Theorem xrre3 9732

Description: A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)

Assertion

Ref	Expression
xrre3	|- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A e. RR )

Proof of Theorem xrre3

Step	Hyp	Ref	Expression
1		mnflt 9696	. . . . . 6 |- ( B e. RR -> -oo < B )
2	1	adantl 275	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> -oo < B )
3		mnfxr 7936	. . . . . . 7 |- -oo e. RR*
4	3	a1i 9	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> -oo e. RR* )
5		rexr 7925	. . . . . . 7 |- ( B e. RR -> B e. RR* )
6	5	adantl 275	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> B e. RR* )
7		simpl 108	. . . . . 6 |- ( ( A e. RR* /\ B e. RR ) -> A e. RR* )
8		xrltletr 9717	. . . . . 6 |- ( ( -oo e. RR* /\ B e. RR* /\ A e. RR* ) -> ( ( -oo < B /\ B <_ A ) -> -oo < A ) )
9	4, 6, 7, 8	syl3anc 1220	. . . . 5 |- ( ( A e. RR* /\ B e. RR ) -> ( ( -oo < B /\ B <_ A ) -> -oo < A ) )
10	2, 9	mpand 426	. . . 4 |- ( ( A e. RR* /\ B e. RR ) -> ( B <_ A -> -oo < A ) )
11	10	imp 123	. . 3 |- ( ( ( A e. RR* /\ B e. RR ) /\ B <_ A ) -> -oo < A )
12	11	adantrr 471	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> -oo < A )
13		simprr 522	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A < +oo )
14		xrrebnd 9729	. . 3 |- ( A e. RR* -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
15	14	ad2antrr 480	. 2 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> ( A e. RR <-> ( -oo < A /\ A < +oo ) ) )
16	12, 13, 15	mpbir2and 929	1 |- ( ( ( A e. RR* /\ B e. RR ) /\ ( B <_ A /\ A < +oo ) ) -> A e. RR )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2128   class class class wbr 3967   RRcr 7733   +oocpnf 7911   -oocmnf 7912   RR*cxr 7913   < clt 7914   <_ cle 7915

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171  ax-un 4395  ax-setind 4498  ax-cnex 7825  ax-resscn 7826  ax-pre-ltirr 7846  ax-pre-ltwlin 7847  ax-pre-lttrn 7848

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-po 4258  df-iso 4259  df-xp 4594  df-cnv 4596  df-pnf 7916  df-mnf 7917  df-xr 7918  df-ltxr 7919  df-le 7920

This theorem is referenced by:  elicore  10175