MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpcogend Structured version   Visualization version   GIF version

Theorem xpcogend 13647
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 3907 . . . . . . 7 ((𝐵𝐶) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐵𝐶))
3 elin 3774 . . . . . . . 8 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
43exbii 1771 . . . . . . 7 (∃𝑦 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑦(𝑦𝐵𝑦𝐶))
52, 4bitri 264 . . . . . 6 ((𝐵𝐶) ≠ ∅ ↔ ∃𝑦(𝑦𝐵𝑦𝐶))
61, 5sylib 208 . . . . 5 (𝜑 → ∃𝑦(𝑦𝐵𝑦𝐶))
76biantrud 528 . . . 4 (𝜑 → ((𝑥𝐴𝑧𝐷) ↔ ((𝑥𝐴𝑧𝐷) ∧ ∃𝑦(𝑦𝐵𝑦𝐶))))
8 brxp 5107 . . . . . . 7 (𝑥(𝐴 × 𝐵)𝑦 ↔ (𝑥𝐴𝑦𝐵))
9 brxp 5107 . . . . . . . 8 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑦𝐶𝑧𝐷))
10 ancom 466 . . . . . . . 8 ((𝑦𝐶𝑧𝐷) ↔ (𝑧𝐷𝑦𝐶))
119, 10bitri 264 . . . . . . 7 (𝑦(𝐶 × 𝐷)𝑧 ↔ (𝑧𝐷𝑦𝐶))
128, 11anbi12i 732 . . . . . 6 ((𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
1312exbii 1771 . . . . 5 (∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧) ↔ ∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)))
14 an4 864 . . . . . 6 (((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
1514exbii 1771 . . . . 5 (∃𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧𝐷𝑦𝐶)) ↔ ∃𝑦((𝑥𝐴𝑧𝐷) ∧ (𝑦𝐵𝑦𝐶)))
16 19.42v 1915 . . . . 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 4678 . 2 (𝜑 → {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)} = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)})
20 df-co 5083 . 2 ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥(𝐴 × 𝐵)𝑦𝑦(𝐶 × 𝐷)𝑧)}
21 df-xp 5080 . 2 (𝐴 × 𝐷) = {⟨𝑥, 𝑧⟩ ∣ (𝑥𝐴𝑧𝐷)}
2219, 20, 213eqtr4g 2680 1 (𝜑 → ((𝐶 × 𝐷) ∘ (𝐴 × 𝐵)) = (𝐴 × 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  cin 3554  c0 3891   class class class wbr 4613  {copab 4672   × cxp 5072  ccom 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-co 5083
This theorem is referenced by:  xpcoidgend  13648
  Copyright terms: Public domain W3C validator