Theorem xpriindi 5861

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 5032	. . . . . . 7 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ 𝑥 ∈ ∅ 𝐵)
2		0iin 5087	. . . . . . 7 ⊢ ∩ 𝑥 ∈ ∅ 𝐵 = V
3	1, 2	eqtrdi 2796	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 𝐵 = V)
4	3	ineq2d 4241	. . . . 5 ⊢ (𝐴 = ∅ → (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = (𝐷 ∩ V))
5		inv1 4421	. . . . 5 ⊢ (𝐷 ∩ V) = 𝐷
6	4, 5	eqtrdi 2796	. . . 4 ⊢ (𝐴 = ∅ → (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵) = 𝐷)
7	6	xpeq2d 5730	. . 3 ⊢ (𝐴 = ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = (𝐶 × 𝐷))
8		iineq1 5032	. . . . . 6 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) = ∩ 𝑥 ∈ ∅ (𝐶 × 𝐵))
9		0iin 5087	. . . . . 6 ⊢ ∩ 𝑥 ∈ ∅ (𝐶 × 𝐵) = V
10	8, 9	eqtrdi 2796	. . . . 5 ⊢ (𝐴 = ∅ → ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) = V)
11	10	ineq2d 4241	. . . 4 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) = ((𝐶 × 𝐷) ∩ V))
12		inv1 4421	. . . 4 ⊢ ((𝐶 × 𝐷) ∩ V) = (𝐶 × 𝐷)
13	11, 12	eqtrdi 2796	. . 3 ⊢ (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) = (𝐶 × 𝐷))
14	7, 13	eqtr4d 2783	. 2 ⊢ (𝐴 = ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
15		xpindi 5858	. . 3 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵))
16		xpiindi 5860	. . . 4 ⊢ (𝐴 ≠ ∅ → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
17	16	ineq2d 4241	. . 3 ⊢ (𝐴 ≠ ∅ → ((𝐶 × 𝐷) ∩ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
18	15, 17	eqtrid 2792	. 2 ⊢ (𝐴 ≠ ∅ → (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
19	14, 18	pm2.61ine 3031	1 ⊢ (𝐶 × (𝐷 ∩ ∩ 𝑥 ∈ 𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Syntax hints:   = wceq 1537   ≠ wne 2946  Vcvv 3488   ∩ cin 3975  ∅c0 4352  ∩ ciin 5016   × cxp 5698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-iin 5018  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by: (None)