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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 2771 . . . 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 fvoveq1 6816 . . . . . . 7 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
7 fveq2 6332 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
87oveq1d 6808 . . . . . . 7 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑦)))
96, 8eqeq12d 2786 . . . . . 6 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦))))
10 oveq2 6801 . . . . . . . 8 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
1110fveq2d 6336 . . . . . . 7 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
12 fveq2 6332 . . . . . . . 8 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1312oveq2d 6809 . . . . . . 7 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑌)))
1411, 13eqeq12d 2786 . . . . . 6 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
159, 14rspc2v 3472 . . . . 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 708 . . 3 (((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
185, 17sylbi 207 . 2 (𝐹 ∈ (𝑆 MgmHom 𝑇) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
19183impib 1108 1 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1071   = wceq 1631  wcel 2145  wral 3061  wf 6027  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  Mgmcmgm 17448   MgmHom cmgmhm 42305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  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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