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

Theorem xpriindi 5700
Description: Distributive law for Cartesian product over relativized indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
xpriindi (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem xpriindi
StepHypRef Expression
1 iineq1 4927 . . . . . . 7 (𝐴 = ∅ → 𝑥𝐴 𝐵 = 𝑥 ∈ ∅ 𝐵)
2 0iin 4978 . . . . . . 7 𝑥 ∈ ∅ 𝐵 = V
31, 2syl6eq 2869 . . . . . 6 (𝐴 = ∅ → 𝑥𝐴 𝐵 = V)
43ineq2d 4186 . . . . 5 (𝐴 = ∅ → (𝐷 𝑥𝐴 𝐵) = (𝐷 ∩ V))
5 inv1 4345 . . . . 5 (𝐷 ∩ V) = 𝐷
64, 5syl6eq 2869 . . . 4 (𝐴 = ∅ → (𝐷 𝑥𝐴 𝐵) = 𝐷)
76xpeq2d 5578 . . 3 (𝐴 = ∅ → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = (𝐶 × 𝐷))
8 iineq1 4927 . . . . . 6 (𝐴 = ∅ → 𝑥𝐴 (𝐶 × 𝐵) = 𝑥 ∈ ∅ (𝐶 × 𝐵))
9 0iin 4978 . . . . . 6 𝑥 ∈ ∅ (𝐶 × 𝐵) = V
108, 9syl6eq 2869 . . . . 5 (𝐴 = ∅ → 𝑥𝐴 (𝐶 × 𝐵) = V)
1110ineq2d 4186 . . . 4 (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)) = ((𝐶 × 𝐷) ∩ V))
12 inv1 4345 . . . 4 ((𝐶 × 𝐷) ∩ V) = (𝐶 × 𝐷)
1311, 12syl6eq 2869 . . 3 (𝐴 = ∅ → ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)) = (𝐶 × 𝐷))
147, 13eqtr4d 2856 . 2 (𝐴 = ∅ → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
15 xpindi 5697 . . 3 (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ (𝐶 × 𝑥𝐴 𝐵))
16 xpiindi 5699 . . . 4 (𝐴 ≠ ∅ → (𝐶 × 𝑥𝐴 𝐵) = 𝑥𝐴 (𝐶 × 𝐵))
1716ineq2d 4186 . . 3 (𝐴 ≠ ∅ → ((𝐶 × 𝐷) ∩ (𝐶 × 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
1815, 17syl5eq 2865 . 2 (𝐴 ≠ ∅ → (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵)))
1914, 18pm2.61ine 3097 1 (𝐶 × (𝐷 𝑥𝐴 𝐵)) = ((𝐶 × 𝐷) ∩ 𝑥𝐴 (𝐶 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wne 3013  Vcvv 3492  cin 3932  c0 4288   ciin 4911   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-iin 4913  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator