Theorem lesub2 8478

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 8452	. . 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 8051	. . . 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 8455	. . . 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 8050	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9	3	recnd 8050	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
10	5	recnd 8050	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
11	8, 9, 10	addsubd 8353	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) - B ) = ( ( C - B ) + A ) )
12	11	breq1d 4040	. . 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 8402	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C - B ) e. RR )
15		leaddsub 8459	. . 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 2164   class class class wbr 4030  (class class class)co 5919   RRcr 7873   + caddc 7877   <_ cle 8057   - cmin 8192

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltadd 7990

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195

This theorem is referenced by:  le2sub  8482  leneg  8486  lesub0  8500  lesub2d  8574  gausslemma2dlem1a  15215