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Theorem xpcogend 13759
Description: The most interesting case of the composition of two cross products. (Contributed by RP, 24-Dec-2019.)
Hypothesis
Ref Expression
xpcogend.1 (𝜑 → (𝐵𝐶) ≠ ∅)
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.
StepHypRef Expression
1 xpcogend.1 . . . . . 6 (𝜑 → (𝐵𝐶) ≠ ∅)
2 n0 3964 . . . . . . 7 ((𝐵𝐶) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐵𝐶))
3 elin 3829 . . . . . . . 8 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
43exbii 1814 . . . . . . 7 (∃𝑦 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑦(𝑦𝐵𝑦𝐶))
52, 4bitri 264 . . . . . 6 ((𝐵𝐶) ≠ ∅ ↔ ∃𝑦(𝑦𝐵𝑦𝐶))
61, 5sylib 208 . . . . 5 (𝜑 → ∃𝑦(𝑦𝐵𝑦𝐶))
76biantrud 527 . . . 4 (𝜑 → ((𝑥𝐴𝑧𝐷) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶))))
8 brxp 5181 . . . . . . 7 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
9 brxp 5181 . . . . . . . 8 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑦𝐶𝑧𝐷))
10 ancom 465 . . . . . . . 8 ((𝑦𝐶𝑧𝐷) ↔ (𝑧𝐷𝑦𝐶))
119, 10bitri 264 . . . . . . 7 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑧𝐷𝑦𝐶))
128, 11anbi12i 733 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
1312exbii 1814 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
14 an4 882 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
1514exbii 1814 . . . . 5 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ∃𝑦((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
16 19.42v 1921 . . . . 5 (∃𝑦((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶)))
1713, 15, 163bitri 286 . . . 4 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶)))
187, 17syl6rbbr 279 . . 3 (𝜑 → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ (𝑥𝐴𝑧𝐷)))
1918opabbidv 4749 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)})
20 df-co 5152 . 2 ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)}
21 df-xp 5149 . 2 (𝐴 × 𝐷) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)}
2219, 20, 213eqtr4g 2710 1 (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  cin 3606  c0 3948   class class class wbr 4685  {copab 4745   × cxp 5141  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-co 5152
This theorem is referenced by:  xpcoidgend  13760
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