Theorem xpima1 5117

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 4677	. . 3 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
2		df-res 4676	. . . 4 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
3	2	rneqi 4895	. . 3 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
4		inxp 4801	. . . 4 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
5	4	rneqi 4895	. . 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 2221	. 2 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
7		xpeq1 4678	. . . 4 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = ( (/) X. ( B i^i _V ) ) )
8		0xp 4744	. . . 4 |- ( (/) X. ( B i^i _V ) ) = (/)
9	7, 8	eqtrdi 2245	. . 3 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = (/) )
10		rneq 4894	. . . 4 |- ( ( ( A i^i C ) X. ( B i^i _V ) ) = (/) -> ran ( ( A i^i C ) X. ( B i^i _V ) ) = ran (/) )
11		rn0 4923	. . . 4 |- ran (/) = (/)
12	10, 11	eqtrdi 2245	. . 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	eqtrid 2241	1 |- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   _Vcvv 2763   i^i cin 3156   (/)c0 3451   X. cxp 4662   ran crn 4665   |` cres 4666   "cima 4667

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-xp 4670  df-rel 4671  df-cnv 4672  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677

This theorem is referenced by: (None)