Theorem lt2add 7923

Description: Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
lt2add	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ B < D ) -> ( A + B ) < ( C + D ) ) )

Proof of Theorem lt2add

Step	Hyp	Ref	Expression
1		simpll 496	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. RR )
2		simprl 498	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR )
3		simplr 497	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. RR )
4		ltadd1 7907	. . . 4 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
5	1, 2, 3, 4	syl3anc 1174	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A < C <-> ( A + B ) < ( C + B ) ) )
6		simprr 499	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR )
7	3, 6, 2	ltadd2d 7899	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B < D <-> ( C + B ) < ( C + D ) ) )
8	5, 7	anbi12d 457	. 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 ) ) ) )
9	1, 3	readdcld 7517	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + B ) e. RR )
10	2, 3	readdcld 7517	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + B ) e. RR )
11	2, 6	readdcld 7517	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + D ) e. RR )
12		lttr 7559	. . 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 ) ) )
13	9, 10, 11, 12	syl3anc 1174	. 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 ) ) )
14	8, 13	sylbid 148	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   <-> wb 103   e. wcel 1438   class class class wbr 3845  (class class class)co 5652   RRcr 7349   + caddc 7353   < clt 7522

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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-1cn 7438  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-addass 7447  ax-i2m1 7450  ax-0id 7453  ax-rnegex 7454  ax-pre-lttrn 7459  ax-pre-ltadd 7461

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-iota 4980  df-fv 5023  df-ov 5655  df-pnf 7524  df-mnf 7525  df-ltxr 7527

This theorem is referenced by:  addgt0  7926  lt2addi  7988  lt2halves  8651