Theorem gsumwmhm 13140

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 2196	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
2		eqid 2196	. . . . 5 |- ( 0g ` N ) = ( 0g ` N )
3	1, 2	mhm0 13110	. . . 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 5931	. . . . . 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 13105	. . . . . . 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 13049	. . . . . 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 2229	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( M gsumg W ) = ( 0g ` M ) )
12	11	fveq2d 5563	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( H ` ( M gsumg W ) ) = ( H ` ( 0g ` M ) ) )
13		coeq2 4825	. . . . . . 7 |- ( W = (/) -> ( H o. W ) = ( H o. (/) ) )
14		co02 5184	. . . . . . 7 |- ( H o. (/) ) = (/)
15	13, 14	eqtrdi 2245	. . . . . 6 |- ( W = (/) -> ( H o. W ) = (/) )
16	15	oveq2d 5939	. . . . 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 13106	. . . . . 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 13049	. . . . 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 2229	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( N gsumg ( H o. W ) ) = ( 0g ` N ) )
23	4, 12, 22	3eqtr4d 2239	. 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 2196	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
27	25, 26	mndcl 13074	. . . . . 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 10943	. . . . . . 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 10956	. . . . . . . . . . . 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 10889	. . . . . . . . . . 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 9449	. . . . . . . 8 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
38		fzoval 10225	. . . . . . . 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 5396	. . . . . 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 5698	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. B )
43		nnm1nn0 9292	. . . . . 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 9638	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
46	44, 45	eleqtrdi 2289	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
47		eqid 2196	. . . . . . 7 |- ( +g ` N ) = ( +g ` N )
48	25, 26, 47	mhmlin 13109	. . . . . 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 5409	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W Fn ( 0 ... ( (♯ ` W ) - 1 ) ) )
52		fvco2 5631	. . . . . 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 2202	. . . 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 5215	. . . . 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 12800	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
59	58	slotex 12715	. . . . . 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 12715	. . . . . 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 10624	. . 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 13050	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( M gsumg W ) = ( seq 0 ( ( +g ` M ) , W ) ` ( (♯ ` W ) - 1 ) ) )
67	66	fveq2d 5563	. . 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 2196	. . . 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 13107	. . . . . 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 5424	. . . . 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 13050	. . 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 2239	. 2 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) )
76		fin0or 6948	. . . 4 |- ( W e. Fin -> ( W = (/) \/ E. j j e. W ) )
77		n0r 3465	. . . . 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 1506   e. wcel 2167   =/= wne 2367   _Vcvv 2763   (/)c0 3451   o. ccom 4668   Fn wfn 5254   -->wf 5255   ` cfv 5259  (class class class)co 5923   Fincfn 6800   0cc0 7881   1c1 7882   - cmin 8199   NNcn 8992   NN0cn0 9251   ZZcz 9328   ZZ>=cuz 9603   ...cfz 10085  ..^cfzo 10219   seqcseq 10541  ♯chash 10869  Word cword 10937   Basecbs 12688   +g cplusg 12765   0gc0g 12937   gsum_g cgsu 12938   Mndcmnd 13067   MndHom cmhm 13099

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7972  ax-resscn 7973  ax-1cn 7974  ax-1re 7975  ax-icn 7976  ax-addcl 7977  ax-addrcl 7978  ax-mulcl 7979  ax-addcom 7981  ax-addass 7983  ax-distr 7985  ax-i2m1 7986  ax-0lt1 7987  ax-0id 7989  ax-rnegex 7990  ax-cnre 7992  ax-pre-ltirr 7993  ax-pre-ltwlin 7994  ax-pre-lttrn 7995  ax-pre-apti 7996  ax-pre-ltadd 7997

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 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-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 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5878  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6199  df-2nd 6200  df-recs 6364  df-frec 6450  df-1o 6475  df-er 6593  df-map 6710  df-en 6801  df-dom 6802  df-fin 6803  df-pnf 8065  df-mnf 8066  df-xr 8067  df-ltxr 8068  df-le 8069  df-sub 8201  df-neg 8202  df-inn 8993  df-2 9051  df-n0 9252  df-z 9329  df-uz 9604  df-fz 10086  df-fzo 10220  df-seqfrec 10542  df-ihash 10870  df-word 10938  df-ndx 12691  df-slot 12692  df-base 12694  df-plusg 12778  df-0g 12939  df-igsum 12940  df-mgm 13009  df-sgrp 13055  df-mnd 13068  df-mhm 13101

This theorem is referenced by: (None)