Theorem xpriindi 5837

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 5015	. . . . . . 7 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ ∅ 𝐵)
2		0iin 5068	. . . . . . 7 ⊢ ∩ 𝑥 ∈ ∅ 𝐵 = V
3	1, 2	eqtrdi 2787	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = V)
4	3	ineq2d 4213	. . . . 5 ⊢ (𝐴 = ∅ → (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = (𝐷 ∩ V))
5		inv1 4395	. . . . 5 ⊢ (𝐷 ∩ V) = 𝐷
6	4, 5	eqtrdi 2787	. . . 4 ⊢ (𝐴 = ∅ → (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = 𝐷)
7	6	xpeq2d 5707	. . 3 ⊢ (𝐴 = ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = (𝐶 × 𝐷))
8		iineq1 5015	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) = ∩ 𝑥 ∈ ∅ (𝐶 × 𝐵))
9		0iin 5068	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ (𝐶 × 𝐵) = V
10	8, 9	eqtrdi 2787	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) = V)
11	10	ineq2d 4213	. . . 4 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) = ((𝐶 × 𝐷) ∩ V))
12		inv1 4395	. . . 4 ⊢ ((𝐶 × 𝐷) ∩ V) = (𝐶 × 𝐷)
13	11, 12	eqtrdi 2787	. . 3 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) = (𝐶 × 𝐷))
14	7, 13	eqtr4d 2774	. 2 ⊢ (𝐴 = ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
15		xpindi 5834	. . 3 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵))
16		xpiindi 5836	. . . 4 ⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
17	16	ineq2d 4213	. . 3 ⊢ (𝐴 ≠ ∅ → ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
18	15, 17	eqtrid 2783	. 2 ⊢ (𝐴 ≠ ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
19	14, 18	pm2.61ine 3024	1 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Syntax hints:   = wceq 1540   ≠ wne 2939  Vcvv 3473   ∩ cin 3948  ∅c0 4323  ∩ ciin 4999   × cxp 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iin 5001  df-opab 5212  df-xp 5683  df-rel 5684

This theorem is referenced by: (None)