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Theorem riotaneg 10558
Description: The negative of the unique real such that  ph. (Contributed by NM, 13-Jun-2005.)
Hypothesis
Ref Expression
riotaneg.1  |-  ( x  =  -u y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
riotaneg  |-  ( E! x  e.  RR  ph  ->  ( iota_ x  e.  RR  ph )  =  -u ( iota_ y  e.  RR  ps ) )
Distinct variable groups:    x, y    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem riotaneg
StepHypRef Expression
1 tru 1409 . 2  |- T.
2 nfriota1 6247 . . . 4  |-  F/_ y
( iota_ y  e.  RR  ps )
32nfneg 9852 . . 3  |-  F/_ y -u ( iota_ y  e.  RR  ps )
4 renegcl 9918 . . . 4  |-  ( y  e.  RR  ->  -u y  e.  RR )
54adantl 464 . . 3  |-  ( ( T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 simpr 459 . . . 4  |-  ( ( T.  /\  ( iota_ y  e.  RR  ps )  e.  RR )  ->  ( iota_ y  e.  RR  ps )  e.  RR )
76renegcld 10027 . . 3  |-  ( ( T.  /\  ( iota_ y  e.  RR  ps )  e.  RR )  ->  -u ( iota_ y  e.  RR  ps )  e.  RR )
8 riotaneg.1 . . 3  |-  ( x  =  -u y  ->  ( ph 
<->  ps ) )
9 negeq 9848 . . 3  |-  ( y  =  ( iota_ y  e.  RR  ps )  ->  -u y  =  -u ( iota_ y  e.  RR  ps ) )
10 renegcl 9918 . . . . 5  |-  ( x  e.  RR  ->  -u x  e.  RR )
11 recn 9612 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  CC )
12 recn 9612 . . . . . 6  |-  ( y  e.  RR  ->  y  e.  CC )
13 negcon2 9908 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1411, 12, 13syl2an 475 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1510, 14reuhyp 4619 . . . 4  |-  ( x  e.  RR  ->  E! y  e.  RR  x  =  -u y )
1615adantl 464 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  E! y  e.  RR  x  =  -u y )
173, 5, 7, 8, 9, 16riotaxfrd 6270 . 2  |-  ( ( T.  /\  E! x  e.  RR  ph )  -> 
( iota_ x  e.  RR  ph )  =  -u ( iota_ y  e.  RR  ps ) )
181, 17mpan 668 1  |-  ( E! x  e.  RR  ph  ->  ( iota_ x  e.  RR  ph )  =  -u ( iota_ y  e.  RR  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405   T. wtru 1406    e. wcel 1842   E!wreu 2756   iota_crio 6239   CCcc 9520   RRcr 9521   -ucneg 9842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-ltxr 9663  df-sub 9843  df-neg 9844
This theorem is referenced by: (None)
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