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Theorem riotaneg 9745
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 1312 . 2  |-  T.
2 nfriota1 6328 . . . 4  |-  F/_ y
( iota_ y  e.  RR ps )
32nfneg 9064 . . 3  |-  F/_ y -u ( iota_ y  e.  RR ps )
4 renegcl 9126 . . . 4  |-  ( y  e.  RR  ->  -u y  e.  RR )
54adantl 452 . . 3  |-  ( (  T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 simpr 447 . . . 4  |-  ( (  T.  /\  ( iota_ y  e.  RR ps )  e.  RR )  ->  ( iota_ y  e.  RR ps )  e.  RR )
76renegcld 9226 . . 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 9060 . . 3  |-  ( y  =  ( iota_ y  e.  RR ps )  ->  -u y  =  -u ( iota_ y  e.  RR ps ) )
10 renegcl 9126 . . . . 5  |-  ( x  e.  RR  ->  -u x  e.  RR )
11 recn 8843 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  CC )
12 recn 8843 . . . . . 6  |-  ( y  e.  RR  ->  y  e.  CC )
13 negcon2 9116 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1411, 12, 13syl2an 463 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1510, 14reuhyp 4578 . . . 4  |-  ( x  e.  RR  ->  E! y  e.  RR  x  =  -u y )
1615adantl 452 . . 3  |-  ( (  T.  /\  x  e.  RR )  ->  E! y  e.  RR  x  =  -u y )
173, 5, 7, 8, 9, 16riotaxfrd 6352 . 2  |-  ( (  T.  /\  E! x  e.  RR  ph )  -> 
( iota_ x  e.  RR ph )  =  -u ( iota_ y  e.  RR ps ) )
181, 17mpan 651 1  |-  ( E! x  e.  RR  ph  ->  ( iota_ x  e.  RR ph )  =  -u ( iota_ y  e.  RR ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    T. wtru 1307    = wceq 1632    e. wcel 1696   E!wreu 2558   iota_crio 6313   CCcc 8751   RRcr 8752   -ucneg 9054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056
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