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Theorem eqord2 9563
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 9305 . . 3  |-  ( x  =  y  ->  -u A  =  -u B )
3 ltord.2 . . . 4  |-  ( x  =  C  ->  A  =  M )
43negeqd 9305 . . 3  |-  ( x  =  C  ->  -u A  =  -u M )
5 ltord.3 . . . 4  |-  ( x  =  D  ->  A  =  N )
65negeqd 9305 . . 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 9469 . . 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 2791 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
121eleq1d 2504 . . . . . . . 8  |-  ( x  =  y  ->  ( A  e.  RR  <->  B  e.  RR ) )
1312rspccva 3053 . . . . . . 7  |-  ( ( A. x  e.  S  A  e.  RR  /\  y  e.  S )  ->  B  e.  RR )
1411, 13sylan 459 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  B  e.  RR )
1514adantrl 698 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  B  e.  RR )
168adantrr 699 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  A  e.  RR )
17 ltneg 9533 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  -u A  <  -u B
) )
1815, 16, 17syl2anc 644 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( B  <  A  <->  -u A  <  -u B
) )
1910, 18sylibd 207 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  -> 
-u A  <  -u B
) )
202, 4, 6, 7, 9, 19eqord1 9560 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <->  -u M  =  -u N
) )
213eleq1d 2504 . . . . . . 7  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
2221rspccva 3053 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
2311, 22sylan 459 . . . . 5  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
2423adantrr 699 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  RR )
2524recnd 9119 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  CC )
265eleq1d 2504 . . . . . . 7  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2726rspccva 3053 . . . . . 6  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2811, 27sylan 459 . . . . 5  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2928adantrl 698 . . . 4  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  RR )
3029recnd 9119 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  CC )
3125, 30neg11ad 9412 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( -u M  =  -u N 
<->  M  =  N ) )
3220, 31bitrd 246 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <-> 
M  =  N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   class class class wbr 4215   RRcr 8994    < clt 9125   -ucneg 9297
This theorem is referenced by:  basellem4  20871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-po 4506  df-so 4507  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-er 6908  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299
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