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Theorem rnxp 5122
Description: The range of a cross product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
Assertion
Ref Expression
rnxp  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )

Proof of Theorem rnxp
StepHypRef Expression
1 df-rn 4716 . . 3  |-  ran  ( A  X.  B )  =  dom  `' ( A  X.  B )
2 cnvxp 5113 . . . 4  |-  `' ( A  X.  B )  =  ( B  X.  A )
32dmeqi 4896 . . 3  |-  dom  `' ( A  X.  B
)  =  dom  ( B  X.  A )
41, 3eqtri 2316 . 2  |-  ran  ( A  X.  B )  =  dom  ( B  X.  A )
5 dmxp 4913 . 2  |-  ( A  =/=  (/)  ->  dom  ( B  X.  A )  =  B )
64, 5syl5eq 2340 1  |-  ( A  =/=  (/)  ->  ran  ( A  X.  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    =/= wne 2459   (/)c0 3468    X. cxp 4703   `'ccnv 4704   dom cdm 4705   ran crn 4706
This theorem is referenced by:  rnxpid  5125  ssxpb  5126  xpexr  5130  xpexr2  5131  unixp  5221  fconst5  5747  fparlem3  6236  fparlem4  6237  frxp  6241  fodomr  7028  dfac5lem3  7768  fpwwe2lem13  8280  vdwlem8  13051  ramz  13088  gsumxp  15243  xkoccn  17329  txindislem  17343  ovolctb  18865  xpima  23217  axlowdimlem13  24654  axlowdim1  24659  ovoliunnfl  25001  prjcp2  25188  rngodmeqrn  25522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716
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