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Theorem mhmlin 17781
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 2795 . . . . . 6 (Base‘𝑇) = (Base‘𝑇)
3 mhmlin.p . . . . . 6 + = (+g𝑆)
4 mhmlin.q . . . . . 6 = (+g𝑇)
5 eqid 2795 . . . . . 6 (0g𝑆) = (0g𝑆)
6 eqid 2795 . . . . . 6 (0g𝑇) = (0g𝑇)
71, 2, 3, 4, 5, 6ismhm 17776 . . . . 5 (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇))))
87simprbi 497 . . . 4 (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ∧ (𝐹‘(0g𝑆)) = (0g𝑇)))
98simp2d 1136 . . 3 (𝐹 ∈ (𝑆 MndHom 𝑇) → ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
10 fvoveq1 7039 . . . . 5 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
11 fveq2 6538 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
1211oveq1d 7031 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑦)))
1310, 12eqeq12d 2810 . . . 4 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦))))
14 oveq2 7024 . . . . . 6 (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
1514fveq2d 6542 . . . . 5 (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
16 fveq2 6538 . . . . . 6 (𝑦 = 𝑌 → (𝐹𝑦) = (𝐹𝑌))
1716oveq2d 7032 . . . . 5 (𝑦 = 𝑌 → ((𝐹𝑋) (𝐹𝑦)) = ((𝐹𝑋) (𝐹𝑌)))
1815, 17eqeq12d 2810 . . . 4 (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹𝑋) (𝐹𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌))))
1913, 18rspc2v 3572 . . 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 1109 1 ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1522  wcel 2081  wral 3105  wf 6221  cfv 6225  (class class class)co 7016  Basecbs 16312  +gcplusg 16394  0gc0g 16542  Mndcmnd 17733   MndHom cmhm 17772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-map 8258  df-mhm 17774
This theorem is referenced by:  mhmf1o  17784  resmhm  17798  resmhm2  17799  resmhm2b  17800  mhmco  17801  mhmima  17802  mhmeql  17803  pwsco2mhm  17810  gsumwmhm  17821  mhmmulg  18022  ghmmhmb  18110  cntzmhm  18210  gsumzmhm  18777  rhmmul  19169  evlslem1  19982  mpfind  20003  mhmvlin  20690  mdetunilem7  20911  dchrzrhmul  25504  dchrmulcl  25507  dchrn0  25508  dchrinvcl  25511  dchrsum2  25526  sum2dchr  25532  mhmhmeotmd  30787
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