Theorem mgmhmlin 18591

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 2729	. . . 4 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		mgmhmlin.p	. . . 4 ⊢ + = (+g‘𝑆)
4		mgmhmlin.q	. . . 4 ⊢ ⨣ = (+g‘𝑇)
5	1, 2, 3, 4	ismgmhm 18588	. . 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 7376	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
7		fveq2 6826	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
8	7	oveq1d 7368	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)))
9	6, 8	eqeq12d 2745	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦))))
10		oveq2 7361	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
11	10	fveq2d 6830	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
12		fveq2 6826	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
13	12	oveq2d 7369	. . . . . . 7 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))
14	11, 13	eqeq12d 2745	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
15	9, 14	rspc2v 3590	. . . . 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 729	. . 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 217	. 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  Basecbs 17138  +_gcplusg 17179  Mgmcmgm 18530   MgmHom cmgmhm 18582

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 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

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 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-map 8762  df-mgmhm 18584

This theorem is referenced by:  mgmhmf1o  18592  resmgmhm  18603  resmgmhm2  18604  resmgmhm2b  18605  mgmhmco  18606  mgmhmima  18607  mgmhmeql  18608