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Theorem ltaddsub 7607
Description: 'Less than' relationship between addition and subtraction. (Contributed by NM, 17-Nov-2004.)
Assertion
Ref Expression
ltaddsub  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  B
)  <  C  <->  A  <  ( C  -  B ) ) )

Proof of Theorem ltaddsub
StepHypRef Expression
1 simp1 939 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
2 simp3 941 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
3 simp2 940 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
42, 3resubcld 7552 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  -  B )  e.  RR )
5 ltadd1 7600 . . 3  |-  ( ( A  e.  RR  /\  ( C  -  B
)  e.  RR  /\  B  e.  RR )  ->  ( A  <  ( C  -  B )  <->  ( A  +  B )  <  ( ( C  -  B )  +  B ) ) )
61, 4, 3, 5syl3anc 1170 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  ( C  -  B )  <->  ( A  +  B )  <  (
( C  -  B
)  +  B ) ) )
72recnd 7209 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  CC )
83recnd 7209 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  CC )
97, 8npcand 7490 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  -  B
)  +  B )  =  C )
109breq2d 3805 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  B
)  <  ( ( C  -  B )  +  B )  <->  ( A  +  B )  <  C
) )
116, 10bitr2d 187 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  +  B
)  <  C  <->  A  <  ( C  -  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    /\ w3a 920    e. wcel 1434   class class class wbr 3793  (class class class)co 5543   RRcr 7042    + caddc 7046    < clt 7215    - cmin 7346
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-cnex 7129  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149  ax-pre-ltadd 7154
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-pnf 7217  df-mnf 7218  df-ltxr 7220  df-sub 7348  df-neg 7349
This theorem is referenced by:  ltaddsub2  7608  ltsub13  7614  ltsub2  7630  ltaddsubi  7677  ltaddsubd  7712  iooshf  9051
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