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Theorem eqord2 10105
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 9833 . . 3  |-  ( x  =  y  ->  -u A  =  -u B )
3 ltord.2 . . . 4  |-  ( x  =  C  ->  A  =  M )
43negeqd 9833 . . 3  |-  ( x  =  C  ->  -u A  =  -u M )
5 ltord.3 . . . 4  |-  ( x  =  D  ->  A  =  N )
65negeqd 9833 . . 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 10007 . . 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 2871 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
121eleq1d 2526 . . . . . . . 8  |-  ( x  =  y  ->  ( A  e.  RR  <->  B  e.  RR ) )
1312rspccva 3209 . . . . . . 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 10073 . . . . 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 10102 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  =  D  <->  -u M  =  -u N
) )
213eleq1d 2526 . . . . . . 7  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
2221rspccva 3209 . . . . . 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 9639 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  CC )
265eleq1d 2526 . . . . . . 7  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2726rspccva 3209 . . . . . 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 9639 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  CC )
3125, 30neg11ad 9946 . 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 1395    e. wcel 1819   A.wral 2807    C_ wss 3471   class class class wbr 4456   RRcr 9508    < clt 9645   -ucneg 9825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827
This theorem is referenced by:  basellem4  23482
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