Theorem mgmhmlin 46546

Description: A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)

Hypotheses

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

Assertion

Ref	Expression
mgmhmlin	⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))

Proof of Theorem mgmhmlin

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

Step	Hyp	Ref	Expression
1		mgmhmlin.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
2		eqid 2732	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		mgmhmlin.p	. . . 4 ⊢ + = (+g‘𝑆)
4		mgmhmlin.q	. . . 4 ⊢ ⨣ = (+g‘𝑇)
5	1, 2, 3, 4	ismgmhm 46543	. . 3 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))))
6		fvoveq1 7431	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
7		fveq2 6891	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
8	7	oveq1d 7423	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)))
9	6, 8	eqeq12d 2748	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦))))
10		oveq2 7416	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
11	10	fveq2d 6895	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
12		fveq2 6891	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
13	12	oveq2d 7424	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))
14	11, 13	eqeq12d 2748	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
15	9, 14	rspc2v 3622	. . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
16	15	com12 32	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
17	16	ad2antll 727	. . 3 ⊢ (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
18	5, 17	sylbi 216	. 2 ⊢ (𝐹 ∈ (𝑆 MgmHom 𝑇) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
19	18	3impib 1116	1 ⊢ ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ⟶wf 6539  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  +_gcplusg 17196  Mgmcmgm 18558   MgmHom cmgmhm 46537

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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-mgmhm 46539

This theorem is referenced by:  mgmhmf1o  46547  resmgmhm  46558  resmgmhm2  46559  resmgmhm2b  46560  mgmhmco  46561  mgmhmima  46562  mgmhmeql  46563