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Theorem mhmlin 18806
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 2737 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
3 mhmlin.p . . . . . 6 + = (+g𝑆)
4 mhmlin.q . . . . . 6 = (+g𝑇)
5 eqid 2737 . . . . . 6 (0g𝑆) = (0g𝑆)
6 eqid 2737 . . . . . 6 (0g𝑇) = (0g𝑇)
71, 2, 3, 4, 5, 6ismhm 18798 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇))))
87simprbi 496 . . . 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 7454 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
11 fveq2 6906 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1211oveq1d 7446 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑦)))
1310, 12eqeq12d 2753 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦))))
14 oveq2 7439 . . . . . 6 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
1514fveq2d 6910 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
16 fveq2 6906 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1716oveq2d 7447 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑌)))
1815, 17eqeq12d 2753 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1913, 18rspc2v 3633 . . 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 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Mndcmnd 18747   MndHom cmhm 18794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-mhm 18796
This theorem is referenced by:  mhmf1o  18809  mhmvlin  18814  resmhm  18833  resmhm2  18834  resmhm2b  18835  mhmco  18836  mhmimalem  18837  mhmeql  18839  pwsco2mhm  18846  gsumwmhm  18858  mhmmulg  19133  ghmmhmb  19245  cntzmhm  19359  gsumzmhm  19955  rhmmul  20486  rhmimasubrnglem  20565  evlslem1  22106  mpfind  22131  mdetunilem7  22624  dchrzrhmul  27290  dchrmulcl  27293  dchrn0  27294  dchrinvcl  27297  dchrsum2  27312  sum2dchr  27318  mhmimasplusg  33043  mhmhmeotmd  33926
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