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Theorem inxp 4863
 Description: The intersection of two cross products. Exercise 9 of [TakeutiZaring] p. 25. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by NM, 3-Aug-1994.) (Revised by set.mm contributors, 27-Aug-2011.)
Assertion
Ref Expression
inxp ((A × B) ∩ (C × D)) = ((AC) × (BD))

Proof of Theorem inxp
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inopab 4862 . . 3 ({x, y (x A y B)} ∩ {x, y (x C y D)}) = {x, y ((x A y B) (x C y D))}
2 an4 797 . . . . 5 (((x A y B) (x C y D)) ↔ ((x A x C) (y B y D)))
3 elin 3219 . . . . . 6 (x (AC) ↔ (x A x C))
4 elin 3219 . . . . . 6 (y (BD) ↔ (y B y D))
53, 4anbi12i 678 . . . . 5 ((x (AC) y (BD)) ↔ ((x A x C) (y B y D)))
62, 5bitr4i 243 . . . 4 (((x A y B) (x C y D)) ↔ (x (AC) y (BD)))
76opabbii 4626 . . 3 {x, y ((x A y B) (x C y D))} = {x, y (x (AC) y (BD))}
81, 7eqtri 2373 . 2 ({x, y (x A y B)} ∩ {x, y (x C y D)}) = {x, y (x (AC) y (BD))}
9 df-xp 4784 . . 3 (A × B) = {x, y (x A y B)}
10 df-xp 4784 . . 3 (C × D) = {x, y (x C y D)}
119, 10ineq12i 3455 . 2 ((A × B) ∩ (C × D)) = ({x, y (x A y B)} ∩ {x, y (x C y D)})
12 df-xp 4784 . 2 ((AC) × (BD)) = {x, y (x (AC) y (BD))}
138, 11, 123eqtr4i 2383 1 ((A × B) ∩ (C × D)) = ((AC) × (BD))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3208  {copab 4622   × cxp 4770 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-xp 4784 This theorem is referenced by:  xpindi  4864  xpindir  4865  dmxpin  4925  xpdisj1  5047  xpdisj2  5048
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