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Theorem infcvgaux2i 12332
Description: Auxiliary theorem for applications of supcvg 12330. (Contributed by NM, 4-Mar-2008.)
Hypotheses
Ref Expression
infcvg.1  |-  R  =  { x  |  E. y  e.  X  x  =  -u A }
infcvg.2  |-  ( y  e.  X  ->  A  e.  RR )
infcvg.3  |-  Z  e.  X
infcvg.4  |-  E. z  e.  RR  A. w  e.  R  w  <_  z
infcvg.5a  |-  S  = 
-u sup ( R ,  RR ,  <  )
infcvg.13  |-  ( y  =  C  ->  A  =  B )
Assertion
Ref Expression
infcvgaux2i  |-  ( C  e.  X  ->  S  <_  B )
Distinct variable groups:    x, A    x, y, B    z, w, R    x, X, y    x, Z, y    y, C
Allowed substitution hints:    A( y, z, w)    B( z, w)    C( x, z, w)    R( x, y)    S( x, y, z, w)    X( z, w)    Z( z, w)

Proof of Theorem infcvgaux2i
StepHypRef Expression
1 infcvg.5a . 2  |-  S  = 
-u sup ( R ,  RR ,  <  )
2 eqid 2296 . . . . . 6  |-  -u B  =  -u B
3 infcvg.13 . . . . . . . . 9  |-  ( y  =  C  ->  A  =  B )
43negeqd 9062 . . . . . . . 8  |-  ( y  =  C  ->  -u A  =  -u B )
54eqeq2d 2307 . . . . . . 7  |-  ( y  =  C  ->  ( -u B  =  -u A  <->  -u B  =  -u B
) )
65rspcev 2897 . . . . . 6  |-  ( ( C  e.  X  /\  -u B  =  -u B
)  ->  E. y  e.  X  -u B  = 
-u A )
72, 6mpan2 652 . . . . 5  |-  ( C  e.  X  ->  E. y  e.  X  -u B  = 
-u A )
8 negex 9066 . . . . . 6  |-  -u B  e.  _V
9 eqeq1 2302 . . . . . . 7  |-  ( x  =  -u B  ->  (
x  =  -u A  <->  -u B  =  -u A
) )
109rexbidv 2577 . . . . . 6  |-  ( x  =  -u B  ->  ( E. y  e.  X  x  =  -u A  <->  E. y  e.  X  -u B  = 
-u A ) )
11 infcvg.1 . . . . . 6  |-  R  =  { x  |  E. y  e.  X  x  =  -u A }
128, 10, 11elab2 2930 . . . . 5  |-  ( -u B  e.  R  <->  E. y  e.  X  -u B  = 
-u A )
137, 12sylibr 203 . . . 4  |-  ( C  e.  X  ->  -u B  e.  R )
14 infcvg.2 . . . . . 6  |-  ( y  e.  X  ->  A  e.  RR )
15 infcvg.3 . . . . . 6  |-  Z  e.  X
16 infcvg.4 . . . . . 6  |-  E. z  e.  RR  A. w  e.  R  w  <_  z
1711, 14, 15, 16infcvgaux1i 12331 . . . . 5  |-  ( R 
C_  RR  /\  R  =/=  (/)  /\  E. z  e.  RR  A. w  e.  R  w  <_  z
)
1817suprubii 9741 . . . 4  |-  ( -u B  e.  R  ->  -u B  <_  sup ( R ,  RR ,  <  ) )
1913, 18syl 15 . . 3  |-  ( C  e.  X  ->  -u B  <_  sup ( R ,  RR ,  <  ) )
203eleq1d 2362 . . . . 5  |-  ( y  =  C  ->  ( A  e.  RR  <->  B  e.  RR ) )
2120, 14vtoclga 2862 . . . 4  |-  ( C  e.  X  ->  B  e.  RR )
2217suprclii 9740 . . . 4  |-  sup ( R ,  RR ,  <  )  e.  RR
23 lenegcon1 9294 . . . 4  |-  ( ( B  e.  RR  /\  sup ( R ,  RR ,  <  )  e.  RR )  ->  ( -u B  <_  sup ( R ,  RR ,  <  )  <->  -u sup ( R ,  RR ,  <  )  <_  B )
)
2421, 22, 23sylancl 643 . . 3  |-  ( C  e.  X  ->  ( -u B  <_  sup ( R ,  RR ,  <  )  <->  -u sup ( R ,  RR ,  <  )  <_  B ) )
2519, 24mpbid 201 . 2  |-  ( C  e.  X  ->  -u sup ( R ,  RR ,  <  )  <_  B )
261, 25syl5eqbr 4072 1  |-  ( C  e.  X  ->  S  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   class class class wbr 4039   supcsup 7209   RRcr 8752    < clt 8883    <_ cle 8884   -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  ax-pre-mulgt0 8830  ax-pre-sup 8831
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-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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