Theorem ofco 7681

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 7059	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐻‘𝑥) ∈ 𝐶)
3	1	feqmptd 6932	. . 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 2731	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) = (𝐹‘𝑦))
10		eqidd 2731	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐺‘𝑦) = (𝐺‘𝑦))
11	4, 5, 6, 7, 8, 9, 10	offval 7665	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑦 ∈ 𝐶 ↦ ((𝐹‘𝑦)𝑅(𝐺‘𝑦))))
12		fveq2 6861	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐹‘𝑦) = (𝐹‘(𝐻‘𝑥)))
13		fveq2 6861	. . . 4 ⊢ (𝑦 = (𝐻‘𝑥) → (𝐺‘𝑦) = (𝐺‘(𝐻‘𝑥)))
14	12, 13	oveq12d 7408	. . 3 ⊢ (𝑦 = (𝐻‘𝑥) → ((𝐹‘𝑦)𝑅(𝐺‘𝑦)) = ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥))))
15	2, 3, 11, 14	fmptco 7104	. 2 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
16		inss1 4203	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
17	8, 16	eqsstrri 3997	. . . . 5 ⊢ 𝐶 ⊆ 𝐴
18		fss 6707	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐴) → 𝐻:𝐷⟶𝐴)
19	1, 17, 18	sylancl 586	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐴)
20		fnfco 6728	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐻:𝐷⟶𝐴) → (𝐹 ∘ 𝐻) Fn 𝐷)
21	4, 19, 20	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐹 ∘ 𝐻) Fn 𝐷)
22		inss2 4204	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
23	8, 22	eqsstrri 3997	. . . . 5 ⊢ 𝐶 ⊆ 𝐵
24		fss 6707	. . . . 5 ⊢ ((𝐻:𝐷⟶𝐶 ∧ 𝐶 ⊆ 𝐵) → 𝐻:𝐷⟶𝐵)
25	1, 23, 24	sylancl 586	. . . 4 ⊢ (𝜑 → 𝐻:𝐷⟶𝐵)
26		fnfco 6728	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐻:𝐷⟶𝐵) → (𝐺 ∘ 𝐻) Fn 𝐷)
27	5, 25, 26	syl2anc 584	. . 3 ⊢ (𝜑 → (𝐺 ∘ 𝐻) Fn 𝐷)
28		ofco.6	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑋)
29		inidm 4193	. . 3 ⊢ (𝐷 ∩ 𝐷) = 𝐷
30	1	ffnd 6692	. . . 4 ⊢ (𝜑 → 𝐻 Fn 𝐷)
31		fvco2 6961	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
32	30, 31	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐹 ∘ 𝐻)‘𝑥) = (𝐹‘(𝐻‘𝑥)))
33		fvco2 6961	. . . 4 ⊢ ((𝐻 Fn 𝐷 ∧ 𝑥 ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
34	30, 33	sylan 580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺 ∘ 𝐻)‘𝑥) = (𝐺‘(𝐻‘𝑥)))
35	21, 27, 28, 28, 29, 32, 34	offval 7665	. 2 ⊢ (𝜑 → ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)) = (𝑥 ∈ 𝐷 ↦ ((𝐹‘(𝐻‘𝑥))𝑅(𝐺‘(𝐻‘𝑥)))))
36	15, 35	eqtr4d 2768	1 ⊢ (𝜑 → ((𝐹 ∘f 𝑅𝐺) ∘ 𝐻) = ((𝐹 ∘ 𝐻) ∘f 𝑅(𝐺 ∘ 𝐻)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3916   ⊆ wss 3917   ↦ cmpt 5191   ∘ ccom 5645   Fn wfn 6509  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∘_f cof 7654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-of 7656

This theorem is referenced by:  gsumzaddlem  19858  coe1add  22157  pf1ind  22249  1arithidomlem2  33514  mendring  43184