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Theorem riotaneg 10613
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 1458 . 2  |- T.
2 nfriota1 6283 . . . 4  |-  F/_ y
( iota_ y  e.  RR  ps )
32nfneg 9896 . . 3  |-  F/_ y -u ( iota_ y  e.  RR  ps )
4 renegcl 9962 . . . 4  |-  ( y  e.  RR  ->  -u y  e.  RR )
54adantl 472 . . 3  |-  ( ( T.  /\  y  e.  RR )  ->  -u y  e.  RR )
6 simpr 467 . . . 4  |-  ( ( T.  /\  ( iota_ y  e.  RR  ps )  e.  RR )  ->  ( iota_ y  e.  RR  ps )  e.  RR )
76renegcld 10073 . . 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 9892 . . 3  |-  ( y  =  ( iota_ y  e.  RR  ps )  ->  -u y  =  -u ( iota_ y  e.  RR  ps ) )
10 renegcl 9962 . . . . 5  |-  ( x  e.  RR  ->  -u x  e.  RR )
11 recn 9654 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  CC )
12 recn 9654 . . . . . 6  |-  ( y  e.  RR  ->  y  e.  CC )
13 negcon2 9952 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1411, 12, 13syl2an 484 . . . . 5  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  =  -u y 
<->  y  =  -u x
) )
1510, 14reuhyp 4641 . . . 4  |-  ( x  e.  RR  ->  E! y  e.  RR  x  =  -u y )
1615adantl 472 . . 3  |-  ( ( T.  /\  x  e.  RR )  ->  E! y  e.  RR  x  =  -u y )
173, 5, 7, 8, 9, 16riotaxfrd 6306 . 2  |-  ( ( T.  /\  E! x  e.  RR  ph )  -> 
( iota_ x  e.  RR  ph )  =  -u ( iota_ y  e.  RR  ps ) )
181, 17mpan 681 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 189    /\ wa 375    = wceq 1454   T. wtru 1455    e. wcel 1897   E!wreu 2750   iota_crio 6275   CCcc 9562   RRcr 9563   -ucneg 9886
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-po 4773  df-so 4774  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-pnf 9702  df-mnf 9703  df-ltxr 9705  df-sub 9887  df-neg 9888
This theorem is referenced by: (None)
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