ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpima1 Unicode version

Theorem xpima1 5070
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 4635 . . 3  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  X.  B )  |`  C )
2 df-res 4634 . . . 4  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
32rneqi 4850 . . 3  |-  ran  (
( A  X.  B
)  |`  C )  =  ran  ( ( A  X.  B )  i^i  ( C  X.  _V ) )
4 inxp 4756 . . . 4  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
54rneqi 4850 . . 3  |-  ran  (
( A  X.  B
)  i^i  ( C  X.  _V ) )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
61, 3, 53eqtri 2202 . 2  |-  ( ( A  X.  B )
" C )  =  ran  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )
7 xpeq1 4636 . . . 4  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  (
(/)  X.  ( B  i^i  _V ) ) )
8 0xp 4702 . . . 4  |-  ( (/)  X.  ( B  i^i  _V ) )  =  (/)
97, 8eqtrdi 2226 . . 3  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  (/) )
10 rneq 4849 . . . 4  |-  ( ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  (/)  ->  ran  (
( A  i^i  C
)  X.  ( B  i^i  _V ) )  =  ran  (/) )
11 rn0 4878 . . . 4  |-  ran  (/)  =  (/)
1210, 11eqtrdi 2226 . . 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, 13eqtrid 2222 1  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )
" C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353   _Vcvv 2737    i^i cin 3128   (/)c0 3422    X. cxp 4620   ran crn 4623    |` cres 4624   "cima 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator