Theorem rnxpid 14409

Description: The range of a square cross product.

Assertion

Ref	Expression
rnxpid	|- ran ( A X. A) = A

Proof of Theorem rnxpid

Step	Hyp	Ref	Expression
1		rn0 4203	. . 3 |- ran (/) = (/)
2		xpeq2 4017	. . . . 5 |- (A = (/) -> (A X. A) = (A X. (/)))
3		xp0 4334	. . . . 5 |- (A X. (/)) = (/)
4	2, 3	syl6eq 1944	. . . 4 |- (A = (/) -> (A X. A) = (/))
5	4	rneqd 4188	. . 3 |- (A = (/) -> ran ( A X. A) = ran (/))
6		id 73	. . 3 |- (A = (/) -> A = (/))
7	1, 5, 6	3eqtr4a 1954	. 2 |- (A = (/) -> ran ( A X. A) = A)
8		rnxp 4342	. 2 |- (A =/= (/) -> ran ( A X. A) = A)
9	7, 8	pm2.61ine 2089	1 |- ran ( A X. A) = A

Syntax hints:   = wceq 1298  (/)c0 2875   X. cxp 3984  ran crn 3987

This theorem is referenced by:  mgmrddd 14727

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005

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