Theorem leltadd 7551

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 7550	. . . . 5 |- ( ( ( B e. RR /\ A e. RR ) /\ ( D e. RR /\ C e. RR ) ) -> ( ( B < D /\ A <_ C ) -> ( B + A ) < ( D + C ) ) )
2	1	ancomsd 265	. . . 4 |- ( ( ( B e. RR /\ A e. RR ) /\ ( D e. RR /\ C e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
3	2	ancom2s 530	. . 3 |- ( ( ( B e. RR /\ A e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
4	3	ancom1s 533	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
5		recn 7106	. . . 4 |- ( A e. RR -> A e. CC )
6		recn 7106	. . . 4 |- ( B e. RR -> B e. CC )
7		addcom 7245	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
8	5, 6, 7	syl2an 283	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
9		recn 7106	. . . 4 |- ( C e. RR -> C e. CC )
10		recn 7106	. . . 4 |- ( D e. RR -> D e. CC )
11		addcom 7245	. . . 4 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) = ( D + C ) )
12	9, 10, 11	syl2an 283	. . 3 |- ( ( C e. RR /\ D e. RR ) -> ( C + D ) = ( D + C ) )
13	8, 12	breqan12d 3800	. 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 167	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 102   = wceq 1284   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   CCcc 6979   RRcr 6980   + caddc 6984   < clt 7153   <_ cle 7154

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-in1 576  ax-in2 577  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-13 1444  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  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-addcom 7076  ax-addass 7078  ax-i2m1 7081  ax-0id 7084  ax-rnegex 7085  ax-pre-ltwlin 7089  ax-pre-ltadd 7092

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  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-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-iota 4887  df-fv 4930  df-ov 5535  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159

This theorem is referenced by:  addgegt0  7553  leltaddd  7666