Theorem leltadd 8617

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 8616	. . . . 5 |- ( ( ( B e. RR /\ A e. RR ) /\ ( D e. RR /\ C e. RR ) ) -> ( ( B < D /\ A <_ C ) -> ( B + A ) < ( D + C ) ) )
2	1	ancomsd 269	. . . 4 |- ( ( ( B e. RR /\ A e. RR ) /\ ( D e. RR /\ C e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
3	2	ancom2s 566	. . 3 |- ( ( ( B e. RR /\ A e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
4	3	ancom1s 569	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
5		recn 8155	. . . 4 |- ( A e. RR -> A e. CC )
6		recn 8155	. . . 4 |- ( B e. RR -> B e. CC )
7		addcom 8306	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) = ( B + A ) )
8	5, 6, 7	syl2an 289	. . 3 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( B + A ) )
9		recn 8155	. . . 4 |- ( C e. RR -> C e. CC )
10		recn 8155	. . . 4 |- ( D e. RR -> D e. CC )
11		addcom 8306	. . . 4 |- ( ( C e. CC /\ D e. CC ) -> ( C + D ) = ( D + C ) )
12	9, 10, 11	syl2an 289	. . 3 |- ( ( C e. RR /\ D e. RR ) -> ( C + D ) = ( D + C ) )
13	8, 12	breqan12d 4102	. 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 169	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 104   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   CCcc 8020   RRcr 8021   + caddc 8025   < clt 8204   <_ cle 8205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0id 8130  ax-rnegex 8131  ax-pre-ltwlin 8135  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-xp 4729  df-cnv 4731  df-iota 5284  df-fv 5332  df-ov 6016  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210

This theorem is referenced by:  addgegt0  8619  leltaddd  8736