Theorem lesub2 8501

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

Assertion

Ref	Expression
lesub2	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( C - B ) <_ ( C - A ) ) )

Proof of Theorem lesub2

Step	Hyp	Ref	Expression
1		leadd2 8475	. . 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 8073	. . . 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		lesubadd 8478	. . . 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 8072	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9	3	recnd 8072	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
10	5	recnd 8072	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
11	8, 9, 10	addsubd 8375	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) - B ) = ( ( C - B ) + A ) )
12	11	breq1d 4044	. . 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 8424	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C - B ) e. RR )
15		leaddsub 8482	. . 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 4034  (class class class)co 5925   RRcr 7895   + caddc 7899   <_ cle 8079   - cmin 8214

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 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltadd 8012

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 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217

This theorem is referenced by:  le2sub  8505  leneg  8509  lesub0  8523  lesub2d  8597  gausslemma2dlem1a  15383