Theorem ofco 7685

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 7065	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐻‘𝑥) ∈ 𝐶)
3	1	feqmptd 6935	. . 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 2763	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
10		eqidd 2763	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) = (𝐺‘𝑦))
11	4, 5, 6, 7, 8, 9, 10	offval 7669	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑦 ∈ 𝐶 ↦ ((𝐹‘𝑦)𝑅(𝐺‘𝑦))))
12		fveq2 6867	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
13		fveq2 6867	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
14	12, 13	oveq12d 7414	. . 3 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
15	2, 3, 11, 14	fmptco 7111	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
16		inss1 4188	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
17	8, 16	eqsstrri 3983	. . . . 5 ⊢ 𝐶 ⊆ 𝐴
18		fss 6708	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐴) → 𝐻:𝐷⟶𝐴)
19	1, 17, 18	sylancl 595	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐴)
20		fnfco 6729	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐻:𝐷⟶𝐴) → (𝐹 ∘ 𝐻) Fn 𝐷)
21	4, 19, 20	syl2anc 593	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐻) Fn 𝐷)
22		inss2 4189	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
23	8, 22	eqsstrri 3983	. . . . 5 ⊢ 𝐶 ⊆ 𝐵
24		fss 6708	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐻:𝐷⟶𝐵)
25	1, 23, 24	sylancl 595	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐵)
26		fnfco 6729	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐻:𝐷⟶𝐵) → (𝐺 ∘ 𝐻) Fn 𝐷)
27	5, 25, 26	syl2anc 593	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐻) Fn 𝐷)
28		ofco.6	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
29		inidm 4178	. . 3 ⊢ (𝐷 ∩ 𝐷) = 𝐷
30	1	ffnd 6692	. . . 4 ⊢ (𝜑 → 𝐻 Fn 𝐷)
31		fvco2 6964	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
32	30, 31	sylan 589	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
33		fvco2 6964	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
34	30, 33	sylan 589	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
35	21, 27, 28, 28, 29, 32, 34	offval 7669	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
36	15, 35	eqtr4d 2800	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560   ∈ wcel 2142   ∩ cin 3903   ⊆ wss 3904   ↦ cmpt 5181   ∘ ccom 5651   Fn wfn 6516  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ∘_f cof 7658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-of 7660

This theorem is referenced by:  gsumzaddlem  19961  coe1add  22327  pf1ind  22418  1arithidomlem2  33732  mplvrpmrhm  33844  mendring  43765