Theorem gsumwmhm 13552

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 2229	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
2		eqid 2229	. . . . 5 |- ( 0g ` N ) = ( 0g ` N )
3	1, 2	mhm0 13522	. . . 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 6018	. . . . . 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 13517	. . . . . . 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 13450	. . . . . 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 2262	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( M gsumg W ) = ( 0g ` M ) )
12	11	fveq2d 5636	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( H ` ( M gsumg W ) ) = ( H ` ( 0g ` M ) ) )
13		coeq2 4883	. . . . . . 7 |- ( W = (/) -> ( H o. W ) = ( H o. (/) ) )
14		co02 5245	. . . . . . 7 |- ( H o. (/) ) = (/)
15	13, 14	eqtrdi 2278	. . . . . 6 |- ( W = (/) -> ( H o. W ) = (/) )
16	15	oveq2d 6026	. . . . 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 13518	. . . . . 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 13450	. . . . 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 2262	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( N gsumg ( H o. W ) ) = ( 0g ` N ) )
23	4, 12, 22	3eqtr4d 2272	. 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 2229	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
27	25, 26	mndcl 13477	. . . . . 6 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
28	27	3expb 1228	. . . . 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 11095	. . . . . . 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 11108	. . . . . . . . . . . 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 11034	. . . . . . . . . . 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 9584	. . . . . . . 8 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
38		fzoval 10361	. . . . . . . 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 5464	. . . . . 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 5775	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. B )
43		nnm1nn0 9426	. . . . . 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 9774	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
46	44, 45	eleqtrdi 2322	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
47		eqid 2229	. . . . . . 7 |- ( +g ` N ) = ( +g ` N )
48	25, 26, 47	mhmlin 13521	. . . . . 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 1228	. . . . 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 5477	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W Fn ( 0 ... ( (♯ ` W ) - 1 ) ) )
52		fvco2 5708	. . . . . 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 2235	. . . 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 5276	. . . . 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 13166	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
59	58	slotex 13080	. . . . . 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 13080	. . . . . 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 10769	. . 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 13451	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( M gsumg W ) = ( seq 0 ( ( +g ` M ) , W ) ` ( (♯ ` W ) - 1 ) ) )
67	66	fveq2d 5636	. . 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 2229	. . . 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 13519	. . . . . 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 5494	. . . . 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 13451	. . 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 2272	. 2 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H ` ( M gsumg W ) ) = ( N gsumg ( H o. W ) ) )
76		fin0or 7061	. . . 4 |- ( W e. Fin -> ( W = (/) \/ E. j j e. W ) )
77		n0r 3505	. . . . 5 |- ( E. j j e. W -> W =/= (/) )
78	77	orim2i 766	. . . 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 803	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 713   = wceq 1395   E.wex 1538   e. wcel 2200   =/= wne 2400   _Vcvv 2799   (/)c0 3491   o. ccom 4724   Fn wfn 5316   -->wf 5317   ` cfv 5321  (class class class)co 6010   Fincfn 6900   0cc0 8015   1c1 8016   - cmin 8333   NNcn 9126   NN0cn0 9385   ZZcz 9462   ZZ>=cuz 9738   ...cfz 10221  ..^cfzo 10355   seqcseq 10686  ♯chash 11014  Word cword 11089   Basecbs 13053   +g cplusg 13131   0gc0g 13310   gsum_g cgsu 13311   Mndcmnd 13470   MndHom cmhm 13511

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-addcom 8115  ax-addass 8117  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-0id 8123  ax-rnegex 8124  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-frec 6548  df-1o 6573  df-er 6693  df-map 6810  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-n0 9386  df-z 9463  df-uz 9739  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-ihash 11015  df-word 11090  df-ndx 13056  df-slot 13057  df-base 13059  df-plusg 13144  df-0g 13312  df-igsum 13313  df-mgm 13410  df-sgrp 13456  df-mnd 13471  df-mhm 13513

This theorem is referenced by: (None)