Theorem ismhm 18603

Description: Property of a monoid homomorphism. (Contributed by Mario Carneiro, 7-Mar-2015.)

Hypotheses

Ref	Expression
ismhm.b	⊢ 𝐵 = (Base‘𝑆)
ismhm.c	⊢ 𝐶 = (Base‘𝑇)
ismhm.p	⊢ + = (+g‘𝑆)
ismhm.q	⊢ ⨣ = (+g‘𝑇)
ismhm.z	⊢ 0 = (0g‘𝑆)
ismhm.y	⊢ 𝑌 = (0g‘𝑇)

Assertion

Ref	Expression
ismhm	⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)   + (𝑥,𝑦)   ⨣ (𝑥,𝑦)   𝑌(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem ismhm

Dummy variables 𝑓 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mhm 18601	. . 3 ⊢ MndHom = (𝑠 ∈ Mnd, 𝑡 ∈ Mnd ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑠)) = (0g‘𝑡))})
2	1	elmpocl 7595	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
3		fveq2 6842	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
4		ismhm.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑇)
5	3, 4	eqtr4di 2794	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
6		fveq2 6842	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
7		ismhm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
8	6, 7	eqtr4di 2794	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
9	5, 8	oveqan12rd 7377	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((Base‘𝑡) ↑m (Base‘𝑠)) = (𝐶 ↑m 𝐵))
10	8	adantr 481	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
11		fveq2 6842	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
12		ismhm.p	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑆)
13	11, 12	eqtr4di 2794	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = + )
14	13	oveqd 7374	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥 + 𝑦))
15	14	fveq2d 6846	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝑓‘(𝑥(+g‘𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
16		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
17		ismhm.q	. . . . . . . . . . . 12 ⊢ ⨣ = (+g‘𝑇)
18	16, 17	eqtr4di 2794	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = ⨣ )
19	18	oveqd 7374	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)))
20	15, 19	eqeqan12d 2750	. . . . . . . . 9 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))))
21	10, 20	raleqbidv 3319	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))))
22	10, 21	raleqbidv 3319	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))))
23		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = (0g‘𝑆))
24		ismhm.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑆)
25	23, 24	eqtr4di 2794	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = 0 )
26	25	fveq2d 6846	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑓‘(0g‘𝑠)) = (𝑓‘ 0 ))
27		fveq2 6842	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = (0g‘𝑇))
28		ismhm.y	. . . . . . . . 9 ⊢ 𝑌 = (0g‘𝑇)
29	27, 28	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = 𝑌)
30	26, 29	eqeqan12d 2750	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑓‘(0g‘𝑠)) = (0g‘𝑡) ↔ (𝑓‘ 0 ) = 𝑌))
31	22, 30	anbi12d 631	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑠)) = (0g‘𝑡)) ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)))
32	9, 31	rabeqbidv 3424	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ∧ (𝑓‘(0g‘𝑠)) = (0g‘𝑡))} = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)})
33		ovex 7390	. . . . . 6 ⊢ (𝐶 ↑m 𝐵) ∈ V
34	33	rabex 5289	. . . . 5 ⊢ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)} ∈ V
35	32, 1, 34	ovmpoa 7510	. . . 4 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝑆 MndHom 𝑇) = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)})
36	35	eleq2d 2823	. . 3 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)}))
37	4	fvexi 6856	. . . . . 6 ⊢ 𝐶 ∈ V
38	7	fvexi 6856	. . . . . 6 ⊢ 𝐵 ∈ V
39	37, 38	elmap 8809	. . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐶)
40	39	anbi1i 624	. . . 4 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐵) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)) ↔ (𝐹:𝐵⟶𝐶 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
41		fveq1 6841	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
42		fveq1 6841	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
43		fveq1 6841	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
44	42, 43	oveq12d 7375	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
45	41, 44	eqeq12d 2752	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
46	45	2ralbidv 3212	. . . . . 6 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
47		fveq1 6841	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘ 0 ) = (𝐹‘ 0 ))
48	47	eqeq1d 2738	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘ 0 ) = 𝑌 ↔ (𝐹‘ 0 ) = 𝑌))
49	46, 48	anbi12d 631	. . . . 5 ⊢ (𝑓 = 𝐹 → ((∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌) ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
50	49	elrab 3645	. . . 4 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)} ↔ (𝐹 ∈ (𝐶 ↑m 𝐵) ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
51		3anass 1095	. . . 4 ⊢ ((𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌) ↔ (𝐹:𝐵⟶𝐶 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
52	40, 50, 51	3bitr4i 302	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ∧ (𝑓‘ 0 ) = 𝑌)} ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌))
53	36, 52	bitrdi 286	. 2 ⊢ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))
54	2, 53	biadanii 820	1 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘ 0 ) = 𝑌)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3064  {crab 3407  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357   ↑_m cmap 8765  Basecbs 17083  +_gcplusg 17133  0_gc0g 17321  Mndcmnd 18556   MndHom cmhm 18599

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-mhm 18601

This theorem is referenced by:  ismhmd  18604  mhmf  18607  mhmpropd  18608  mhmlin  18609  mhm0  18610  idmhm  18611  mhmf1o  18612  0mhm  18630  resmhm  18631  resmhm2  18632  resmhm2b  18633  mhmco  18634  prdspjmhm  18639  pwsdiagmhm  18641  pwsco1mhm  18642  pwsco2mhm  18643  frmdup1  18674  mhmfmhm  18870  ghmmhm  19018  frgpmhm  19547  mulgmhm  19606  srglmhm  19952  srgrmhm  19953  dfrhm2  20148  isrhm2d  20160  expmhm  20866  mat1mhm  21833  scmatmhm  21883  mat2pmatmhm  22082  pm2mpmhm  22169  dchrelbas3  26586  xrge0iifmhm  32520  esumcocn  32679  elmrsubrn  34114  deg1mhm  41520  ismhm0  46089  mhmismgmhm  46090  c0mhm  46198