Theorem le2sub 8608

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 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		lesub1 8603	. . . 4 |- ( ( A e. RR /\ C e. RR /\ B e. RR ) -> ( A <_ C <-> ( A - B ) <_ ( C - B ) ) )
5	1, 2, 3, 4	syl3anc 1271	. . 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		lesub2 8604	. . . 4 |- ( ( D e. RR /\ B e. RR /\ C e. RR ) -> ( D <_ B <-> ( C - B ) <_ ( C - D ) ) )
8	6, 3, 2, 7	syl3anc 1271	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( D <_ B <-> ( C - B ) <_ ( C - D ) ) )
9	5, 8	anbi12d 473	. 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 8410	. . . 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	2, 3	resubcld 8527	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - B ) e. RR )
13		resubcl 8410	. . . 4 |- ( ( C e. RR /\ D e. RR ) -> ( C - D ) e. RR )
14	13	adantl 277	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - D ) e. RR )
15		letr 8229	. . 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 1271	. 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 150	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 104   <-> wb 105   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RRcr 7998   <_ cle 8182   - cmin 8317

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltwlin 8112  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320

This theorem is referenced by:  le2subd  8711