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Theorem mgmhmlin 42314
Description: A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)
Hypotheses
Ref Expression
mgmhmlin.b 𝐵 = (Base‘𝑆)
mgmhmlin.p + = (+g𝑆)
mgmhmlin.q = (+g𝑇)
Assertion
Ref Expression
mgmhmlin ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))

Proof of Theorem mgmhmlin
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmlin.b . . . 4 𝐵 = (Base‘𝑆)
2 eqid 2760 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 mgmhmlin.p . . . 4 + = (+g𝑆)
4 mgmhmlin.q . . . 4 = (+g𝑇)
51, 2, 3, 4ismgmhm 42311 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
6 oveq1 6821 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 + 𝑦) = (𝑋 + 𝑦))
76fveq2d 6357 . . . . . . 7 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
8 fveq2 6353 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
98oveq1d 6829 . . . . . . 7 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑦)))
107, 9eqeq12d 2775 . . . . . 6 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦))))
11 oveq2 6822 . . . . . . . 8 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
1211fveq2d 6357 . . . . . . 7 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
13 fveq2 6353 . . . . . . . 8 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1413oveq2d 6830 . . . . . . 7 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑌)))
1512, 14eqeq12d 2775 . . . . . 6 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1610, 15rspc2v 3461 . . . . 5 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1716com12 32 . . . 4 (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1817ad2antll 767 . . 3 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
195, 18sylbi 207 . 2 (𝐹 ∈ (𝑆 MgmHom 𝑇) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
20193impib 1109 1 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wf 6045  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  Mgmcmgm 17461   MgmHom cmgmhm 42305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-mgmhm 42307
This theorem is referenced by:  mgmhmf1o  42315  resmgmhm  42326  resmgmhm2  42327  resmgmhm2b  42328  mgmhmco  42329  mgmhmima  42330  mgmhmeql  42331
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