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Theorem infcvgaux1i 12523
Description: Auxiliary theorem for applications of supcvg 12522. 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 9357 . . . . . 6  |-  ( y  e.  X  ->  -u A  e.  RR )
4 eleq1 2426 . . . . . 6  |-  ( x  =  -u A  ->  (
x  e.  RR  <->  -u A  e.  RR ) )
53, 4syl5ibrcom 213 . . . . 5  |-  ( y  e.  X  ->  (
x  =  -u A  ->  x  e.  RR ) )
65rexlimiv 2746 . . . 4  |-  ( E. y  e.  X  x  =  -u A  ->  x  e.  RR )
76abssi 3334 . . 3  |-  { x  |  E. y  e.  X  x  =  -u A }  C_  RR
81, 7eqsstri 3294 . 2  |-  R  C_  RR
9 infcvg.3 . . . . . 6  |-  Z  e.  X
10 eqid 2366 . . . . . 6  |-  -u [_ Z  /  y ]_ A  =  -u [_ Z  / 
y ]_ A
1110nfth 1558 . . . . . . 7  |-  F/ y
-u [_ Z  /  y ]_ A  =  -u [_ Z  /  y ]_ A
12 csbeq1a 3175 . . . . . . . . 9  |-  ( y  =  Z  ->  A  =  [_ Z  /  y ]_ A )
1312negeqd 9193 . . . . . . . 8  |-  ( y  =  Z  ->  -u A  =  -u [_ Z  / 
y ]_ A )
1413eqeq2d 2377 . . . . . . 7  |-  ( y  =  Z  ->  ( -u
[_ Z  /  y ]_ A  =  -u A  <->  -u
[_ Z  /  y ]_ A  =  -u [_ Z  /  y ]_ A
) )
1511, 14rspce 2964 . . . . . 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 653 . . . . 5  |-  E. y  e.  X  -u [_ Z  /  y ]_ A  =  -u A
17 negex 9197 . . . . . 6  |-  -u [_ Z  /  y ]_ A  e.  _V
18 nfcsb1v 3199 . . . . . . . . 9  |-  F/_ y [_ Z  /  y ]_ A
1918nfneg 9195 . . . . . . . 8  |-  F/_ y -u
[_ Z  /  y ]_ A
2019nfeq2 2513 . . . . . . 7  |-  F/ y  x  =  -u [_ Z  /  y ]_ A
21 eqeq1 2372 . . . . . . 7  |-  ( x  =  -u [_ Z  / 
y ]_ A  ->  (
x  =  -u A  <->  -u
[_ Z  /  y ]_ A  =  -u A
) )
2220, 21rexbid 2647 . . . . . 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 2999 . . . . 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 200 . . . 4  |-  -u [_ Z  /  y ]_ A  e.  { x  |  E. y  e.  X  x  =  -u A }
2524, 1eleqtrri 2439 . . 3  |-  -u [_ Z  /  y ]_ A  e.  R
26 ne0i 3549 . . 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 1131 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 935    = wceq 1647    e. wcel 1715   {cab 2352    =/= wne 2529   A.wral 2628   E.wrex 2629   [_csb 3167    C_ wss 3238   (/)c0 3543   class class class wbr 4125   RRcr 8883    <_ cle 9015   -ucneg 9185
This theorem is referenced by:  infcvgaux2i  12524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-ltxr 9019  df-sub 9186  df-neg 9187
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