Theorem ismgmhm 18655

Description: Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.)

Hypotheses

Ref	Expression
ismgmhm.b	⊢ 𝐵 = (Base‘𝑆)
ismgmhm.c	⊢ 𝐶 = (Base‘𝑇)
ismgmhm.p	⊢ + = (+g‘𝑆)
ismgmhm.q	⊢ ⨣ = (+g‘𝑇)

Assertion

Ref	Expression
ismgmhm	⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘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)

Proof of Theorem ismgmhm

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

Step	Hyp	Ref	Expression
1		mgmhmrcl 18653	. 2 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
2		fveq2 6827	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
3		ismgmhm.c	. . . . . . . 8 ⊢ 𝐶 = (Base‘𝑇)
4	2, 3	eqtr4di 2792	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
5		fveq2 6827	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
6		ismgmhm.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑆)
7	5, 6	eqtr4di 2792	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
8	4, 7	oveqan12rd 7376	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((Base‘𝑡) ↑m (Base‘𝑠)) = (𝐶 ↑m 𝐵))
9	7	adantr 481	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
10		fveq2 6827	. . . . . . . . . . . 12 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = (+g‘𝑆))
11		ismgmhm.p	. . . . . . . . . . . 12 ⊢ + = (+g‘𝑆)
12	10, 11	eqtr4di 2792	. . . . . . . . . . 11 ⊢ (𝑠 = 𝑆 → (+g‘𝑠) = + )
13	12	oveqd 7373	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (𝑥(+g‘𝑠)𝑦) = (𝑥 + 𝑦))
14	13	fveq2d 6831	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (𝑓‘(𝑥(+g‘𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
15		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = (+g‘𝑇))
16		ismgmhm.q	. . . . . . . . . . 11 ⊢ ⨣ = (+g‘𝑇)
17	15, 16	eqtr4di 2792	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (+g‘𝑡) = ⨣ )
18	17	oveqd 7373	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)))
19	14, 18	eqeqan12d 2753	. . . . . . . 8 ⊢ ((𝑠 = 𝑆 ∧ 𝑡 = 𝑇) → ((𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))))
20	9, 19	raleqbidv 3313	. . . . . . 7 ⊢ ((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))))
21	9, 20	raleqbidv 3313	. . . . . 6 ⊢ ((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))))
22	8, 21	rabeqbidv 3409	. . . . 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 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))})
23		df-mgmhm 18651	. . . . 5 ⊢ MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑m (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g‘𝑠)𝑦)) = ((𝑓‘𝑥)(+g‘𝑡)(𝑓‘𝑦))})
24		ovex 7389	. . . . . 6 ⊢ (𝐶 ↑m 𝐵) ∈ V
25	24	rabex 5267	. . . . 5 ⊢ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))} ∈ V
26	22, 23, 25	ovmpoa 7511	. . . 4 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝑆 MgmHom 𝑇) = {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))})
27	26	eleq2d 2825	. . 3 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))}))
28		fveq1 6826	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
29		fveq1 6826	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
30		fveq1 6826	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
31	29, 30	oveq12d 7374	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
32	28, 31	eqeq12d 2755	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
33	32	2ralbidv 3203	. . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
34	33	elrab 3629	. . . 4 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))} ↔ (𝐹 ∈ (𝐶 ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
35	3	fvexi 6841	. . . . . 6 ⊢ 𝐶 ∈ V
36	6	fvexi 6841	. . . . . 6 ⊢ 𝐵 ∈ V
37	35, 36	elmap 8809	. . . . 5 ⊢ (𝐹 ∈ (𝐶 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐶)
38	37	anbi1i 630	. . . 4 ⊢ ((𝐹 ∈ (𝐶 ↑m 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
39	34, 38	bitri 276	. . 3 ⊢ (𝐹 ∈ {𝑓 ∈ (𝐶 ↑m 𝐵) ∣ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓‘𝑥) ⨣ (𝑓‘𝑦))} ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))))
40	27, 39	bitrdi 288	. 2 ⊢ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))))
41	1, 40	biadanii 827	1 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶𝐶 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))))

Syntax hints:   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763  Basecbs 17170  +_gcplusg 17211  Mgmcmgm 18597   MgmHom cmgmhm 18649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8765  df-mgmhm 18651

This theorem is referenced by:  mgmhmf  18656  mgmhmpropd  18657  mgmhmlin  18658  mgmhmf1o  18659  idmgmhm  18660  resmgmhm  18670  resmgmhm2  18671  resmgmhm2b  18672  mgmhmco  18673  ismhm0  18749  mhmismgmhm  18750  isrnghmmul  20413  c0mgm  20430  c0snmgmhm  20433