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Theorem mhmlin 18569
Description: A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)
Hypotheses
Ref Expression
mhmlin.b 𝐵 = (Base‘𝑆)
mhmlin.p + = (+g𝑆)
mhmlin.q = (+g𝑇)
Assertion
Ref Expression
mhmlin ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))

Proof of Theorem mhmlin
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhmlin.b . . . . . 6 𝐵 = (Base‘𝑆)
2 eqid 2738 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
3 mhmlin.p . . . . . 6 + = (+g𝑆)
4 mhmlin.q . . . . . 6 = (+g𝑇)
5 eqid 2738 . . . . . 6 (0g𝑆) = (0g𝑆)
6 eqid 2738 . . . . . 6 (0g𝑇) = (0g𝑇)
71, 2, 3, 4, 5, 6ismhm 18563 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇))))
87simprbi 498 . . . 4 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇)))
98simp2d 1144 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑇) → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
10 fvoveq1 7375 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
11 fveq2 6840 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1211oveq1d 7367 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑦)))
1310, 12eqeq12d 2754 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦))))
14 oveq2 7360 . . . . . 6 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
1514fveq2d 6844 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
16 fveq2 6840 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1716oveq2d 7368 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑌)))
1815, 17eqeq12d 2754 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1913, 18rspc2v 3589 . . 3 ((𝑋𝐵𝑌𝐵) → (∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
209, 19syl5com 31 . 2 (𝐹 ∈ (𝑆 MndHom 𝑇) → ((𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
21203impib 1117 1 ((𝐹 ∈ (𝑆 MndHom 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 3063  wf 6490  cfv 6494  (class class class)co 7352  Basecbs 17043  +gcplusg 17093  0gc0g 17281  Mndcmnd 18516   MndHom cmhm 18559
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 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8726  df-mhm 18561
This theorem is referenced by:  mhmf1o  18572  resmhm  18591  resmhm2  18592  resmhm2b  18593  mhmco  18594  mhmima  18595  mhmeql  18596  pwsco2mhm  18603  gsumwmhm  18615  mhmmulg  18876  ghmmhmb  18978  cntzmhm  19078  gsumzmhm  19673  rhmmul  20112  evlslem1  21444  mpfind  21469  mhmvlin  21698  mdetunilem7  21919  dchrzrhmul  26546  dchrmulcl  26549  dchrn0  26550  dchrinvcl  26553  dchrsum2  26568  sum2dchr  26574  mhmhmeotmd  32312
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