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Theorem xpcogend 14329
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 ndisj 4284 . . . . . 6 ((𝐵𝐶) ≠ ∅ ↔ ∃𝑦(𝑦𝐵𝑦𝐶))
31, 2sylib 221 . . . . 5 (𝜑 → ∃𝑦(𝑦𝐵𝑦𝐶))
43biantrud 535 . . . 4 (𝜑 → ((𝑥𝐴𝑧𝐷) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶))))
5 brxp 5569 . . . . . . 7 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
6 brxp 5569 . . . . . . . 8 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑦𝐶𝑧𝐷))
76biancomi 466 . . . . . . 7 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑧𝐷𝑦𝐶))
85, 7anbi12i 629 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
98exbii 1849 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
10 an4 655 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
1110exbii 1849 . . . . 5 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ∃𝑦((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
12 19.42v 1954 . . . . 5 (∃𝑦((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶)))
139, 11, 123bitri 300 . . . 4 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶)))
144, 13syl6rbbr 293 . . 3 (𝜑 → (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ (𝑥𝐴𝑧𝐷)))
1514opabbidv 5099 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)})
16 df-co 5532 . 2 ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)}
17 df-xp 5529 . 2 (𝐴 × 𝐷) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)}
1815, 16, 173eqtr4g 2861 1 (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wex 1781  wcel 2112  wne 2990  cin 3883  c0 4246   class class class wbr 5033  {copab 5095   × cxp 5521  ccom 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-xp 5529  df-co 5532
This theorem is referenced by:  xpcoidgend  14330
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