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Theorem mhmlin 17958
 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 2798 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
3 mhmlin.p . . . . . 6 + = (+g𝑆)
4 mhmlin.q . . . . . 6 = (+g𝑇)
5 eqid 2798 . . . . . 6 (0g𝑆) = (0g𝑆)
6 eqid 2798 . . . . . 6 (0g𝑇) = (0g𝑇)
71, 2, 3, 4, 5, 6ismhm 17953 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇))))
87simprbi 500 . . . 4 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇)))
98simp2d 1140 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑇) → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
10 fvoveq1 7159 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
11 fveq2 6646 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1211oveq1d 7151 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑦)))
1310, 12eqeq12d 2814 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦))))
14 oveq2 7144 . . . . . 6 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
1514fveq2d 6650 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
16 fveq2 6646 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1716oveq2d 7152 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑌)))
1815, 17eqeq12d 2814 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1913, 18rspc2v 3581 . . 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 1113 1 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  Basecbs 16478  +gcplusg 16560  0gc0g 16708  Mndcmnd 17906   MndHom cmhm 17949 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-map 8394  df-mhm 17951 This theorem is referenced by:  mhmf1o  17961  resmhm  17980  resmhm2  17981  resmhm2b  17982  mhmco  17983  mhmima  17984  mhmeql  17985  pwsco2mhm  17992  gsumwmhm  18005  mhmmulg  18264  ghmmhmb  18365  cntzmhm  18465  gsumzmhm  19054  rhmmul  19479  evlslem1  20760  mpfind  20785  mhmvlin  21014  mdetunilem7  21233  dchrzrhmul  25840  dchrmulcl  25843  dchrn0  25844  dchrinvcl  25847  dchrsum2  25862  sum2dchr  25868  mhmhmeotmd  31295
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