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Theorem xpcom 4807
Description: Composition of two cross products. (Contributed by Jim Kingdon, 20-Dec-2018.)
Assertion
Ref Expression
xpcom  X.  C  o.  X.  X.  C
Distinct variable groups:   ,   ,   , C

Proof of Theorem xpcom
Dummy variables  a  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ibar 285 . . . 4  a  c  C  a  c  C
2 ancom 253 . . . . . . . 8  a  a
32anbi1i 431 . . . . . . 7  a  c  C  a  c  C
4 brxp 4318 . . . . . . . 8  a  X.  a
5 brxp 4318 . . . . . . . 8  X.  C c  c  C
64, 5anbi12i 433 . . . . . . 7  a  X.  X.  C c  a  c  C
7 anandi 524 . . . . . . 7  a  c  C  a  c  C
83, 6, 73bitr4i 201 . . . . . 6  a  X.  X.  C c 
a  c  C
98exbii 1493 . . . . 5  a  X.  X.  C c 
a  c  C
10 19.41v 1779 . . . . 5  a  c  C  a  c  C
119, 10bitr2i 174 . . . 4  a  c  C  a  X.  X.  C c
121, 11syl6rbb 186 . . 3  a  X.  X.  C c  a  c  C
1312opabbidv 3814 . 2  { <. a ,  c
>.  |  a  X.  X.  C c }  { <. a ,  c >.  |  a  c  C }
14 df-co 4297 . 2  X.  C  o.  X.  { <. a ,  c >.  |  a  X.  X.  C c }
15 df-xp 4294 . 2  X.  C  { <. a ,  c
>.  |  a  c  C }
1613, 14, 153eqtr4g 2094 1  X.  C  o.  X.  X.  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242  wex 1378   wcel 1390   class class class wbr 3755   {copab 3808    X. cxp 4286    o. ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-co 4297
This theorem is referenced by: (None)
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