Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mgmhmlin Structured version   Visualization version   GIF version

Theorem mgmhmlin 44359
 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 2824 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 mgmhmlin.p . . . 4 + = (+g𝑆)
4 mgmhmlin.q . . . 4 = (+g𝑇)
51, 2, 3, 4ismgmhm 44356 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
6 fvoveq1 7174 . . . . . . 7 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
7 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
87oveq1d 7166 . . . . . . 7 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑦)))
96, 8eqeq12d 2840 . . . . . 6 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦))))
10 oveq2 7159 . . . . . . . 8 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
1110fveq2d 6667 . . . . . . 7 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
12 fveq2 6663 . . . . . . . 8 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1312oveq2d 7167 . . . . . . 7 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑌)))
1411, 13eqeq12d 2840 . . . . . 6 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
159, 14rspc2v 3619 . . . . 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 220 . 2 (𝐹 ∈ (𝑆 MgmHom 𝑇) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
19183impib 1113 1 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2115  ∀wral 3133  ⟶wf 6341  ‘cfv 6345  (class class class)co 7151  Basecbs 16485  +gcplusg 16567  Mgmcmgm 17852   MgmHom cmgmhm 44350 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-map 8406  df-mgmhm 44352 This theorem is referenced by:  mgmhmf1o  44360  resmgmhm  44371  resmgmhm2  44372  resmgmhm2b  44373  mgmhmco  44374  mgmhmima  44375  mgmhmeql  44376
 Copyright terms: Public domain W3C validator