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Theorem xpima1 4817
Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)
Assertion
Ref Expression
xpima1  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )
" C )  =  (/) )

Proof of Theorem xpima1
StepHypRef Expression
1 df-ima 4404 . . 3  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  |`  C )
2 df-res 4403 . . . 4  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
32rneqi 4610 . . 3  |-  ran  (
( A  X.  B
)  |`  C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
4 inxp 4518 . . . 4  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
54rneqi 4610 . . 3  |-  ran  (
( A  X.  B
)  i^i  ( C  X.  _V ) )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
61, 3, 53eqtri 2107 . 2  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
7 xpeq1 4405 . . . 4  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  (
(/)  X.  ( B  i^i  _V ) ) )
8 0xp 4466 . . . 4  |-  ( (/)  X.  ( B  i^i  _V ) )  =  (/)
97, 8syl6eq 2131 . . 3  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  (/) )
10 rneq 4609 . . . 4  |-  ( ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  ran  (/) )
11 rn0 4636 . . . 4  |-  ran  (/)  =  (/)
1210, 11syl6eq 2131 . . 3  |-  ( ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  (/) )
139, 12syl 14 . 2  |-  ( ( A  i^i  C )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  (/) )
146, 13syl5eq 2127 1  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )
" C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   _Vcvv 2610    i^i cin 2981   (/)c0 3267    X. cxp 4389   ran crn 4392    |` cres 4393   "cima 4394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404
This theorem is referenced by: (None)
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