Theorem xpriindi 5806

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 4966	. . . . . . 7 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ ∅ 𝐵)
2		0iin 5020	. . . . . . 7 ⊢ ∩ 𝑥 ∈ ∅ 𝐵 = V
3	1, 2	eqtrdi 2812	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = V)
4	3	ineq2d 4172	. . . . 5 ⊢ (𝐴 = ∅ → (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = (𝐷 ∩ V))
5		inv1 4351	. . . . 5 ⊢ (𝐷 ∩ V) = 𝐷
6	4, 5	eqtrdi 2812	. . . 4 ⊢ (𝐴 = ∅ → (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = 𝐷)
7	6	xpeq2d 5675	. . 3 ⊢ (𝐴 = ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = (𝐶 × 𝐷))
8		iineq1 4966	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) = ∩ 𝑥 ∈ ∅ (𝐶 × 𝐵))
9		0iin 5020	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ (𝐶 × 𝐵) = V
10	8, 9	eqtrdi 2812	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) = V)
11	10	ineq2d 4172	. . . 4 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) = ((𝐶 × 𝐷) ∩ V))
12		inv1 4351	. . . 4 ⊢ ((𝐶 × 𝐷) ∩ V) = (𝐶 × 𝐷)
13	11, 12	eqtrdi 2812	. . 3 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) = (𝐶 × 𝐷))
14	7, 13	eqtr4d 2799	. 2 ⊢ (𝐴 = ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
15		xpindi 5803	. . 3 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵))
16		xpiindi 5805	. . . 4 ⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
17	16	ineq2d 4172	. . 3 ⊢ (𝐴 ≠ ∅ → ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
18	15, 17	eqtrid 2808	. 2 ⊢ (𝐴 ≠ ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
19	14, 18	pm2.61ine 3039	1 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Syntax hints:   = wceq 1559   ≠ wne 2956  Vcvv 3453   ∩ cin 3903  ∅c0 4285  ∩ ciin 4949   × cxp 5643

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-iin 4951  df-opab 5162  df-xp 5651  df-rel 5652

This theorem is referenced by: (None)