Theorem mhmlin 17550

Description: A monoid homomorphism commutes with composition. (Contributed by Mario Carneiro, 7-Mar-2015.)

Hypotheses

Ref	Expression
mhmlin.b	⊢ 𝐵 = (Base‘𝑆)
mhmlin.p	⊢ + = (+g‘𝑆)
mhmlin.q	⊢ ⨣ = (+g‘𝑇)

Assertion

Ref	Expression
mhmlin	⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))

Proof of Theorem mhmlin

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhmlin.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
2		eqid 2771	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		mhmlin.p	. . . . . 6 ⊢ + = (+g‘𝑆)
4		mhmlin.q	. . . . . 6 ⊢ ⨣ = (+g‘𝑇)
5		eqid 2771	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		eqid 2771	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
7	1, 2, 3, 4, 5, 6	ismhm 17545	. . . . 5 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇))))
8	7	simprbi 484	. . . 4 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → (𝐹:𝐵⟶(Base‘𝑇) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ∧ (𝐹‘(0g‘𝑆)) = (0g‘𝑇)))
9	8	simp2d 1137	. . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
10		fvoveq1 6816	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
11		fveq2 6332	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
12	11	oveq1d 6808	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)))
13	10, 12	eqeq12d 2786	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦))))
14		oveq2 6801	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
15	14	fveq2d 6336	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
16		fveq2 6332	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
17	16	oveq2d 6809	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))
18	15, 17	eqeq12d 2786	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
19	13, 18	rspc2v 3472	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
20	9, 19	syl5com 31	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
21	20	3impib 1108	1 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  +_gcplusg 16149  0_gc0g 16308  Mndcmnd 17502   MndHom cmhm 17541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-map 8011  df-mhm 17543

This theorem is referenced by:  mhmf1o  17553  resmhm  17567  resmhm2  17568  resmhm2b  17569  mhmco  17570  mhmima  17571  mhmeql  17572  pwsco2mhm  17579  gsumwmhm  17590  mhmmulg  17791  ghmmhmb  17879  cntzmhm  17978  gsumzmhm  18544  rhmmul  18937  evlslem1  19730  mpfind  19751  mhmvlin  20420  mdetunilem7  20642  dchrzrhmul  25192  dchrmulcl  25195  dchrn0  25196  dchrinvcl  25199  dchrsum2  25214  sum2dchr  25220  mhmhmeotmd  30313