Theorem xpima1 5113

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 4673	. . 3 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
2		df-res 4672	. . . 4 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
3	2	rneqi 4891	. . 3 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
4		inxp 4797	. . . 4 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
5	4	rneqi 4891	. . 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 2218	. 2 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
7		xpeq1 4674	. . . 4 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = ( (/) X. ( B i^i _V ) ) )
8		0xp 4740	. . . 4 |- ( (/) X. ( B i^i _V ) ) = (/)
9	7, 8	eqtrdi 2242	. . 3 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = (/) )
10		rneq 4890	. . . 4 |- ( ( ( A i^i C ) X. ( B i^i _V ) ) = (/) -> ran ( ( A i^i C ) X. ( B i^i _V ) ) = ran (/) )
11		rn0 4919	. . . 4 |- ran (/) = (/)
12	10, 11	eqtrdi 2242	. . 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 2238	1 |- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1364   _Vcvv 2760   i^i cin 3153   (/)c0 3447   X. cxp 4658   ran crn 4661   |` cres 4662   "cima 4663

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673

This theorem is referenced by: (None)