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Theorem infcvgaux2i 13623
Description: Auxiliary theorem for applications of supcvg 13621. (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 2462 . . . . . 6  |-  -u B  =  -u B
3 infcvg.13 . . . . . . . . 9  |-  ( y  =  C  ->  A  =  B )
43negeqd 9805 . . . . . . . 8  |-  ( y  =  C  ->  -u A  =  -u B )
54eqeq2d 2476 . . . . . . 7  |-  ( y  =  C  ->  ( -u B  =  -u A  <->  -u B  =  -u B
) )
65rspcev 3209 . . . . . 6  |-  ( ( C  e.  X  /\  -u B  =  -u B
)  ->  E. y  e.  X  -u B  = 
-u A )
72, 6mpan2 671 . . . . 5  |-  ( C  e.  X  ->  E. y  e.  X  -u B  = 
-u A )
8 negex 9809 . . . . . 6  |-  -u B  e.  _V
9 eqeq1 2466 . . . . . . 7  |-  ( x  =  -u B  ->  (
x  =  -u A  <->  -u B  =  -u A
) )
109rexbidv 2968 . . . . . 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 3248 . . . . 5  |-  ( -u B  e.  R  <->  E. y  e.  X  -u B  = 
-u A )
137, 12sylibr 212 . . . 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 13622 . . . . 5  |-  ( R 
C_  RR  /\  R  =/=  (/)  /\  E. z  e.  RR  A. w  e.  R  w  <_  z
)
1817suprubii 10505 . . . 4  |-  ( -u B  e.  R  ->  -u B  <_  sup ( R ,  RR ,  <  ) )
1913, 18syl 16 . . 3  |-  ( C  e.  X  ->  -u B  <_  sup ( R ,  RR ,  <  ) )
203eleq1d 2531 . . . . 5  |-  ( y  =  C  ->  ( A  e.  RR  <->  B  e.  RR ) )
2120, 14vtoclga 3172 . . . 4  |-  ( C  e.  X  ->  B  e.  RR )
2217suprclii 10504 . . . 4  |-  sup ( R ,  RR ,  <  )  e.  RR
23 lenegcon1 10047 . . . 4  |-  ( ( B  e.  RR  /\  sup ( R ,  RR ,  <  )  e.  RR )  ->  ( -u B  <_  sup ( R ,  RR ,  <  )  <->  -u sup ( R ,  RR ,  <  )  <_  B )
)
2421, 22, 23sylancl 662 . . 3  |-  ( C  e.  X  ->  ( -u B  <_  sup ( R ,  RR ,  <  )  <->  -u sup ( R ,  RR ,  <  )  <_  B ) )
2519, 24mpbid 210 . 2  |-  ( C  e.  X  ->  -u sup ( R ,  RR ,  <  )  <_  B )
261, 25syl5eqbr 4475 1  |-  ( C  e.  X  ->  S  <_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374    e. wcel 1762   {cab 2447   A.wral 2809   E.wrex 2810   class class class wbr 4442   supcsup 7891   RRcr 9482    < clt 9619    <_ cle 9620   -ucneg 9797
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 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799
This theorem is referenced by: (None)
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