Theorem xpriindi 5820

Description: Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
xpriindi	⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem xpriindi

Step	Hyp	Ref	Expression
1		iineq1 4972	. . . . . . 7 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ ∅ 𝐵)
2		0iin 5025	. . . . . . 7 ⊢ ∩ 𝑥 ∈ ∅ 𝐵 = V
3	1, 2	eqtrdi 2814	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = V)
4	3	ineq2d 4120	. . . . 5 ⊢ (𝐴 = ∅ → (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = (𝐷 ∩ V))
5		inv1 4351	. . . . 5 ⊢ (𝐷 ∩ V) = 𝐷
6	4, 5	eqtrdi 2814	. . . 4 ⊢ (𝐴 = ∅ → (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = 𝐷)
7	6	xpeq2d 5686	. . 3 ⊢ (𝐴 = ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = (𝐶 × 𝐷))
8		iineq1 4972	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) = ∩ 𝑥 ∈ ∅ (𝐶 × 𝐵))
9		0iin 5025	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ (𝐶 × 𝐵) = V
10	8, 9	eqtrdi 2814	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) = V)
11	10	ineq2d 4120	. . . 4 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) = ((𝐶 × 𝐷) ∩ V))
12		inv1 4351	. . . 4 ⊢ ((𝐶 × 𝐷) ∩ V) = (𝐶 × 𝐷)
13	11, 12	eqtrdi 2814	. . 3 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) = (𝐶 × 𝐷))
14	7, 13	eqtr4d 2801	. 2 ⊢ (𝐴 = ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
15		xpindi 5817	. . 3 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵))
16		xpiindi 5819	. . . 4 ⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
17	16	ineq2d 4120	. . 3 ⊢ (𝐴 ≠ ∅ → ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
18	15, 17	eqtrid 2810	. 2 ⊢ (𝐴 ≠ ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
19	14, 18	pm2.61ine 3042	1 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

This proof depends on syntax axioms:   = wceq 1570   ≠ wne 2959  Vcvv 3454   ∩ cin 4092  ∅c0 4281  ∩ ciin 4955   × cxp 5654

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-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

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-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  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 4839  df-iin 4957  df-opab 5172  df-xp 5662  df-rel 5663

This theorem is used by: (None)