Theorem gzsumwmhm 13780

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

Hypothesis

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

Assertion

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

Proof of Theorem gzsumwmhm

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

Step	Hyp	Ref	Expression
1		eqid 2238	. . . . 5 |- ( 0g ` M ) = ( 0g ` M )
2		eqid 2238	. . . . 5 |- ( 0g ` N ) = ( 0g ` N )
3	1, 2	mhm0 13752	. . . 4 |- ( H e. ( M MndHom N ) -> ( H ` ( 0g ` M ) ) = ( 0g ` N ) )
4	3	ad2antrr 492	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( H ` ( 0g ` M ) ) = ( 0g ` N ) )
5		oveq2 6083	. . . . . 6 |- ( W = (/) -> ( M gzsumgz W ) = ( M gzsumgz (/) ) )
6	5	adantl 277	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( M gzsumgz W ) = ( M gzsumgz (/) ) )
7		mhmrcl1 13747	. . . . . . 7 |- ( H e. ( M MndHom N ) -> M e. Mnd )
8	7	ad2antrr 492	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> M e. Mnd )
9	1	gzsum0 13690	. . . . . 6 |- ( M e. Mnd -> ( M gzsumgz (/) ) = ( 0g ` M ) )
10	8, 9	syl 14	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( M gzsumgz (/) ) = ( 0g ` M ) )
11	6, 10	eqtrd 2271	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( M gzsumgz W ) = ( 0g ` M ) )
12	11	fveq2d 5694	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( H ` ( M gzsumgz W ) ) = ( H ` ( 0g ` M ) ) )
13		coeq2 4933	. . . . . . 7 |- ( W = (/) -> ( H o. W ) = ( H o. (/) ) )
14		co02 5296	. . . . . . 7 |- ( H o. (/) ) = (/)
15	13, 14	eqtrdi 2287	. . . . . 6 |- ( W = (/) -> ( H o. W ) = (/) )
16	15	oveq2d 6091	. . . . 5 |- ( W = (/) -> ( N gzsumgz ( H o. W ) ) = ( N gzsumgz (/) ) )
17	16	adantl 277	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( N gzsumgz ( H o. W ) ) = ( N gzsumgz (/) ) )
18		mhmrcl2 13748	. . . . . 6 |- ( H e. ( M MndHom N ) -> N e. Mnd )
19	18	ad2antrr 492	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> N e. Mnd )
20	2	gzsum0 13690	. . . . 5 |- ( N e. Mnd -> ( N gzsumgz (/) ) = ( 0g ` N ) )
21	19, 20	syl 14	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( N gzsumgz (/) ) = ( 0g ` N ) )
22	17, 21	eqtrd 2271	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( N gzsumgz ( H o. W ) ) = ( 0g ` N ) )
23	4, 12, 22	3eqtr4d 2281	. 2 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W = (/) ) -> ( H ` ( M gzsumgz W ) ) = ( N gzsumgz ( H o. W ) ) )
24	7	ad2antrr 492	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> M e. Mnd )
25		gzsumwmhm.b	. . . . . . 7 |- B = ( Base ` M )
26		eqid 2238	. . . . . . 7 |- ( +g ` M ) = ( +g ` M )
27	25, 26	mndcl 13713	. . . . . 6 |- ( ( M e. Mnd /\ x e. B /\ y e. B ) -> ( x ( +g ` M ) y ) e. B )
28	27	3expb 1235	. . . . 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 11288	. . . . . . 7 |- ( W e. Word B -> W : ( 0..^ (♯ ` W ) ) --> B )
31	30	ad2antlr 493	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W : ( 0..^ (♯ ` W ) ) --> B )
32		wrdfin 11301	. . . . . . . . . . . 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 11212	. . . . . . . . . . 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 9746	. . . . . . . 8 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> (♯ ` W ) e. ZZ )
38		fzoval 10533	. . . . . . . 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 5516	. . . . . 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 5834	. . . 4 |- ( ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) /\ x e. ( 0 ... ( (♯ ` W ) - 1 ) ) ) -> ( W ` x ) e. B )
43		nnm1nn0 9583	. . . . . 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 9936	. . . . 5 |- NN0 = ( ZZ>= ` 0 )
46	44, 45	eleqtrdi 2331	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( (♯ ` W ) - 1 ) e. ( ZZ>= ` 0 ) )
47		eqid 2238	. . . . . . 7 |- ( +g ` N ) = ( +g ` N )
48	25, 26, 47	mhmlin 13751	. . . . . 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 1235	. . . . 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 518	. . . 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 5529	. . . . . 6 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W Fn ( 0 ... ( (♯ ` W ) - 1 ) ) )
52		fvco2 5768	. . . . . 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 2244	. . . 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 533	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> W e. Word B )
56		coexg 5327	. . . . 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 13443	. . . . . . 7 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
59	58	slotex 13357	. . . . . 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 492	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( +g ` M ) e. _V )
62	58	slotex 13357	. . . . . 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 492	. . . 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 10945	. . 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	gzsumval2 13691	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( M gzsumgz W ) = ( seq 0 ( ( +g ` M ) , W ) ` ( (♯ ` W ) - 1 ) ) )
67	66	fveq2d 5694	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H ` ( M gzsumgz W ) ) = ( H ` ( seq 0 ( ( +g ` M ) , W ) ` ( (♯ ` W ) - 1 ) ) ) )
68		eqid 2238	. . . 4 |- ( Base ` N ) = ( Base ` N )
69	18	ad2antrr 492	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> N e. Mnd )
70	25, 68	mhmf 13749	. . . . . 6 |- ( H e. ( M MndHom N ) -> H : B --> ( Base ` N ) )
71	70	ad2antrr 492	. . . . 5 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> H : B --> ( Base ` N ) )
72	71, 41	fcod 5548	. . . 4 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H o. W ) : ( 0 ... ( (♯ ` W ) - 1 ) ) --> ( Base ` N ) )
73	68, 47, 69, 46, 72	gzsumval2 13691	. . 3 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( N gzsumgz ( H o. W ) ) = ( seq 0 ( ( +g ` N ) , ( H o. W ) ) ` ( (♯ ` W ) - 1 ) ) )
74	65, 67, 73	3eqtr4d 2281	. 2 |- ( ( ( H e. ( M MndHom N ) /\ W e. Word B ) /\ W =/= (/) ) -> ( H ` ( M gzsumgz W ) ) = ( N gzsumgz ( H o. W ) ) )
75		fin0or 7180	. . . 4 |- ( W e. Fin -> ( W = (/) \/ E. j j e. W ) )
76		n0r 3535	. . . . 5 |- ( E. j j e. W -> W =/= (/) )
77	76	orim2i 773	. . . 4 |- ( ( W = (/) \/ E. j j e. W ) -> ( W = (/) \/ W =/= (/) ) )
78	75, 77	syl 14	. . 3 |- ( W e. Fin -> ( W = (/) \/ W =/= (/) ) )
79	33, 78	syl 14	. 2 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( W = (/) \/ W =/= (/) ) )
80	23, 74, 79	mpjaodan 810	1 |- ( ( H e. ( M MndHom N ) /\ W e. Word B ) -> ( H ` ( M gzsumgz W ) ) = ( N gzsumgz ( H o. W ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   _Vcvv 2821   (/)c0 3520   o. ccom 4773   Fn wfn 5367   -->wf 5368   ` cfv 5372  (class class class)co 6075   Fincfn 7012   0cc0 8169   1c1 8170   - cmin 8487   NNcn 9283   NN0cn0 9542   ZZcz 9623   ZZ>=cuz 9900   ...cfz 10390  ..^cfzo 10527   seqcseq 10862  ♯chash 11192  Word cword 11282   Basecbs 13330   +g cplusg 13408   0gc0g 13587   gzsum_gz cgzsu 13588   Mndcmnd 13706   MndHom cmhm 13741

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-2 9342  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-ihash 11193  df-word 11283  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-0g 13589  df-gzsum 13590  df-mgm 13653  df-sgrp 13694  df-mnd 13707  df-mhm 13743

This theorem is referenced by: (None)