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Theorem riotaneg 10507
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 1378 . 2  |- T.
2 nfriota1 6243 . . . 4  |-  F/_ y
( iota_ y  e.  RR  ps )
32nfneg 9805 . . 3  |-  F/_ y -u ( iota_ y  e.  RR  ps )
4 renegcl 9871 . . . 4  |-  ( y  e.  RR  ->  -u y  e.  RR )
54adantl 466 . . 3  |-  ( ( T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 simpr 461 . . . 4  |-  ( ( T.  /\  ( iota_ y  e.  RR  ps )  e.  RR )  ->  ( iota_ y  e.  RR  ps )  e.  RR )
76renegcld 9975 . . 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 9801 . . 3  |-  ( y  =  ( iota_ y  e.  RR  ps )  ->  -u y  =  -u ( iota_ y  e.  RR  ps ) )
10 renegcl 9871 . . . . 5  |-  ( x  e.  RR  ->  -u x  e.  RR )
11 recn 9571 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  CC )
12 recn 9571 . . . . . 6  |-  ( y  e.  RR  ->  y  e.  CC )
13 negcon2 9861 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1411, 12, 13syl2an 477 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1510, 14reuhyp 4668 . . . 4  |-  ( x  e.  RR  ->  E! y  e.  RR  x  =  -u y )
1615adantl 466 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  E! y  e.  RR  x  =  -u y )
173, 5, 7, 8, 9, 16riotaxfrd 6267 . 2  |-  ( ( T.  /\  E! x  e.  RR  ph )  -> 
( iota_ x  e.  RR  ph )  =  -u ( iota_ y  e.  RR  ps ) )
181, 17mpan 670 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 369    = wceq 1374   T. wtru 1375    e. wcel 1762   E!wreu 2809   iota_crio 6235   CCcc 9479   RRcr 9480   -ucneg 9795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9796  df-neg 9797
This theorem is referenced by: (None)
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