MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqord1 Structured version   Unicode version

Theorem eqord1 9868
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
eqord1  |-  ( (
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 eqord1
StepHypRef Expression
1 ltord.1 . . . 4  |-  ( x  =  y  ->  A  =  B )
2 ltord.2 . . . 4  |-  ( x  =  C  ->  A  =  M )
3 ltord.3 . . . 4  |-  ( x  =  D  ->  A  =  N )
4 ltord.4 . . . 4  |-  S  C_  RR
5 ltord.5 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
6 ltord.6 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  A  <  B ) )
71, 2, 3, 4, 5, 6leord1 9867 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  M  <_  N ) )
81, 3, 2, 4, 5, 6leord1 9867 . . . 4  |-  ( (
ph  /\  ( D  e.  S  /\  C  e.  S ) )  -> 
( D  <_  C  <->  N  <_  M ) )
98ancom2s 800 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( D  <_  C  <->  N  <_  M ) )
107, 9anbi12d 710 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( C  <_  D  /\  D  <_  C
)  <->  ( M  <_  N  /\  N  <_  M
) ) )
114sseli 3352 . . . 4  |-  ( C  e.  S  ->  C  e.  RR )
124sseli 3352 . . . 4  |-  ( D  e.  S  ->  D  e.  RR )
13 letri3 9460 . . . 4  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
1411, 12, 13syl2an 477 . . 3  |-  ( ( C  e.  S  /\  D  e.  S )  ->  ( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
1514adantl 466 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
( C  <_  D  /\  D  <_  C ) ) )
165ralrimiva 2799 . . . . 5  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
172eleq1d 2509 . . . . . 6  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
1817rspccva 3072 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
1916, 18sylan 471 . . . 4  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
2019adantrr 716 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  RR )
213eleq1d 2509 . . . . . 6  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 3072 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2316, 22sylan 471 . . . 4  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2423adantrl 715 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  RR )
2520, 24letri3d 9516 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
2610, 15, 253bitr4d 285 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
M  =  N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715    C_ wss 3328   class class class wbr 4292   RRcr 9281    < clt 9418    <_ cle 9419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-resscn 9339  ax-pre-lttri 9356  ax-pre-lttrn 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424
This theorem is referenced by:  eqord2  9871  expcan  11916  ovolicc2lem3  21002  rmyeq0  29296  rmyeq  29297
  Copyright terms: Public domain W3C validator