Theorem ltsub2 8638

Description: Subtraction of both sides of 'less than'. (Contributed by NM, 29-Sep-2005.) (Proof shortened by Mario Carneiro, 27-May-2016.)

Assertion

Ref	Expression
ltsub2	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C - B ) < ( C - A ) ) )

Proof of Theorem ltsub2

Step	Hyp	Ref	Expression
1		ltadd2 8598	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )
2		simp3 1025	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		simp1 1023	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
4	2, 3	readdcld 8208	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C + A ) e. RR )
5		simp2 1024	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
6		ltsubadd 8611	. . . 4 |- ( ( ( C + A ) e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C + A ) - B ) < C <-> ( C + A ) < ( C + B ) ) )
7	4, 5, 2, 6	syl3anc 1273	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C + A ) - B ) < C <-> ( C + A ) < ( C + B ) ) )
8	2	recnd 8207	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9	3	recnd 8207	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
10	5	recnd 8207	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
11	8, 9, 10	addsubd 8510	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) - B ) = ( ( C - B ) + A ) )
12	11	breq1d 4098	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C + A ) - B ) < C <-> ( ( C - B ) + A ) < C ) )
13	1, 7, 12	3bitr2d 216	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( ( C - B ) + A ) < C ) )
14	2, 5	resubcld 8559	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C - B ) e. RR )
15		ltaddsub 8615	. . 3 |- ( ( ( C - B ) e. RR /\ A e. RR /\ C e. RR ) -> ( ( ( C - B ) + A ) < C <-> ( C - B ) < ( C - A ) ) )
16	14, 3, 2, 15	syl3anc 1273	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C - B ) + A ) < C <-> ( C - B ) < ( C - A ) ) )
17	13, 16	bitrd 188	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C - B ) < ( C - A ) ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1004   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   + caddc 8034   < clt 8213   - cmin 8349

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-sub 8351  df-neg 8352

This theorem is referenced by:  lt2sub  8639  ltneg  8641  ltsub2d  8734  ltm1  9025