Theorem ismhm 18688

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 18686	. . 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 7610	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd))
3		fveq2 6840	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
4		ismhm.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑇)
5	3, 4	eqtr4di 2782	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
6		fveq2 6840	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
7		ismhm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
8	6, 7	eqtr4di 2782	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
9	5, 8	oveqan12rd 7389	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((Base‘𝑡) ↑m (Base‘𝑠)) = (𝐶 ↑m 𝐵))
10	8	adantr 480	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
11		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
12		ismhm.p	. . . . . . . . . . . . 13 ⊢ + = (+g‘𝑆)
13	11, 12	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = + )
14	13	oveqd 7386	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥 + 𝑦))
15	14	fveq2d 6844	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝑓‘(𝑥(+g‘𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
16		fveq2 6840	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
17		ismhm.q	. . . . . . . . . . . 12 ⊢ ⨣ = (+g‘𝑇)
18	16, 17	eqtr4di 2782	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = ⨣ )
19	18	oveqd 7386	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)))
20	15, 19	eqeqan12d 2743	. . . . . . . . 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 3316	. . . . . . . 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 3316	. . . . . . 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 6840	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = (0g‘𝑆))
24		ismhm.z	. . . . . . . . . 10 ⊢ 0 = (0g‘𝑆)
25	23, 24	eqtr4di 2782	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (0g‘𝑠) = 0 )
26	25	fveq2d 6844	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (𝑓‘(0g‘𝑠)) = (𝑓‘ 0 ))
27		fveq2 6840	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = (0g‘𝑇))
28		ismhm.y	. . . . . . . . 9 ⊢ 𝑌 = (0g‘𝑇)
29	27, 28	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (0g‘𝑡) = 𝑌)
30	26, 29	eqeqan12d 2743	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑓‘(0g‘𝑠)) = (0g‘𝑡) ↔ (𝑓‘ 0 ) = 𝑌))
31	22, 30	anbi12d 632	. . . . . 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 3421	. . . . 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 7402	. . . . . 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 7524	. . . 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 2814	. . 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 6854	. . . . . 6 ⊢ 𝐶 ∈ V
38	7	fvexi 6854	. . . . . 6 ⊢ 𝐵 ∈ V
39	37, 38	elmap 8821	. . . . 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 6839	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
42		fveq1 6839	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
43		fveq1 6839	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
44	42, 43	oveq12d 7387	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
45	41, 44	eqeq12d 2745	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
46	45	2ralbidv 3199	. . . . . 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 6839	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘ 0 ) = (𝐹‘ 0 ))
48	47	eqeq1d 2731	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘ 0 ) = 𝑌 ↔ (𝐹‘ 0 ) = 𝑌))
49	46, 48	anbi12d 632	. . . . 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 3656	. . . 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 1094	. . . 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 303	. . 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 287	. 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 821	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ↑_m cmap 8776  Basecbs 17155  +_gcplusg 17196  0_gc0g 17378  Mndcmnd 18637   MndHom cmhm 18684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778  df-mhm 18686

This theorem is referenced by:  ismhmd  18689  mhmf  18692  ismhm0  18693  mhmismgmhm  18694  mhmpropd  18695  mhmlin  18696  mhm0  18697  idmhm  18698  mhmf1o  18699  0mhm  18722  resmhm  18723  resmhm2  18724  resmhm2b  18725  mhmco  18726  prdspjmhm  18732  pwsdiagmhm  18734  pwsco1mhm  18735  pwsco2mhm  18736  frmdup1  18767  mhmfmhm  18973  ghmmhm  19134  frgpmhm  19671  mulgmhm  19733  srglmhm  20106  srgrmhm  20107  c0mhm  20345  dfrhm2  20359  isrhm2d  20372  expmhm  21329  mat1mhm  22347  scmatmhm  22397  mat2pmatmhm  22596  pm2mpmhm  22683  dchrelbas3  27125  zringfrac  33498  xrge0iifmhm  33902  esumcocn  34043  elmrsubrn  35480  deg1mhm  43162