Theorem xpima1 4993

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 4560	. . 3 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
2		df-res 4559	. . . 4 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
3	2	rneqi 4775	. . 3 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
4		inxp 4681	. . . 4 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
5	4	rneqi 4775	. . 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 2165	. 2 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
7		xpeq1 4561	. . . 4 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = ( (/) X. ( B i^i _V ) ) )
8		0xp 4627	. . . 4 |- ( (/) X. ( B i^i _V ) ) = (/)
9	7, 8	eqtrdi 2189	. . 3 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. ( B i^i _V ) ) = (/) )
10		rneq 4774	. . . 4 |- ( ( ( A i^i C ) X. ( B i^i _V ) ) = (/) -> ran ( ( A i^i C ) X. ( B i^i _V ) ) = ran (/) )
11		rn0 4803	. . . 4 |- ran (/) = (/)
12	10, 11	eqtrdi 2189	. . 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 2185	1 |- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )

Syntax hints:   -> wi 4   = wceq 1332   _Vcvv 2689   i^i cin 3075   (/)c0 3368   X. cxp 4545   ran crn 4548   |` cres 4549   "cima 4550

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by: (None)