Theorem lesub2 8355

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 8329	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( C + A ) <_ ( C + B ) ) )
2		simp3 989	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. RR )
3		simp1 987	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. RR )
4	2, 3	readdcld 7928	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C + A ) e. RR )
5		simp2 988	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. RR )
6		lesubadd 8332	. . . 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 1228	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C + A ) - B ) <_ C <-> ( C + A ) <_ ( C + B ) ) )
8	2	recnd 7927	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> C e. CC )
9	3	recnd 7927	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A e. CC )
10	5	recnd 7927	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> B e. CC )
11	8, 9, 10	addsubd 8230	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( C + A ) - B ) = ( ( C - B ) + A ) )
12	11	breq1d 3992	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C + A ) - B ) <_ C <-> ( ( C - B ) + A ) <_ C ) )
13	1, 7, 12	3bitr2d 215	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( ( C - B ) + A ) <_ C ) )
14	2, 5	resubcld 8279	. . 3 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( C - B ) e. RR )
15		leaddsub 8336	. . 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 1228	. 2 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( C - B ) + A ) <_ C <-> ( C - B ) <_ ( C - A ) ) )
17	13, 16	bitrd 187	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A <_ B <-> ( C - B ) <_ ( C - A ) ) )

Syntax hints:   -> wi 4   <-> wb 104   /\ w3a 968   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   + caddc 7756   <_ cle 7934   - cmin 8069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072

This theorem is referenced by:  le2sub  8359  leneg  8363  lesub0  8377  lesub2d  8451