Theorem gsumwmhm 13106

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 13076	. . . 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 5930	. . . . . 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 13071	. . . . . . 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 13015	. . . . . 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 5562	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( H ` ( M gsumg W ) ) = ( H ` ( 0g ` M ) ) )
13		coeq2 4824	. . . . . . 7 |- ( W = (/) -> ( H o. W ) = ( H o. (/) ) )
14		co02 5183	. . . . . . 7 |- ( H o. (/) ) = (/)
15	13, 14	eqtrdi 2245	. . . . . 6 |- ( W = (/) -> ( H o. W ) = (/) )
16	15	oveq2d 5938	. . . . 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 13072	. . . . . 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 13015	. . . . 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 13040	. . . . . 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 10926	. . . . . . 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 10939	. . . . . . . . . . . 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 10872	. . . . . . . . . . 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 9444	. . . . . . . 8 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
38		fzoval 10220	. . . . . . . 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 5395	. . . . . 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 5697	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. B )
43		nnm1nn0 9287	. . . . . 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 9633	. . . . 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 13075	. . . . . 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 5408	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W Fn ( 0 ... ( (♯ ` W ) - 1 ) ) )
52		fvco2 5630	. . . . . 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 5214	. . . . 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 12766	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
59	58	slotex 12681	. . . . . 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 12681	. . . . . 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 10607	. . 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 13016	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( M gsumg W ) = ( seq 0 ( ( +g ` M ) , W ) ` ( (♯ ` W ) - 1 ) ) )
67	66	fveq2d 5562	. . 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 13073	. . . . . 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 5423	. . . . 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 13016	. . 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 6947	. . . 4 |- ( W e. Fin -> ( W = (/) \/ E. j j e. W ) )
77		n0r 3464	. . . . 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 3450   o. ccom 4667   Fn wfn 5253   -->wf 5254   ` cfv 5258  (class class class)co 5922   Fincfn 6799   0cc0 7877   1c1 7878   - cmin 8195   NNcn 8987   NN0cn0 9246   ZZcz 9323   ZZ>=cuz 9598   ...cfz 10080  ..^cfzo 10214   seqcseq 10524  ♯chash 10852  Word cword 10920   Basecbs 12654   +g cplusg 12731   0gc0g 12903   gsum_g cgsu 12904   Mndcmnd 13033   MndHom cmhm 13065

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7968  ax-resscn 7969  ax-1cn 7970  ax-1re 7971  ax-icn 7972  ax-addcl 7973  ax-addrcl 7974  ax-mulcl 7975  ax-addcom 7977  ax-addass 7979  ax-distr 7981  ax-i2m1 7982  ax-0lt1 7983  ax-0id 7985  ax-rnegex 7986  ax-cnre 7988  ax-pre-ltirr 7989  ax-pre-ltwlin 7990  ax-pre-lttrn 7991  ax-pre-apti 7992  ax-pre-ltadd 7993

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 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-er 6592  df-map 6709  df-en 6800  df-dom 6801  df-fin 6802  df-pnf 8061  df-mnf 8062  df-xr 8063  df-ltxr 8064  df-le 8065  df-sub 8197  df-neg 8198  df-inn 8988  df-2 9046  df-n0 9247  df-z 9324  df-uz 9599  df-fz 10081  df-fzo 10215  df-seqfrec 10525  df-ihash 10853  df-word 10921  df-ndx 12657  df-slot 12658  df-base 12660  df-plusg 12744  df-0g 12905  df-igsum 12906  df-mgm 12975  df-sgrp 13021  df-mnd 13034  df-mhm 13067

This theorem is referenced by: (None)