Theorem ltleadd 8490

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 8473	. . . . . 6 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
2	1	3com23 1211	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
3	2	3expa 1205	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ C e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
4	3	adantrr 479	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
5		leadd2 8475	. . . . . 6 |- ( ( B e. RR /\ D e. RR /\ C e. RR ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
6	5	3com23 1211	. . . . 5 |- ( ( B e. RR /\ C e. RR /\ D e. RR ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
7	6	3expb 1206	. . . 4 |- ( ( B e. RR /\ ( C e. RR /\ D e. RR ) ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
8	7	adantll 476	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
9	4, 8	anbi12d 473	. 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 8022	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
11	10	adantr 276	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + B ) e. RR )
12		readdcl 8022	. . . . 5 |- ( ( C e. RR /\ B e. RR ) -> ( C + B ) e. RR )
13	12	ancoms 268	. . . 4 |- ( ( B e. RR /\ C e. RR ) -> ( C + B ) e. RR )
14	13	ad2ant2lr 510	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + B ) e. RR )
15		readdcl 8022	. . . 4 |- ( ( C e. RR /\ D e. RR ) -> ( C + D ) e. RR )
16	15	adantl 277	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + D ) e. RR )
17		ltletr 8133	. . 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 1249	. 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 150	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   <-> wb 105   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   RRcr 7895   + caddc 7899   < clt 8078   <_ cle 8079

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-pre-ltwlin 8009  ax-pre-ltadd 8012

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-xp 4670  df-cnv 4672  df-iota 5220  df-fv 5267  df-ov 5928  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084

This theorem is referenced by:  leltadd  8491  addgtge0  8494  ltleaddd  8609