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Theorem ltadd2 9011
Description: Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
ltadd2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
) )

Proof of Theorem ltadd2
StepHypRef Expression
1 axltadd 8983 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  ->  ( C  +  A )  <  ( C  +  B
) ) )
2 oveq2 5950 . . . . . 6  |-  ( A  =  B  ->  ( C  +  A )  =  ( C  +  B ) )
32a1i 10 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  =  B  ->  ( C  +  A )  =  ( C  +  B ) ) )
4 axltadd 8983 . . . . . 6  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) ) )
543com12 1155 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( B  <  A  ->  ( C  +  B )  <  ( C  +  A
) ) )
63, 5orim12d 811 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  =  B  \/  B  <  A
)  ->  ( ( C  +  A )  =  ( C  +  B )  \/  ( C  +  B )  <  ( C  +  A
) ) ) )
76con3d 125 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -.  ( ( C  +  A )  =  ( C  +  B )  \/  ( C  +  B )  <  ( C  +  A )
)  ->  -.  ( A  =  B  \/  B  <  A ) ) )
8 simp3 957 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  C  e.  RR )
9 simp1 955 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  A  e.  RR )
108, 9readdcld 8949 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  A )  e.  RR )
11 simp2 956 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  B  e.  RR )
128, 11readdcld 8949 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  +  B )  e.  RR )
13 axlttri 8981 . . . 4  |-  ( ( ( C  +  A
)  e.  RR  /\  ( C  +  B
)  e.  RR )  ->  ( ( C  +  A )  < 
( C  +  B
)  <->  -.  ( ( C  +  A )  =  ( C  +  B )  \/  ( C  +  B )  <  ( C  +  A
) ) ) )
1410, 12, 13syl2anc 642 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  <->  -.  (
( C  +  A
)  =  ( C  +  B )  \/  ( C  +  B
)  <  ( C  +  A ) ) ) )
15 axlttri 8981 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A
) ) )
169, 11, 15syl2anc 642 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  -.  ( A  =  B  \/  B  <  A ) ) )
177, 14, 163imtr4d 259 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  +  A
)  <  ( C  +  B )  ->  A  <  B ) )
181, 17impbid 183 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  B  <->  ( C  +  A )  <  ( C  +  B )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ w3a 934    = wceq 1642    e. wcel 1710   class class class wbr 4102  (class class class)co 5942   RRcr 8823    + caddc 8827    < clt 8954
This theorem is referenced by:  ltadd2d  9059  readdcan  9073  ltadd1  9328  ltaddpos  9351  avglt1  10038  flbi2  11036  stoweidlem11  27083  stoweidlem13  27085  stoweidlem14  27086  stoweidlem26  27098  stoweidlem44  27116
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-resscn 8881  ax-addrcl 8885  ax-pre-lttri 8898  ax-pre-ltadd 8900
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-ltxr 8959
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