Theorem xpima2m 4788

Description: The image by a cross product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
xpima2m	|- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = B )

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem xpima2m

Step	Hyp	Ref	Expression
1		df-ima 4376	. . . 4 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
2		df-res 4375	. . . . 5 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
3	2	rneqi 4580	. . . 4 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
4		inxp 4488	. . . . 5 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
5	4	rneqi 4580	. . . 4 |- ran ( ( A X. B ) i^i ( C X. _V ) ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
6	1, 3, 5	3eqtri 2105	. . 3 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
7		rnxpm 4772	. . 3 |- ( E. x x e. ( A i^i C ) -> ran ( ( A i^i C ) X. ( B i^i _V ) ) = ( B i^i _V ) )
8	6, 7	syl5eq 2125	. 2 |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = ( B i^i _V ) )
9		inv1 3280	. 2 |- ( B i^i _V ) = B
10	8, 9	syl6eq 2129	1 |- ( E. x x e. ( A i^i C ) -> ( ( A X. B ) " C ) = B )

Syntax hints:   -> wi 4   = wceq 1284   E.wex 1421   e. wcel 1433   _Vcvv 2601   i^i cin 2972   X. cxp 4361   ran crn 4364   |` cres 4365   "cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by:  xpimasn  4789