Theorem leltadd 7988

Description: Adding both sides of two orderings. (Contributed by NM, 15-Aug-2008.)

Assertion

Ref	Expression
leltadd	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( A + B ) < ( C + D ) ) )

Proof of Theorem leltadd

Step	Hyp	Ref	Expression
1		ltleadd 7987	. . . . 5 |- ( ( ( B e. RR /\ A e. RR ) /\ ( D e. RR /\ C e. RR ) ) -> ( ( B < D /\ A <_ C ) -> ( B + A ) < ( D + C ) ) )
2	1	ancomsd 266	. . . 4 |- ( ( ( B e. RR /\ A e. RR ) /\ ( D e. RR /\ C e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
3	2	ancom2s 534	. . 3 |- ( ( ( B e. RR /\ A e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
4	3	ancom1s 537	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
5		recn 7538	. . . 4 |- ( A e. RR -> A e. CC )
6		recn 7538	. . . 4 |- ( B e. RR -> B e. CC )
7		addcom 7682	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
8	5, 6, 7	syl2an 284	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
9		recn 7538	. . . 4 |- ( C e. RR -> C e. CC )
10		recn 7538	. . . 4 |- ( D e. RR -> D e. CC )
11		addcom 7682	. . . 4 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) = ( D + C ) )
12	9, 10, 11	syl2an 284	. . 3 |- ( ( C e. RR /\ D e. RR ) -> ( C + D ) = ( D + C ) )
13	8, 12	breqan12d 3868	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A + B ) < ( C + D ) <-> ( B + A ) < ( D + C ) ) )
14	4, 13	sylibrd 168	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( A + B ) < ( C + D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1290   e. wcel 1439   class class class wbr 3853  (class class class)co 5668   CCcc 7411   RRcr 7412   + caddc 7416   < clt 7585   <_ cle 7586

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-cnex 7499  ax-resscn 7500  ax-1cn 7501  ax-icn 7503  ax-addcl 7504  ax-addrcl 7505  ax-mulcl 7506  ax-addcom 7508  ax-addass 7510  ax-i2m1 7513  ax-0id 7516  ax-rnegex 7517  ax-pre-ltwlin 7521  ax-pre-ltadd 7524

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2624  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-xp 4460  df-cnv 4462  df-iota 4995  df-fv 5038  df-ov 5671  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591

This theorem is referenced by:  addgegt0  7990  leltaddd  8106