Theorem lt2sub 7936

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

Assertion

Ref	Expression
lt2sub	|- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( ( A < C /\ D < B ) -> ( A - B ) < ( C - D ) ) )

Proof of Theorem lt2sub

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		ltsub1 7934	. . . 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		ltsub2 7935	. . . 4 |- ( ( D e. RR /\ B e. RR /\ C e. RR ) -> ( D < B <-> ( C - B ) < ( C - D ) ) )
8	6, 3, 2, 7	syl3anc 1174	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( D < B <-> ( C - B ) < ( C - D ) ) )
9	5, 8	anbi12d 457	. 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 7744	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( A - B ) e. RR )
11	10	adantr 270	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( A - B ) e. RR )
12	2, 3	resubcld 7857	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - B ) e. RR )
13		resubcl 7744	. . . 4 |- ( ( C e. RR /\ D e. RR ) -> ( C - D ) e. RR )
14	13	adantl 271	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - D ) e. RR )
15		lttr 7557	. . 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 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 ) ) )
17	9, 16	sylbid 148	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 102   <-> wb 103   e. wcel 1438   class class class wbr 3845  (class class class)co 5652   RRcr 7347   < clt 7520   - cmin 7651

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 7434  ax-resscn 7435  ax-1cn 7436  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-lttrn 7457  ax-pre-ltadd 7459

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-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  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-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-ltxr 7525  df-sub 7653  df-neg 7654

This theorem is referenced by:  lt2subd  8043