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Theorem leord1 9300
Description: Infer an ordering relation from a proof in only one direction. (Contributed by Mario Carneiro, 14-Jun-2014.)
Hypotheses
Ref Expression
ltord.1  |-  ( x  =  y  ->  A  =  B )
ltord.2  |-  ( x  =  C  ->  A  =  M )
ltord.3  |-  ( x  =  D  ->  A  =  N )
ltord.4  |-  S  C_  RR
ltord.5  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
ltord.6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
Assertion
Ref Expression
leord1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  M  <_  N ) )
Distinct variable groups:    x, B    x, y, C    x, D, y    x, M, y    x, N, y    ph, x, y   
x, S, y
Allowed substitution hints:    A( x, y)    B( y)

Proof of Theorem leord1
StepHypRef Expression
1 ltord.1 . . . . 5  |-  ( x  =  y  ->  A  =  B )
2 ltord.3 . . . . 5  |-  ( x  =  D  ->  A  =  N )
3 ltord.2 . . . . 5  |-  ( x  =  C  ->  A  =  M )
4 ltord.4 . . . . 5  |-  S  C_  RR
5 ltord.5 . . . . 5  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
6 ltord.6 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
71, 2, 3, 4, 5, 6ltord1 9299 . . . 4  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <  C  <->  N  <  M ) )
87ancom2s 777 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <  C  <->  N  <  M ) )
98notbid 285 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( -.  D  < 
C  <->  -.  N  <  M ) )
104sseli 3176 . . . 4  |-  ( C  e.  S  ->  C  e.  RR )
114sseli 3176 . . . 4  |-  ( D  e.  S  ->  D  e.  RR )
12 lenlt 8901 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  <_  D  <->  -.  D  <  C ) )
1310, 11, 12syl2an 463 . . 3  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  <_  D  <->  -.  D  <  C ) )
1413adantl 452 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  -.  D  <  C ) )
155ralrimiva 2626 . . . . 5  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
163eleq1d 2349 . . . . . 6  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1716rspccva 2883 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
1815, 17sylan 457 . . . 4  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
1918adantrr 697 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  RR )
202eleq1d 2349 . . . . . 6  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2120rspccva 2883 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2215, 21sylan 457 . . . 4  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2322adantrl 696 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  RR )
2419, 23lenltd 8965 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <_  N  <->  -.  N  <  M ) )
259, 14, 243bitr4d 276 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  M  <_  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023   RRcr 8736    < clt 8867    <_ cle 8868
This theorem is referenced by:  eqord1  9301  leord2  9303  lermxnn0  27037  lermy  27042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-pre-lttri 8811
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873
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