Users' Mathboxes 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 41074
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 2621 . . . 4 (Base‘𝑇) = (Base‘𝑇)
3 mgmhmlin.p . . . 4 + = (+g𝑆)
4 mgmhmlin.q . . . 4 = (+g𝑇)
51, 2, 3, 4ismgmhm 41071 . . 3 (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
6 oveq1 6611 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 + 𝑦) = (𝑋 + 𝑦))
76fveq2d 6152 . . . . . . 7 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
8 fveq2 6148 . . . . . . . 8 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
98oveq1d 6619 . . . . . . 7 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑦)))
107, 9eqeq12d 2636 . . . . . 6 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦))))
11 oveq2 6612 . . . . . . . 8 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
1211fveq2d 6152 . . . . . . 7 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
13 fveq2 6148 . . . . . . . 8 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1413oveq2d 6620 . . . . . . 7 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑌)))
1512, 14eqeq12d 2636 . . . . . 6 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1610, 15rspc2v 3306 . . . . 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 764 . . 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 1259 1 ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Mgmcmgm 17161   MgmHom cmgmhm 41065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-map 7804  df-mgmhm 41067
This theorem is referenced by:  mgmhmf1o  41075  resmgmhm  41086  resmgmhm2  41087  resmgmhm2b  41088  mgmhmco  41089  mgmhmima  41090  mgmhmeql  41091
  Copyright terms: Public domain W3C validator