Theorem xpcogend 14168

Description: The most interesting case of the composition of two cross products. (Contributed by RP, 24-Dec-2019.)

Hypothesis

Ref	Expression
xpcogend.1	⊢ (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅)

Assertion

Ref	Expression
xpcogend	⊢ (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))

Proof of Theorem xpcogend

Dummy variables 𝑥 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpcogend.1	. . . . . 6 ⊢ (𝜑 → (𝐵 ∩ 𝐶) ≠ ∅)
2		ndisj 4247	. . . . . 6 ⊢ ((𝐵 ∩ 𝐶) ≠ ∅ ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
3	1, 2	sylib 219	. . . . 5 ⊢ (𝜑 → ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
4	3	biantrud 532	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
5		brxp 5489	. . . . . . 7 ⊢ (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
6		brxp 5489	. . . . . . . 8 ⊢ (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑦 ∈ 𝐶 ∧ 𝑧 ∈ 𝐷))
7	6	biancomi 463	. . . . . . 7 ⊢ (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶))
8	5, 7	anbi12i 626	. . . . . 6 ⊢ ((𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)))
9	8	exbii 1829	. . . . 5 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)))
10		an4 652	. . . . . 6 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
11	10	exbii 1829	. . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑧 ∈ 𝐷 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
12		19.42v 1931	. . . . 5 ⊢ (∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
13	9, 11, 12	3bitri 298	. . . 4 ⊢ (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷) ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
14	4, 13	syl6rbbr 291	. . 3 ⊢ (𝜑 → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧) ↔ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷)))
15	14	opabbidv 5028	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷)})
16		df-co 5452	. 2 ⊢ ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦 ∧ 𝑦(𝐶 × 𝐷)𝑧)}
17		df-xp 5449	. 2 ⊢ (𝐴 × 𝐷) = {⟨𝑥, 𝑧⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐷)}
18	15, 16, 17	3eqtr4g 2856	1 ⊢ (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522  ∃wex 1761   ∈ wcel 2081   ≠ wne 2984   ∩ cin 3858  ∅c0 4211   class class class wbr 4962  {copab 5024   × cxp 5441   ∘ ccom 5447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-br 4963  df-opab 5025  df-xp 5449  df-co 5452

This theorem is referenced by:  xpcoidgend  14169