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Theorem leord2 9348
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
leord2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  N  <_  M ) )
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 leord2
StepHypRef Expression
1 ltord.1 . . . 4  |-  ( x  =  y  ->  A  =  B )
21negeqd 9091 . . 3  |-  ( x  =  y  ->  -u A  =  -u B )
3 ltord.2 . . . 4  |-  ( x  =  C  ->  A  =  M )
43negeqd 9091 . . 3  |-  ( x  =  C  ->  -u A  =  -u M )
5 ltord.3 . . . 4  |-  ( x  =  D  ->  A  =  N )
65negeqd 9091 . . 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 9255 . . 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 2660 . . . . . . 7  |-  ( ph  ->  A. x  e.  S  A  e.  RR )
121eleq1d 2382 . . . . . . . 8  |-  ( x  =  y  ->  ( A  e.  RR  <->  B  e.  RR ) )
1312rspccva 2917 . . . . . . 7  |-  ( ( A. x  e.  S  A  e.  RR  /\  y  e.  S )  ->  B  e.  RR )
1411, 13sylan 457 . . . . . 6  |-  ( (
ph  /\  y  e.  S )  ->  B  e.  RR )
1514adantrl 696 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  B  e.  RR )
168adantrr 697 . . . . 5  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  ->  A  e.  RR )
17 ltneg 9319 . . . . 5  |-  ( ( B  e.  RR  /\  A  e.  RR )  ->  ( B  <  A  <->  -u A  <  -u B
) )
1815, 16, 17syl2anc 642 . . . 4  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( B  <  A  <->  -u A  <  -u B
) )
1910, 18sylibd 205 . . 3  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  <  y  -> 
-u A  <  -u B
) )
202, 4, 6, 7, 9, 19leord1 9345 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  -u M  <_  -u N ) )
215eleq1d 2382 . . . . . 6  |-  ( x  =  D  ->  ( A  e.  RR  <->  N  e.  RR ) )
2221rspccva 2917 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  D  e.  S )  ->  N  e.  RR )
2311, 22sylan 457 . . . 4  |-  ( (
ph  /\  D  e.  S )  ->  N  e.  RR )
2423adantrl 696 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  N  e.  RR )
253eleq1d 2382 . . . . . 6  |-  ( x  =  C  ->  ( A  e.  RR  <->  M  e.  RR ) )
2625rspccva 2917 . . . . 5  |-  ( ( A. x  e.  S  A  e.  RR  /\  C  e.  S )  ->  M  e.  RR )
2711, 26sylan 457 . . . 4  |-  ( (
ph  /\  C  e.  S )  ->  M  e.  RR )
2827adantrr 697 . . 3  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  ->  M  e.  RR )
29 leneg 9322 . . 3  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( N  <_  M  <->  -u M  <_  -u N ) )
3024, 28, 29syl2anc 642 . 2  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( N  <_  M  <->  -u M  <_  -u N ) )
3120, 30bitr4d 247 1  |-  ( (
ph  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( C  <_  D  <->  N  <_  M ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   A.wral 2577    C_ wss 3186   class class class wbr 4060   RRcr 8781    < clt 8912    <_ cle 8913   -ucneg 9083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-po 4351  df-so 4352  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-riota 6346  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085
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