Theorem leltadd 8740

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 8739	. . . . 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 568	. . 3 |- ( ( ( B e. RR /\ A e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
4	3	ancom1s 571	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B < D ) -> ( B + A ) < ( D + C ) ) )
5		recn 8277	. . . 4 |- ( A e. RR -> A e. CC )
6		recn 8277	. . . 4 |- ( B e. RR -> B e. CC )
7		addcom 8428	. . . 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 8277	. . . 4 |- ( C e. RR -> C e. CC )
10		recn 8277	. . . 4 |- ( D e. RR -> D e. CC )
11		addcom 8428	. . . 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 4131	. 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 1398   e. wcel 2205   class class class wbr 4115  (class class class)co 6059   CCcc 8142   RRcr 8143   + caddc 8147   < clt 8325   <_ cle 8326

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-setind 4665  ax-cnex 8235  ax-resscn 8236  ax-1cn 8237  ax-icn 8239  ax-addcl 8240  ax-addrcl 8241  ax-mulcl 8242  ax-addcom 8244  ax-addass 8246  ax-i2m1 8249  ax-0id 8252  ax-rnegex 8253  ax-pre-ltwlin 8257  ax-pre-ltadd 8260

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-br 4116  df-opab 4178  df-xp 4761  df-cnv 4763  df-iota 5318  df-fv 5366  df-ov 6062  df-pnf 8327  df-mnf 8328  df-xr 8329  df-ltxr 8330  df-le 8331

This theorem is referenced by:  addgegt0  8742  leltaddd  8859