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Theorem riotaneg 10409
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 1374 . 2  |- T.
2 nfriota1 6161 . . . 4  |-  F/_ y
( iota_ y  e.  RR  ps )
32nfneg 9710 . . 3  |-  F/_ y -u ( iota_ y  e.  RR  ps )
4 renegcl 9776 . . . 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 9879 . . 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 9706 . . 3  |-  ( y  =  ( iota_ y  e.  RR  ps )  ->  -u y  =  -u ( iota_ y  e.  RR  ps ) )
10 renegcl 9776 . . . . 5  |-  ( x  e.  RR  ->  -u x  e.  RR )
11 recn 9476 . . . . . 6  |-  ( x  e.  RR  ->  x  e.  CC )
12 recn 9476 . . . . . 6  |-  ( y  e.  RR  ->  y  e.  CC )
13 negcon2 9766 . . . . . 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 4621 . . . 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 6185 . 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 1370   T. wtru 1371    e. wcel 1758   E!wreu 2797   iota_crio 6153   CCcc 9384   RRcr 9385   -ucneg 9700
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-ltxr 9527  df-sub 9701  df-neg 9702
This theorem is referenced by: (None)
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