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Theorem riotaneg 10588
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 1442 . 2  |- T.
2 nfriota1 6272 . . . 4  |-  F/_ y
( iota_ y  e.  RR  ps )
32nfneg 9873 . . 3  |-  F/_ y -u ( iota_ y  e.  RR  ps )
4 renegcl 9939 . . . 4  |-  ( y  e.  RR  ->  -u y  e.  RR )
54adantl 468 . . 3  |-  ( ( T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 simpr 463 . . . 4  |-  ( ( T.  /\  ( iota_ y  e.  RR  ps )  e.  RR )  ->  ( iota_ y  e.  RR  ps )  e.  RR )
76renegcld 10048 . . 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 9869 . . 3  |-  ( y  =  ( iota_ y  e.  RR  ps )  ->  -u y  =  -u ( iota_ y  e.  RR  ps ) )
10 renegcl 9939 . . . . 5  |-  ( x  e.  RR  ->  -u x  e.  RR )
11 recn 9631 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  CC )
12 recn 9631 . . . . . 6  |-  ( y  e.  RR  ->  y  e.  CC )
13 negcon2 9929 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1411, 12, 13syl2an 480 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1510, 14reuhyp 4647 . . . 4  |-  ( x  e.  RR  ->  E! y  e.  RR  x  =  -u y )
1615adantl 468 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  E! y  e.  RR  x  =  -u y )
173, 5, 7, 8, 9, 16riotaxfrd 6295 . 2  |-  ( ( T.  /\  E! x  e.  RR  ph )  -> 
( iota_ x  e.  RR  ph )  =  -u ( iota_ y  e.  RR  ps ) )
181, 17mpan 675 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 188    /\ wa 371    = wceq 1438   T. wtru 1439    e. wcel 1869   E!wreu 2778   iota_crio 6264   CCcc 9539   RRcr 9540   -ucneg 9863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-po 4772  df-so 4773  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-ltxr 9682  df-sub 9864  df-neg 9865
This theorem is referenced by: (None)
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