Theorem ltsub2 8486

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 8446	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < B <-> ( C + A ) < ( C + B ) ) )
2		simp3 1001	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		simp1 999	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
4	2, 3	readdcld 8056	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C + A ) e. RR )
5		simp2 1000	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
6		ltsubadd 8459	. . . 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 1249	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C + A ) - B ) < C <-> ( C + A ) < ( C + B ) ) )
8	2	recnd 8055	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9	3	recnd 8055	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
10	5	recnd 8055	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
11	8, 9, 10	addsubd 8358	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) - B ) = ( ( C - B ) + A ) )
12	11	breq1d 4043	. . 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 8407	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C - B ) e. RR )
15		ltaddsub 8463	. . 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 1249	. 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 980   e. wcel 2167   class class class wbr 4033  (class class class)co 5922   RRcr 7878   + caddc 7882   < clt 8061   - cmin 8197

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-sub 8199  df-neg 8200

This theorem is referenced by:  lt2sub  8487  ltneg  8489  ltsub2d  8582  ltm1  8873