Theorem xpima1 5057

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 4624	. . 3 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
2		df-res 4623	. . . 4 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
3	2	rneqi 4839	. . 3 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
4		inxp 4745	. . . 4 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
5	4	rneqi 4839	. . 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 2195	. 2 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
7		xpeq1 4625	. . . 4 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = ( (/) X. ( B i^i _V ) ) )
8		0xp 4691	. . . 4 |- ( (/) X. ( B i^i _V ) ) = (/)
9	7, 8	eqtrdi 2219	. . 3 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = (/) )
10		rneq 4838	. . . 4 |- ( ( ( A i^i C ) X. ( B i^i _V ) ) = (/) -> ran ( ( A i^i C ) X. ( B i^i _V ) ) = ran (/) )
11		rn0 4867	. . . 4 |- ran (/) = (/)
12	10, 11	eqtrdi 2219	. . 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 2215	1 |- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1348   _Vcvv 2730   i^i cin 3120   (/)c0 3414   X. cxp 4609   ran crn 4612   |` cres 4613   "cima 4614

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624

This theorem is referenced by: (None)