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Theorem infcvgaux1i 12591
Description: Auxiliary theorem for applications of supcvg 12590. Hypothesis for several supremum theorems. (Contributed by NM, 8-Feb-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
Assertion
Ref Expression
infcvgaux1i  |-  ( R 
C_  RR  /\  R  =/=  (/)  /\  E. z  e.  RR  A. w  e.  R  w  <_  z
)
Distinct variable groups:    x, A    x, y    z, w, R   
x, X, y    x, Z, y
Allowed substitution hints:    A( y, z, w)    R( x, y)    X( z, w)    Z( z, w)

Proof of Theorem infcvgaux1i
StepHypRef Expression
1 infcvg.1 . . 3  |-  R  =  { x  |  E. y  e.  X  x  =  -u A }
2 infcvg.2 . . . . . . 7  |-  ( y  e.  X  ->  A  e.  RR )
32renegcld 9420 . . . . . 6  |-  ( y  e.  X  ->  -u A  e.  RR )
4 eleq1 2464 . . . . . 6  |-  ( x  =  -u A  ->  (
x  e.  RR  <->  -u A  e.  RR ) )
53, 4syl5ibrcom 214 . . . . 5  |-  ( y  e.  X  ->  (
x  =  -u A  ->  x  e.  RR ) )
65rexlimiv 2784 . . . 4  |-  ( E. y  e.  X  x  =  -u A  ->  x  e.  RR )
76abssi 3378 . . 3  |-  { x  |  E. y  e.  X  x  =  -u A }  C_  RR
81, 7eqsstri 3338 . 2  |-  R  C_  RR
9 infcvg.3 . . . . . 6  |-  Z  e.  X
10 eqid 2404 . . . . . 6  |-  -u [_ Z  /  y ]_ A  =  -u [_ Z  / 
y ]_ A
1110nfth 1559 . . . . . . 7  |-  F/ y
-u [_ Z  /  y ]_ A  =  -u [_ Z  /  y ]_ A
12 csbeq1a 3219 . . . . . . . . 9  |-  ( y  =  Z  ->  A  =  [_ Z  /  y ]_ A )
1312negeqd 9256 . . . . . . . 8  |-  ( y  =  Z  ->  -u A  =  -u [_ Z  / 
y ]_ A )
1413eqeq2d 2415 . . . . . . 7  |-  ( y  =  Z  ->  ( -u
[_ Z  /  y ]_ A  =  -u A  <->  -u
[_ Z  /  y ]_ A  =  -u [_ Z  /  y ]_ A
) )
1511, 14rspce 3007 . . . . . 6  |-  ( ( Z  e.  X  /\  -u
[_ Z  /  y ]_ A  =  -u [_ Z  /  y ]_ A
)  ->  E. y  e.  X  -u [_ Z  /  y ]_ A  =  -u A )
169, 10, 15mp2an 654 . . . . 5  |-  E. y  e.  X  -u [_ Z  /  y ]_ A  =  -u A
17 negex 9260 . . . . . 6  |-  -u [_ Z  /  y ]_ A  e.  _V
18 nfcsb1v 3243 . . . . . . . . 9  |-  F/_ y [_ Z  /  y ]_ A
1918nfneg 9258 . . . . . . . 8  |-  F/_ y -u
[_ Z  /  y ]_ A
2019nfeq2 2551 . . . . . . 7  |-  F/ y  x  =  -u [_ Z  /  y ]_ A
21 eqeq1 2410 . . . . . . 7  |-  ( x  =  -u [_ Z  / 
y ]_ A  ->  (
x  =  -u A  <->  -u
[_ Z  /  y ]_ A  =  -u A
) )
2220, 21rexbid 2685 . . . . . 6  |-  ( x  =  -u [_ Z  / 
y ]_ A  ->  ( E. y  e.  X  x  =  -u A  <->  E. y  e.  X  -u [_ Z  /  y ]_ A  =  -u A ) )
2317, 22elab 3042 . . . . 5  |-  ( -u [_ Z  /  y ]_ A  e.  { x  |  E. y  e.  X  x  =  -u A }  <->  E. y  e.  X  -u [_ Z  /  y ]_ A  =  -u A )
2416, 23mpbir 201 . . . 4  |-  -u [_ Z  /  y ]_ A  e.  { x  |  E. y  e.  X  x  =  -u A }
2524, 1eleqtrri 2477 . . 3  |-  -u [_ Z  /  y ]_ A  e.  R
26 ne0i 3594 . . 3  |-  ( -u [_ Z  /  y ]_ A  e.  R  ->  R  =/=  (/) )
2725, 26ax-mp 8 . 2  |-  R  =/=  (/)
28 infcvg.4 . 2  |-  E. z  e.  RR  A. w  e.  R  w  <_  z
298, 27, 283pm3.2i 1132 1  |-  ( R 
C_  RR  /\  R  =/=  (/)  /\  E. z  e.  RR  A. w  e.  R  w  <_  z
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   A.wral 2666   E.wrex 2667   [_csb 3211    C_ wss 3280   (/)c0 3588   class class class wbr 4172   RRcr 8945    <_ cle 9077   -ucneg 9248
This theorem is referenced by:  infcvgaux2i  12592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250
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