . . . 9 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> (♯ ` W ) e. NN )
37	36	nnzd 9428	. . . . . . . 8 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
38		fzoval 10204	. . . . . . . 8 |- ( (♯ ` W ) e. ZZ -> ( 0..^ (♯ ` W ) ) = ( 0 ... ( (♯ ` W ) - 1 ) ) )
39	37, 38	syl 14	. . . . . . 7 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( 0..^ (♯ ` W ) ) = ( 0 ... ( (♯ ` W ) - 1 ) ) )
40	39	feq2d 5383	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( W : ( 0..^ (♯ ` W ) ) --> B <-> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> B ) )
41	31, 40	mpbid 147	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> B )
42	41	ffvelcdmda 5685	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. B )
43		nnm1nn0 9271	. . . . . 6 |- ( (♯ ` W ) e. NN -> ( (♯ ` W ) - 1 ) e. NN0 )
44	36, 43	syl 14	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. NN0 )
45		nn0uz 9617	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
46	44, 45	eleqtrdi 2286	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
47		eqid 2193	. . . . . . 7 |- ( +g ` N ) = ( +g ` N )
48	25, 26, 47	mhmlin 13029	. . . . . 6 |- ( ( H e. ( M MndHom N ) /\ x e. B /\ y e. B ) -> ( H ` ( x ( +g ` M ) y ) ) = ( ( H ` x ) ( +g ` N ) ( H ` y ) ) )
49	48	3expb 1206	. . . . 5 |- ( ( H e. ( M MndHom N ) /\ ( x e. B /\ y e. B ) ) -> ( H ` ( x ( +g ` M ) y ) ) = ( ( H ` x ) ( +g ` N ) ( H ` y ) ) )
50	49	ad4ant14 514	. . . 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 ) ) )
51	41	ffnd 5396	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W Fn ( 0 ... ( (♯ ` W ) - 1 ) ) )
52		fvco2 5618	. . . . . 6 |- ( ( W Fn ( 0 ... ( (♯ ` W ) - 1 ) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( ( H o. W ) ` x ) = ( H ` ( W ` x ) ) )
53	51, 52	sylan 283	. . . . 5 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( ( H o. W ) ` x ) = ( H ` ( W ` x ) ) )
54	53	eqcomd 2199	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( H ` ( W ` x ) ) = ( ( H o. W ) ` x ) )
55		simplr 528	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W e. Word B )
56		coexg 5202	. . . . 5 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( H o. W ) e. _V )
57	56	adantr 276	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H o. W ) e. _V )
58		plusgslid 12720	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
59	58	slotex 12635	. . . . . 6 |- ( M e. Mnd -> ( +g ` M ) e. _V )
60	7, 59	syl 14	. . . . 5 |- ( H e. ( M MndHom N ) -> ( +g ` M ) e. _V )
61	60	ad2antrr 488	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( +g ` M ) e. _V )
62	58	slotex 12635	. . . . . 6 |- ( N e. Mnd -> ( +g ` N ) e. _V )
63	18, 62	syl 14	. . . . 5 |- ( H e. ( M MndHom N ) -> ( +g ` N ) e. _V )
64	63	ad2antrr 488	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( +g ` N ) e. _V )
65	29, 42, 46, 50, 54, 55, 57, 61, 64	seqhomog 10591	. . 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 ) ) )
66	25, 26, 24, 46, 41	gsumval2 12970	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( M gsumg W ) = ( seq 0 ( ( +g ` M ) , W ) ` ( (♯ ` W ) - 1 ) ) )
67	66	fveq2d 5550	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H ` ( M gsumg W ) ) = ( H ` ( seq 0 ( ( +g ` M ) , W ) ` ( (♯ ` W ) - 1 ) ) ) )
68		eqid 2193	. . . 4 |- ( Base ` N ) = ( Base ` N )
69	18	ad2antrr 488	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> N e. Mnd )
70	25, 68	mhmf 13027	. . . . . 6 |- ( H e. ( M MndHom N ) -> H : B --> ( Base ` N ) )
71	70	ad2antrr 488	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> H : B --> ( Base ` N ) )
72		fco 5411	. . . . 5 |- ( ( H : B --> ( Base ` N ) /\ W : ( 0 ... ( (♯ ` W ) - 1 ) ) --> B ) -> ( H o. W ) : ( 0 ... ( (♯ ` W ) - 1 ) ) --> ( Base ` N ) )
73	71, 41, 72	syl2anc 411	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H o. W ) : ( 0 ... ( (♯ ` W ) - 1 ) ) --> ( Base ` N ) )
74	68, 47, 69, 46, 73	gsumval2 12970	. . 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 ) ) )
75	65, 67, 74	3eqtr4d 2236	. 2 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) )
76		fin0or 6933	. . . 4 |- ( W e. Fin -> ( W = (/) \/ E. j j e. W ) )
77		n0r 3460	. . . . 5 |- ( E. j j e. W -> W =/= (/) )
78	77	orim2i 762	. . . 4 |- ( ( W = (/) \/ E. j j e. W ) -> ( W = (/) \/ W =/= (/) ) )
79	76, 78	syl 14	. . 3 |- ( W e. Fin -> ( W = (/) \/ W =/= (/) ) )
80	33, 79	syl 14	. 2 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( W = (/) \/ W =/= (/) ) )
81	23, 75, 80	mpjaodan 799	1 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   =/= wne 2364   _Vcvv 2760   (/)c0 3446   o. ccom 4659   Fn wfn 5241   -->wf 5242   ` cfv 5246  (class class class)co 5910   Fincfn 6785   0cc0 7862   1c1 7863   - cmin 8180   NNcn 8972   NN0cn0 9230   ZZcz 9307   ZZ>=cuz 9582   ...cfz 10064  ..^cfzo 10198   seqcseq 10508  ♯chash 10836  Word cword 10904   Basecbs 12608   +g cplusg 12685   0gc0g 12857   gsum_g cgsu 12858   Mndcmnd 12987   MndHom cmhm 13019

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-iinf 4616  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-distr 7966  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-cnre 7973  ax-pre-ltirr 7974  ax-pre-ltwlin 7975  ax-pre-lttrn 7976  ax-pre-apti 7977  ax-pre-ltadd 7978

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4322  df-iord 4395  df-on 4397  df-ilim 4398  df-suc 4400  df-iom 4619  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-1st 6184  df-2nd 6185  df-recs 6349  df-frec 6435  df-1o 6460  df-er 6578  df-map 6695  df-en 6786  df-dom 6787  df-fin 6788  df-pnf 8046  df-mnf 8047  df-xr 8048  df-ltxr 8049  df-le 8050  df-sub 8182  df-neg 8183  df-inn 8973  df-2 9031  df-n0 9231  df-z 9308  df-uz 9583  df-fz 10065  df-fzo 10199  df-seqfrec 10509  df-ihash 10837  df-word 10905  df-ndx 12611  df-slot 12612  df-base 12614  df-plusg 12698  df-0g 12859  df-igsum 12860  df-mgm 12929  df-sgrp 12975  df-mnd 12988  df-mhm 13021

This theorem is referenced by: (None)