Theorem le2sub 8216

Description: Subtracting both sides of two 'less than or equal to' relations. (Contributed by Mario Carneiro, 14-Apr-2016.)

Assertion

Ref	Expression
le2sub	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ D <_ B ) -> ( A - B ) <_ ( C - D ) ) )

Proof of Theorem le2sub

Step	Hyp	Ref	Expression
1		simpll 518	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> A e. RR )
2		simprl 520	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> C e. RR )
3		simplr 519	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> B e. RR )
4		lesub1 8211	. . . 4 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A <_ C <-> ( A - B ) <_ ( C - B ) ) )
5	1, 2, 3, 4	syl3anc 1216	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A <_ C <-> ( A - B ) <_ ( C - B ) ) )
6		simprr 521	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> D e. RR )
7		lesub2 8212	. . . 4 |- ( ( D e. RR /\ B e. RR /\ C e. RR ) -> ( D <_ B <-> ( C - B ) <_ ( C - D ) ) )
8	6, 3, 2, 7	syl3anc 1216	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( D <_ B <-> ( C - B ) <_ ( C - D ) ) )
9	5, 8	anbi12d 464	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ D <_ B ) <-> ( ( A - B ) <_ ( C - B ) /\ ( C - B ) <_ ( C - D ) ) ) )
10		resubcl 8019	. . . 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	2, 3	resubcld 8136	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - B ) e. RR )
13		resubcl 8019	. . . 4 |- ( ( C e. RR /\ D e. RR ) -> ( C - D ) e. RR )
14	13	adantl 275	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - D ) e. RR )
15		letr 7840	. . 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 ) ) )
16	11, 12, 14, 15	syl3anc 1216	. 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 ) ) )
17	9, 16	sylbid 149	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A <_ C /\ D <_ B ) -> ( A - B ) <_ ( C - D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 1480   class class class wbr 3924  (class class class)co 5767   RRcr 7612   <_ cle 7794   - cmin 7926

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltwlin 7726  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929

This theorem is referenced by:  le2subd  8319