Theorem le2add 8471

Description: Adding both sides of two 'less than or equal to' relations. (Contributed by NM, 17-Apr-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
le2add	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ B <_ D ) -> ( A + B ) <_ ( C + D ) ) )

Proof of Theorem le2add

Step	Hyp	Ref	Expression
1		simpll 527	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. RR )
2		simprl 529	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR )
3		simplr 528	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. RR )
4		leadd1 8457	. . . 4 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A <_ C <-> ( A + B ) <_ ( C + B ) ) )
5	1, 2, 3, 4	syl3anc 1249	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A <_ C <-> ( A + B ) <_ ( C + B ) ) )
6		simprr 531	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR )
7		leadd2 8458	. . . 4 |- ( ( B e. RR /\ D e. RR /\ C e. RR ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
8	3, 6, 2, 7	syl3anc 1249	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( B <_ D <-> ( C + B ) <_ ( C + D ) ) )
9	5, 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	1, 3	readdcld 8056	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A + B ) e. RR )
11	2, 3	readdcld 8056	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + B ) e. RR )
12	2, 6	readdcld 8056	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C + D ) e. RR )
13		letr 8109	. . 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 ) ) )
14	10, 11, 12, 13	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 ) ) )
15	9, 14	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 4033  (class class class)co 5922   RRcr 7878   + caddc 7882   <_ cle 8062

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 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-pre-ltwlin 7992  ax-pre-ltadd 7995

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 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-xp 4669  df-cnv 4671  df-iota 5219  df-fv 5266  df-ov 5925  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067

This theorem is referenced by:  addge0  8478  le2addi  8538  le2addd  8590