MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltord1 Structured version   Unicode version

Theorem ltord1 10080
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
ltord1  |-  ( (
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 ltord1
StepHypRef Expression
1 ltord.1 . . 3  |-  ( x  =  y  ->  A  =  B )
2 ltord.2 . . 3  |-  ( x  =  C  ->  A  =  M )
3 ltord.3 . . 3  |-  ( x  =  D  ->  A  =  N )
4 ltord.4 . . 3  |-  S  C_  RR
5 ltord.5 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
6 ltord.6 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
71, 2, 3, 4, 5, 6ltordlem 10079 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  ->  M  <  N ) )
8 eqeq1 2445 . . . . . . . 8  |-  ( x  =  C  ->  (
x  =  D  <->  C  =  D ) )
92eqeq1d 2443 . . . . . . . 8  |-  ( x  =  C  ->  ( A  =  N  <->  M  =  N ) )
108, 9imbi12d 320 . . . . . . 7  |-  ( x  =  C  ->  (
( x  =  D  ->  A  =  N )  <->  ( C  =  D  ->  M  =  N ) ) )
1110, 3vtoclg 3151 . . . . . 6  |-  ( C  e.  S  ->  ( C  =  D  ->  M  =  N ) )
1211ad2antrl 727 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  ->  M  =  N ) )
131, 3, 2, 4, 5, 6ltordlem 10079 . . . . . 6  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1413ancom2s 800 . . . . 5  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <  C  ->  N  <  M ) )
1512, 14orim12d 836 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( C  =  D  \/  D  < 
C )  ->  ( M  =  N  \/  N  <  M ) ) )
1615con3d 133 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( -.  ( M  =  N  \/  N  <  M )  ->  -.  ( C  =  D  \/  D  <  C ) ) )
175ralrimiva 2855 . . . . . 6  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
182eleq1d 2510 . . . . . . 7  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1918rspccva 3193 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
2017, 19sylan 471 . . . . 5  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
213eleq1d 2510 . . . . . . 7  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 3193 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2317, 22sylan 471 . . . . 5  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2420, 23anim12dan 835 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  e.  RR  /\  N  e.  RR ) )
25 axlttri 9654 . . . 4  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <  N  <->  -.  ( M  =  N  \/  N  <  M
) ) )
2624, 25syl 16 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <  N  <->  -.  ( M  =  N  \/  N  <  M
) ) )
274sseli 3482 . . . . 5  |-  ( C  e.  S  ->  C  e.  RR )
284sseli 3482 . . . . 5  |-  ( D  e.  S  ->  D  e.  RR )
29 axlttri 9654 . . . . 5  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3027, 28, 29syl2an 477 . . . 4  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3130adantl 466 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  <->  -.  ( C  =  D  \/  D  <  C
) ) )
3216, 26, 313imtr4d 268 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  <  N  ->  C  <  D ) )
337, 32impbid 191 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <  D  <->  M  <  N ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791    C_ wss 3458   class class class wbr 4433   RRcr 9489    < clt 9626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-resscn 9547  ax-pre-lttri 9564
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-ltxr 9631
This theorem is referenced by:  leord1  10081  ltord2  10083  ltexp2  12193  eflt  13724  tanord1  22789  tanord  22790  monotuz  30845  monotoddzzfi  30846  rpexpmord  30852
  Copyright terms: Public domain W3C validator