Theorem ltsub2 7884

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 7844	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )
2		simp3 943	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		simp1 941	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
4	2, 3	readdcld 7464	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C + A ) e. RR )
5		simp2 942	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
6		ltsubadd 7857	. . . 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 1172	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C + A ) - B ) < C <-> ( C + A ) < ( C + B ) ) )
8	2	recnd 7463	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9	3	recnd 7463	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
10	5	recnd 7463	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
11	8, 9, 10	addsubd 7761	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) - B ) = ( ( C - B ) + A ) )
12	11	breq1d 3832	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C + A ) - B ) < C <-> ( ( C - B ) + A ) < C ) )
13	1, 7, 12	3bitr2d 214	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( ( C - B ) + A ) < C ) )
14	2, 5	resubcld 7806	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C - B ) e. RR )
15		ltaddsub 7861	. . 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 1172	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C - B ) + A ) < C <-> ( C - B ) < ( C - A ) ) )
17	13, 16	bitrd 186	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C - B ) < ( C - A ) ) )

Syntax hints:   -> wi 4   <-> wb 103   /\ w3a 922   e. wcel 1436   class class class wbr 3822  (class class class)co 5615   RRcr 7296   + caddc 7300   < clt 7469   - cmin 7600

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 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-cnex 7383  ax-resscn 7384  ax-1cn 7385  ax-icn 7387  ax-addcl 7388  ax-addrcl 7389  ax-mulcl 7390  ax-addcom 7392  ax-addass 7394  ax-distr 7396  ax-i2m1 7397  ax-0id 7400  ax-rnegex 7401  ax-cnre 7403  ax-pre-ltadd 7408

This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-id 4096  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-iota 4948  df-fun 4985  df-fv 4991  df-riota 5571  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-pnf 7471  df-mnf 7472  df-ltxr 7474  df-sub 7602  df-neg 7603

This theorem is referenced by:  lt2sub  7885  ltneg  7887  ltsub2d  7976  ltm1  8245