Theorem ltleadd 8344

Description: Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.)

Assertion

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

Proof of Theorem ltleadd

Step	Hyp	Ref	Expression
1		ltadd1 8327	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
2	1	3com23 1199	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
3	2	3expa 1193	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
4	3	adantrr 471	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
5		leadd2 8329	. . . . . 6 |- ( ( B e. RR /\ D e. RR /\ C e. RR ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
6	5	3com23 1199	. . . . 5 |- ( ( B e. RR /\ C e. RR /\ D e. RR ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
7	6	3expb 1194	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR ) ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
8	7	adantll 468	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
9	4, 8	anbi12d 465	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ B <_ D ) <-> ( ( A + B ) < ( C + B ) /\ ( C + B ) <_ ( C + D ) ) ) )
10		readdcl 7879	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
11	10	adantr 274	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + B ) e. RR )
12		readdcl 7879	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
13	12	ancoms 266	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( C + B ) e. RR )
14	13	ad2ant2lr 502	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + B ) e. RR )
15		readdcl 7879	. . . 4 |- ( ( C e. RR /\ D e. RR ) -> ( C + D ) e. RR )
16	15	adantl 275	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + D ) e. RR )
17		ltletr 7988	. . 3 |- ( ( ( A + B ) e. RR /\ ( C + B ) e. RR /\ ( C + D ) e. RR ) -> ( ( ( A + B ) < ( C + B ) /\ ( C + B ) <_ ( C + D ) ) -> ( A + B ) < ( C + D ) ) )
18	11, 14, 16, 17	syl3anc 1228	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( ( A + B ) < ( C + B ) /\ ( C + B ) <_ ( C + D ) ) -> ( A + B ) < ( C + D ) ) )
19	9, 18	sylbid 149	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   <-> wb 104   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   + caddc 7756   < clt 7933   <_ cle 7934

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-pre-ltwlin 7866  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-iota 5153  df-fv 5196  df-ov 5845  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939

This theorem is referenced by:  leltadd  8345  addgtge0  8348  ltleaddd  8463