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Theorem eqord2 9522
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 )
ltord2.6  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  B  <  A ) )
Assertion
Ref Expression
eqord2  |-  ( (
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 eqord2
StepHypRef Expression
1 ltord.1 . . . 4  |-  ( x  =  y  ->  A  =  B )
21negeqd 9264 . . 3  |-  ( x  =  y  ->  -u A  =  -u B )
3 ltord.2 . . . 4  |-  ( x  =  C  ->  A  =  M )
43negeqd 9264 . . 3  |-  ( x  =  C  ->  -u A  =  -u M )
5 ltord.3 . . . 4  |-  ( x  =  D  ->  A  =  N )
65negeqd 9264 . . 3  |-  ( x  =  D  ->  -u A  =  -u N )
7 ltord.4 . . 3  |-  S  C_  RR
8 ltord.5 . . . 4  |-  ( (
ph  /\  x  e.  S )  ->  A  e.  RR )
98renegcld 9428 . . 3  |-  ( (
ph  /\  x  e.  S )  ->  -u A  e.  RR )
10 ltord2.6 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  ->  B  <  A ) )
118ralrimiva 2757 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
121eleq1d 2478 . . . . . . . 8  |-  ( x  =  y  ->  ( A  e.  RR  <->  B  e.  RR ) )
1312rspccva 3019 . . . . . . 7  |-  ( ( A. x  e.  S  A  e.  RR  /\  y  e.  S )  ->  B  e.  RR )
1411, 13sylan 458 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  B  e.  RR )
1514adantrl 697 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  B  e.  RR )
168adantrr 698 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  A  e.  RR )
17 ltneg 9492 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  -u A  <  -u B
) )
1815, 16, 17syl2anc 643 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( B  <  A  <->  -u A  <  -u B
) )
1910, 18sylibd 206 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  -> 
-u A  <  -u B
) )
202, 4, 6, 7, 9, 19eqord1 9519 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <->  -u M  =  -u N
) )
213eleq1d 2478 . . . . . . 7  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
2221rspccva 3019 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
2311, 22sylan 458 . . . . 5  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
2423adantrr 698 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  RR )
2524recnd 9078 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  CC )
265eleq1d 2478 . . . . . . 7  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2726rspccva 3019 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2811, 27sylan 458 . . . . 5  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2928adantrl 697 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  RR )
3029recnd 9078 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  CC )
3125, 30neg11ad 9371 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( -u M  =  -u N 
<->  M  =  N ) )
3220, 31bitrd 245 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
M  =  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674    C_ wss 3288   class class class wbr 4180   RRcr 8953    < clt 9084   -ucneg 9256
This theorem is referenced by:  basellem4  20827
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-riota 6516  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258
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