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Theorem xpcogend 13113
 Description: The most interesting case of the composition of two cross products. (Contributed by RP, 24-Dec-2019.)
Hypothesis
Ref Expression
xpcogend.1
Assertion
Ref Expression
xpcogend

Proof of Theorem xpcogend
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpcogend.1 . . . . . 6
2 n0 3732 . . . . . . 7
3 elin 3608 . . . . . . . 8
43exbii 1726 . . . . . . 7
52, 4bitri 257 . . . . . 6
61, 5sylib 201 . . . . 5
76biantrud 515 . . . 4
8 brxp 4870 . . . . . . 7
9 brxp 4870 . . . . . . . 8
10 ancom 457 . . . . . . . 8
119, 10bitri 257 . . . . . . 7
128, 11anbi12i 711 . . . . . 6
1312exbii 1726 . . . . 5
14 an4 840 . . . . . 6
1514exbii 1726 . . . . 5
16 19.42v 1842 . . . . 5
1713, 15, 163bitri 279 . . . 4
187, 17syl6rbbr 272 . . 3
1918opabbidv 4459 . 2
20 df-co 4848 . 2
21 df-xp 4845 . 2
2219, 20, 213eqtr4g 2530 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452  wex 1671   wcel 1904   wne 2641   cin 3389  c0 3722   class class class wbr 4395  copab 4453   cxp 4837   ccom 4843 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-co 4848 This theorem is referenced by:  xpcoidgend  13114
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