Theorem ofco 7750

Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)

Hypotheses

Ref	Expression
ofco.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofco.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
ofco.3	⊢ (𝜑 → 𝐻:𝐷⟶𝐶)
ofco.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofco.5	⊢ (𝜑 → 𝐵 ∈ 𝑊)
ofco.6	⊢ (𝜑 → 𝐷 ∈ 𝑋)
ofco.7	⊢ (𝐴 ∩ 𝐵) = 𝐶

Assertion

Ref	Expression
ofco	⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)))

Proof of Theorem ofco

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

Step	Hyp	Ref	Expression
1		ofco.3	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐶)
2	1	ffvelcdmda 7116	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐻‘𝑥) ∈ 𝐶)
3	1	feqmptd 6984	. . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐷 ↦ (𝐻‘𝑥)))
4		ofco.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
5		ofco.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
6		ofco.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		ofco.5	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
8		ofco.7	. . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝐶
9		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
10		eqidd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) = (𝐺‘𝑦))
11	4, 5, 6, 7, 8, 9, 10	offval 7734	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑦 ∈ 𝐶 ↦ ((𝐹‘𝑦)𝑅(𝐺‘𝑦))))
12		fveq2 6916	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
13		fveq2 6916	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
14	12, 13	oveq12d 7469	. . 3 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
15	2, 3, 11, 14	fmptco 7162	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
16		inss1 4136	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
17	8, 16	eqsstrri 3960	. . . . 5 ⊢ 𝐶 ⊆ 𝐴
18		fss 6756	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐴) → 𝐻:𝐷⟶𝐴)
19	1, 17, 18	sylancl 598	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐴)
20		fnfco 6777	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐻:𝐷⟶𝐴) → (𝐹 ∘ 𝐻) Fn 𝐷)
21	4, 19, 20	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐻) Fn 𝐷)
22		inss2 4137	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
23	8, 22	eqsstrri 3960	. . . . 5 ⊢ 𝐶 ⊆ 𝐵
24		fss 6756	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐻:𝐷⟶𝐵)
25	1, 23, 24	sylancl 598	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐵)
26		fnfco 6777	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐻:𝐷⟶𝐵) → (𝐺 ∘ 𝐻) Fn 𝐷)
27	5, 25, 26	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐻) Fn 𝐷)
28		ofco.6	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
29		inidm 4126	. . 3 ⊢ (𝐷 ∩ 𝐷) = 𝐷
30	1	ffnd 6740	. . . 4 ⊢ (𝜑 → 𝐻 Fn 𝐷)
31		fvco2 7013	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
32	30, 31	sylan 592	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
33		fvco2 7013	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
34	30, 33	sylan 592	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
35	21, 27, 28, 28, 29, 32, 34	offval 7734	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
36	15, 35	eqtr4d 2801	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092   ↦ cmpt 5192   ∘ ccom 5662   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∘_f cof 7723

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-of 7725

This theorem is used by:  gsumzaddlem  20186  coe1add  22638  pf1ind  22728  1arithidomlem2  34172  mplvrpmrhm  34290  mendring  44451