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Theorem mgmhmlin 46154
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 2737 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 mgmhmlin.p . . . 4 + = (+g𝑆)
4 mgmhmlin.q . . . 4 = (+g𝑇)
51, 2, 3, 4ismgmhm 46151 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
6 fvoveq1 7385 . . . . . . 7 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
7 fveq2 6847 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
87oveq1d 7377 . . . . . . 7 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑦)))
96, 8eqeq12d 2753 . . . . . 6 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦))))
10 oveq2 7370 . . . . . . . 8 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
1110fveq2d 6851 . . . . . . 7 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
12 fveq2 6847 . . . . . . . 8 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1312oveq2d 7378 . . . . . . 7 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑌)))
1411, 13eqeq12d 2753 . . . . . 6 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
159, 14rspc2v 3593 . . . . 5 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1615com12 32 . . . 4 (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1716ad2antll 728 . . 3 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
185, 17sylbi 216 . 2 (𝐹 ∈ (𝑆 MgmHom 𝑇) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
19183impib 1117 1 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3065  wf 6497  cfv 6501  (class class class)co 7362  Basecbs 17090  +gcplusg 17140  Mgmcmgm 18502   MgmHom cmgmhm 46145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-map 8774  df-mgmhm 46147
This theorem is referenced by:  mgmhmf1o  46155  resmgmhm  46166  resmgmhm2  46167  resmgmhm2b  46168  mgmhmco  46169  mgmhmima  46170  mgmhmeql  46171
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