Theorem lt2sub 8447

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 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		ltsub1 8445	. . . 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		ltsub2 8446	. . . 4 |- ( ( D e. RR /\ B e. RR /\ C e. RR ) -> ( D < B <-> ( C - B ) < ( C - D ) ) )
8	6, 3, 2, 7	syl3anc 1249	. . 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 8251	. . . 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 8368	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( C e. RR /\ D e. RR ) ) -> ( C - B ) e. RR )
13		resubcl 8251	. . . 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		lttr 8061	. . 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 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 ) ) )
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 2160   class class class wbr 4018  (class class class)co 5896   RRcr 7840   < clt 8022   - cmin 8158

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7932  ax-resscn 7933  ax-1cn 7934  ax-icn 7936  ax-addcl 7937  ax-addrcl 7938  ax-mulcl 7939  ax-addcom 7941  ax-addass 7943  ax-distr 7945  ax-i2m1 7946  ax-0id 7949  ax-rnegex 7950  ax-cnre 7952  ax-pre-lttrn 7955  ax-pre-ltadd 7957

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5899  df-oprab 5900  df-mpo 5901  df-pnf 8024  df-mnf 8025  df-ltxr 8027  df-sub 8160  df-neg 8161

This theorem is referenced by:  lt2subd  8555