Theorem mhmlin 18850

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 2761	. . . . . 6 ⊢ (Base‘𝑇) = (Base‘𝑇)
3		mhmlin.p	. . . . . 6 ⊢ + = (+g‘𝑆)
4		mhmlin.q	. . . . . 6 ⊢ ⨣ = (+g‘𝑇)
5		eqid 2761	. . . . . 6 ⊢ (0g‘𝑆) = (0g‘𝑆)
6		eqid 2761	. . . . . 6 ⊢ (0g‘𝑇) = (0g‘𝑇)
7	1, 2, 3, 4, 5, 6	ismhm 18842	. . . . 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 502	. . . 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 1159	. . 3 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
10		fvoveq1 7433	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑋 + 𝑦)))
11		fveq2 6881	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
12	11	oveq1d 7425	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)))
13	10, 12	eqeq12d 2777	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦))))
14		oveq2 7418	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑋 + 𝑦) = (𝑋 + 𝑌))
15	14	fveq2d 6885	. . . . 5 ⊢ (𝑦 = 𝑌 → (𝐹‘(𝑋 + 𝑦)) = (𝐹‘(𝑋 + 𝑌)))
16		fveq2 6881	. . . . . 6 ⊢ (𝑦 = 𝑌 → (𝐹‘𝑦) = (𝐹‘𝑌))
17	16	oveq2d 7426	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))
18	15, 17	eqeq12d 2777	. . . 4 ⊢ (𝑦 = 𝑌 → ((𝐹‘(𝑋 + 𝑦)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
19	13, 18	rspc2v 3591	. . 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 32	. 2 ⊢ (𝐹 ∈ (𝑆 MndHom 𝑇) → ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌))))
21	20	3impib 1132	1 ⊢ ((𝐹 ∈ (𝑆 MndHom 𝑇) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹‘𝑋) ⨣ (𝐹‘𝑌)))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  +_gcplusg 17309  0_gc0g 17491  Mndcmnd 18791   MndHom cmhm 18838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-map 8825  df-mhm 18840

This theorem is referenced by:  mhmf1o  18853  mhmvlin  18858  resmhm  18878  resmhm2  18879  resmhm2b  18880  mhmco  18881  mhmimalem  18882  mhmeql  18884  pwsco2mhm  18891  gsumwmhm  18903  mhmmulg  19180  ghmmhmb  19296  cntzmhm  19410  gsumzmhm  20006  rhmmul  20567  rhmimasubrnglem  20649  evlslem1  22212  mpfind  22245  mdetunilem7  22754  dchrzrhmul  27386  dchrmulcl  27389  dchrn0  27390  dchrinvcl  27393  dchrsum2  27408  sum2dchr  27414  mhmimasplusg  33323  fxpsubm  33458  mplvrpmrhm  33903  mhmhmeotmd  34283