Theorem xpima1 4980

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

Step	Hyp	Ref	Expression
1		df-ima 4547	. . 3 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
2		df-res 4546	. . . 4 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
3	2	rneqi 4762	. . 3 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
4		inxp 4668	. . . 4 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
5	4	rneqi 4762	. . 3 |- ran ( ( A X. B ) i^i ( C X. _V ) ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
6	1, 3, 5	3eqtri 2162	. 2 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
7		xpeq1 4548	. . . 4 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = ( (/) X. ( B i^i _V ) ) )
8		0xp 4614	. . . 4 |- ( (/) X. ( B i^i _V ) ) = (/)
9	7, 8	syl6eq 2186	. . 3 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = (/) )
10		rneq 4761	. . . 4 |- ( ( ( A i^i C ) X. ( B i^i _V ) ) = (/) -> ran ( ( A i^i C ) X. ( B i^i _V ) ) = ran (/) )
11		rn0 4790	. . . 4 |- ran (/) = (/)
12	10, 11	syl6eq 2186	. . 3 |- ( ( ( A i^i C ) X. ( B i^i _V ) ) = (/) -> ran ( ( A i^i C ) X. ( B i^i _V ) ) = (/) )
13	9, 12	syl 14	. 2 |- ( ( A i^i C ) = (/) -> ran ( ( A i^i C ) X. ( B i^i _V ) ) = (/) )
14	6, 13	syl5eq 2182	1 |- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1331   _Vcvv 2681   i^i cin 3065   (/)c0 3358   X. cxp 4532   ran crn 4535   |` cres 4536   "cima 4537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547

This theorem is referenced by: (None)