Theorem offres 7812

Description: Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Assertion

Ref	Expression
offres	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘f 𝑅𝐺) ↾ 𝐷) = ((𝐹 ↾ 𝐷) ∘f 𝑅(𝐺 ↾ 𝐷)))

Proof of Theorem offres

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elinel2 4134	. . . . 5 ⊢ (𝑥 ∈ ((dom 𝐹 ∩ dom 𝐺) ∩ 𝐷) → 𝑥 ∈ 𝐷)
2		fvres 6787	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → ((𝐹 ↾ 𝐷)‘𝑥) = (𝐹‘𝑥))
3		fvres 6787	. . . . . 6 ⊢ (𝑥 ∈ 𝐷 → ((𝐺 ↾ 𝐷)‘𝑥) = (𝐺‘𝑥))
4	2, 3	oveq12d 7286	. . . . 5 ⊢ (𝑥 ∈ 𝐷 → (((𝐹 ↾ 𝐷)‘𝑥)𝑅((𝐺 ↾ 𝐷)‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
5	1, 4	syl 17	. . . 4 ⊢ (𝑥 ∈ ((dom 𝐹 ∩ dom 𝐺) ∩ 𝐷) → (((𝐹 ↾ 𝐷)‘𝑥)𝑅((𝐺 ↾ 𝐷)‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
6	5	mpteq2ia 5181	. . 3 ⊢ (𝑥 ∈ ((dom 𝐹 ∩ dom 𝐺) ∩ 𝐷) ↦ (((𝐹 ↾ 𝐷)‘𝑥)𝑅((𝐺 ↾ 𝐷)‘𝑥))) = (𝑥 ∈ ((dom 𝐹 ∩ dom 𝐺) ∩ 𝐷) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
7		inindi 4165	. . . . 5 ⊢ (𝐷 ∩ (dom 𝐹 ∩ dom 𝐺)) = ((𝐷 ∩ dom 𝐹) ∩ (𝐷 ∩ dom 𝐺))
8		incom 4139	. . . . 5 ⊢ ((dom 𝐹 ∩ dom 𝐺) ∩ 𝐷) = (𝐷 ∩ (dom 𝐹 ∩ dom 𝐺))
9		dmres 5910	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐷) = (𝐷 ∩ dom 𝐹)
10		dmres 5910	. . . . . 6 ⊢ dom (𝐺 ↾ 𝐷) = (𝐷 ∩ dom 𝐺)
11	9, 10	ineq12i 4149	. . . . 5 ⊢ (dom (𝐹 ↾ 𝐷) ∩ dom (𝐺 ↾ 𝐷)) = ((𝐷 ∩ dom 𝐹) ∩ (𝐷 ∩ dom 𝐺))
12	7, 8, 11	3eqtr4ri 2778	. . . 4 ⊢ (dom (𝐹 ↾ 𝐷) ∩ dom (𝐺 ↾ 𝐷)) = ((dom 𝐹 ∩ dom 𝐺) ∩ 𝐷)
13	12	mpteq1i 5174	. . 3 ⊢ (𝑥 ∈ (dom (𝐹 ↾ 𝐷) ∩ dom (𝐺 ↾ 𝐷)) ↦ (((𝐹 ↾ 𝐷)‘𝑥)𝑅((𝐺 ↾ 𝐷)‘𝑥))) = (𝑥 ∈ ((dom 𝐹 ∩ dom 𝐺) ∩ 𝐷) ↦ (((𝐹 ↾ 𝐷)‘𝑥)𝑅((𝐺 ↾ 𝐷)‘𝑥)))
14		resmpt3 5943	. . 3 ⊢ ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ↾ 𝐷) = (𝑥 ∈ ((dom 𝐹 ∩ dom 𝐺) ∩ 𝐷) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
15	6, 13, 14	3eqtr4ri 2778	. 2 ⊢ ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ↾ 𝐷) = (𝑥 ∈ (dom (𝐹 ↾ 𝐷) ∩ dom (𝐺 ↾ 𝐷)) ↦ (((𝐹 ↾ 𝐷)‘𝑥)𝑅((𝐺 ↾ 𝐷)‘𝑥)))
16		offval3 7811	. . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
17	16	reseq1d 5887	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘f 𝑅𝐺) ↾ 𝐷) = ((𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ↾ 𝐷))
18		resexg 5934	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝐹 ↾ 𝐷) ∈ V)
19		resexg 5934	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ↾ 𝐷) ∈ V)
20		offval3 7811	. . 3 ⊢ (((𝐹 ↾ 𝐷) ∈ V ∧ (𝐺 ↾ 𝐷) ∈ V) → ((𝐹 ↾ 𝐷) ∘f 𝑅(𝐺 ↾ 𝐷)) = (𝑥 ∈ (dom (𝐹 ↾ 𝐷) ∩ dom (𝐺 ↾ 𝐷)) ↦ (((𝐹 ↾ 𝐷)‘𝑥)𝑅((𝐺 ↾ 𝐷)‘𝑥))))
21	18, 19, 20	syl2an 595	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ↾ 𝐷) ∘f 𝑅(𝐺 ↾ 𝐷)) = (𝑥 ∈ (dom (𝐹 ↾ 𝐷) ∩ dom (𝐺 ↾ 𝐷)) ↦ (((𝐹 ↾ 𝐷)‘𝑥)𝑅((𝐺 ↾ 𝐷)‘𝑥))))
22	15, 17, 21	3eqtr4a 2805	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → ((𝐹 ∘f 𝑅𝐺) ↾ 𝐷) = ((𝐹 ↾ 𝐷) ∘f 𝑅(𝐺 ↾ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109  Vcvv 3430   ∩ cin 3890   ↦ cmpt 5161  dom cdm 5588   ↾ cres 5590  ‘cfv 6430  (class class class)co 7268   ∘_f cof 7522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-of 7524

This theorem is referenced by:  pwssplit2  20303  pwssplit3  20304  islindf4  21026  tsmsadd  23279  jensen  26119  fdivmpt  45838