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Theorem eqord2 9883
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 9616 . . 3  |-  ( x  =  y  ->  -u A  =  -u B )
3 ltord.2 . . . 4  |-  ( x  =  C  ->  A  =  M )
43negeqd 9616 . . 3  |-  ( x  =  C  ->  -u A  =  -u M )
5 ltord.3 . . . 4  |-  ( x  =  D  ->  A  =  N )
65negeqd 9616 . . 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 9787 . . 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 2811 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
121eleq1d 2509 . . . . . . . 8  |-  ( x  =  y  ->  ( A  e.  RR  <->  B  e.  RR ) )
1312rspccva 3084 . . . . . . 7  |-  ( ( A. x  e.  S  A  e.  RR  /\  y  e.  S )  ->  B  e.  RR )
1411, 13sylan 471 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  B  e.  RR )
1514adantrl 715 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  B  e.  RR )
168adantrr 716 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  A  e.  RR )
17 ltneg 9851 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  -u A  <  -u B
) )
1815, 16, 17syl2anc 661 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( B  <  A  <->  -u A  <  -u B
) )
1910, 18sylibd 214 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  -> 
-u A  <  -u B
) )
202, 4, 6, 7, 9, 19eqord1 9880 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <->  -u M  =  -u N
) )
213eleq1d 2509 . . . . . . 7  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
2221rspccva 3084 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
2311, 22sylan 471 . . . . 5  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
2423adantrr 716 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  RR )
2524recnd 9424 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  CC )
265eleq1d 2509 . . . . . . 7  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2726rspccva 3084 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2811, 27sylan 471 . . . . 5  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2928adantrl 715 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  RR )
3029recnd 9424 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  CC )
3125, 30neg11ad 9727 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( -u M  =  -u N 
<->  M  =  N ) )
3220, 31bitrd 253 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 2727    C_ wss 3340   class class class wbr 4304   RRcr 9293    < clt 9430   -ucneg 9608
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610
This theorem is referenced by:  basellem4  22433
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