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Theorem esumpfinvallem 26459
Description: Lemma for esumpfinval 26460 (Contributed by Thierry Arnoux, 28-Jun-2017.)
Assertion
Ref Expression
esumpfinvallem  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (fld 
gsumg  F )  =  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  F ) )

Proof of Theorem esumpfinvallem
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fex 5947 . . . 4  |-  ( ( F : A --> ( 0 [,) +oo )  /\  A  e.  V )  ->  F  e.  _V )
21ancoms 450 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  F  e.  _V )
3 ovex 6115 . . . 4  |-  (flds  ( 0 [,) +oo ) )  e.  _V
43a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (flds  ( 0 [,) +oo )
)  e.  _V )
5 ovex 6115 . . . 4  |-  ( RR*ss  ( 0 [,) +oo ) )  e.  _V
65a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( RR*ss  (
0 [,) +oo )
)  e.  _V )
7 mnfxr 11090 . . . . . . . . 9  |- -oo  e.  RR*
8 pnfxr 11088 . . . . . . . . 9  |- +oo  e.  RR*
9 mnflt0 11101 . . . . . . . . 9  |- -oo  <  0
10 pnfge 11106 . . . . . . . . . 10  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
118, 10ax-mp 5 . . . . . . . . 9  |- +oo  <_ +oo
12 icossioo 25995 . . . . . . . . 9  |-  ( ( ( -oo  e.  RR*  /\ +oo  e.  RR* )  /\  ( -oo  <  0  /\ +oo  <_ +oo ) )  -> 
( 0 [,) +oo )  C_  ( -oo (,) +oo ) )
137, 8, 9, 11, 12mp4an 668 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  ( -oo (,) +oo )
14 ioomax 11366 . . . . . . . 8  |-  ( -oo (,) +oo )  =  RR
1513, 14sseqtri 3385 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
16 ax-resscn 9335 . . . . . . 7  |-  RR  C_  CC
1715, 16sstri 3362 . . . . . 6  |-  ( 0 [,) +oo )  C_  CC
18 eqid 2441 . . . . . . 7  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
19 cnfldbas 17781 . . . . . . 7  |-  CC  =  ( Base ` fld )
2018, 19ressbas2 14225 . . . . . 6  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
2117, 20ax-mp 5 . . . . 5  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
22 icossxr 11376 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR*
23 eqid 2441 . . . . . . 7  |-  ( RR*ss  ( 0 [,) +oo ) )  =  (
RR*ss  ( 0 [,) +oo ) )
24 xrsbas 17791 . . . . . . 7  |-  RR*  =  ( Base `  RR*s )
2523, 24ressbas2 14225 . . . . . 6  |-  ( ( 0 [,) +oo )  C_ 
RR*  ->  ( 0 [,) +oo )  =  ( Base `  ( RR*ss  (
0 [,) +oo )
) ) )
2622, 25ax-mp 5 . . . . 5  |-  ( 0 [,) +oo )  =  ( Base `  ( RR*ss  ( 0 [,) +oo ) ) )
2721, 26eqtr3i 2463 . . . 4  |-  ( Base `  (flds  ( 0 [,) +oo )
) )  =  (
Base `  ( RR*ss  ( 0 [,) +oo ) ) )
2827a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( Base `  (flds  ( 0 [,) +oo ) ) )  =  ( Base `  ( RR*ss  ( 0 [,) +oo ) ) ) )
29 simprl 750 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  /\  y  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  ->  x  e.  ( Base `  (flds  ( 0 [,) +oo )
) ) )
3029, 21syl6eleqr 2532 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  /\  y  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  ->  x  e.  ( 0 [,) +oo ) )
31 simprr 751 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  /\  y  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  -> 
y  e.  ( Base `  (flds  ( 0 [,) +oo )
) ) )
3231, 21syl6eleqr 2532 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  /\  y  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  -> 
y  e.  ( 0 [,) +oo ) )
33 ge0addcl 11393 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  e.  ( 0 [,) +oo )
)
34 ovex 6115 . . . . . . 7  |-  ( 0 [,) +oo )  e. 
_V
35 cnfldadd 17782 . . . . . . . 8  |-  +  =  ( +g  ` fld )
3618, 35ressplusg 14276 . . . . . . 7  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
3734, 36ax-mp 5 . . . . . 6  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
3837oveqi 6103 . . . . 5  |-  ( x  +  y )  =  ( x ( +g  `  (flds  ( 0 [,) +oo )
) ) y )
3933, 38, 213eltr3g 2523 . . . 4  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x ( +g  `  (flds  ( 0 [,) +oo ) ) ) y )  e.  ( Base `  (flds  ( 0 [,) +oo )
) ) )
4030, 32, 39syl2anc 656 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  /\  y  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  -> 
( x ( +g  `  (flds  ( 0 [,) +oo )
) ) y )  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
41 simpl 454 . . . . . . 7  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  x  e.  ( 0 [,) +oo )
)
4215, 41sseldi 3351 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  x  e.  RR )
43 simpr 458 . . . . . . 7  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  y  e.  ( 0 [,) +oo )
)
4415, 43sseldi 3351 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  y  e.  RR )
45 rexadd 11198 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x +e
y )  =  ( x  +  y ) )
4645eqcomd 2446 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  =  ( x +e y ) )
4742, 44, 46syl2anc 656 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  =  ( x +e y ) )
48 xrsadd 17792 . . . . . . . 8  |-  +e 
=  ( +g  `  RR*s
)
4923, 48ressplusg 14276 . . . . . . 7  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) )
5034, 49ax-mp 5 . . . . . 6  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )
5150oveqi 6103 . . . . 5  |-  ( x +e y )  =  ( x ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) y )
5247, 38, 513eqtr3g 2496 . . . 4  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x ( +g  `  (flds  ( 0 [,) +oo ) ) ) y )  =  ( x ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) y ) )
5330, 32, 52syl2anc 656 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  /\  y  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  -> 
( x ( +g  `  (flds  ( 0 [,) +oo )
) ) y )  =  ( x ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) y ) )
54 simpr 458 . . . 4  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  F : A --> ( 0 [,) +oo ) )
55 ffun 5558 . . . 4  |-  ( F : A --> ( 0 [,) +oo )  ->  Fun  F )
5654, 55syl 16 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  Fun  F )
57 frn 5562 . . . . 5  |-  ( F : A --> ( 0 [,) +oo )  ->  ran  F  C_  ( 0 [,) +oo ) )
5854, 57syl 16 . . . 4  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ran  F  C_  (
0 [,) +oo )
)
5958, 21syl6sseq 3399 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ran  F  C_  ( Base `  (flds  ( 0 [,) +oo )
) ) )
602, 4, 6, 28, 40, 53, 56, 59gsumpropd2 26178 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( (flds  ( 0 [,) +oo ) )  gsumg  F )  =  ( ( RR*ss  ( 0 [,) +oo ) ) 
gsumg  F ) )
61 cnfldex 17780 . . . 4  |-fld  e.  _V
6261a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->fld 
e.  _V )
63 simpl 454 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  A  e.  V
)
6417a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( 0 [,) +oo )  C_  CC )
65 0e0icopnf 11391 . . . 4  |-  0  e.  ( 0 [,) +oo )
6665a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  0  e.  ( 0 [,) +oo )
)
67 simpr 458 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  x  e.  CC )
6867addid2d 9566 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  ( 0  +  x )  =  x )
6967addid1d 9565 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  ( x  + 
0 )  =  x )
7068, 69jca 529 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  ( ( 0  +  x )  =  x  /\  ( x  +  0 )  =  x ) )
7119, 35, 18, 62, 63, 64, 54, 66, 70gsumress 15500 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (fld 
gsumg  F )  =  ( (flds  ( 0 [,) +oo )
)  gsumg  F ) )
72 xrge0base 26079 . . 3  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
73 xrge0plusg 26081 . . 3  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
74 ovex 6115 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
75 ressress 14231 . . . . 5  |-  ( ( ( 0 [,] +oo )  e.  _V  /\  (
0 [,) +oo )  e.  _V )  ->  (
( RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )  =  (
RR*ss  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) ) ) )
7674, 34, 75mp2an 667 . . . 4  |-  ( (
RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )  =  (
RR*ss  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) ) )
77 incom 3540 . . . . . 6  |-  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) )  =  ( ( 0 [,) +oo )  i^i  ( 0 [,] +oo ) )
78 icossicc 25991 . . . . . . 7  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
79 dfss 3340 . . . . . . 7  |-  ( ( 0 [,) +oo )  C_  ( 0 [,] +oo ) 
<->  ( 0 [,) +oo )  =  ( (
0 [,) +oo )  i^i  ( 0 [,] +oo ) ) )
8078, 79mpbi 208 . . . . . 6  |-  ( 0 [,) +oo )  =  ( ( 0 [,) +oo )  i^i  (
0 [,] +oo )
)
8177, 80eqtr4i 2464 . . . . 5  |-  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) )  =  ( 0 [,) +oo )
8281oveq2i 6101 . . . 4  |-  ( RR*ss  ( ( 0 [,] +oo )  i^i  (
0 [,) +oo )
) )  =  (
RR*ss  ( 0 [,) +oo ) )
8376, 82eqtr2i 2462 . . 3  |-  ( RR*ss  ( 0 [,) +oo ) )  =  ( ( RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo )
)
84 ovex 6115 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  _V
8584a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( RR*ss  (
0 [,] +oo )
)  e.  _V )
8678a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )
87 iccssxr 11374 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
88 simpr 458 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  x  e.  ( 0 [,] +oo ) )
8987, 88sseldi 3351 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  x  e.  RR* )
90 xaddid2 11206 . . . . 5  |-  ( x  e.  RR*  ->  ( 0 +e x )  =  x )
9189, 90syl 16 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  ( 0 +e x )  =  x )
92 xaddid1 11205 . . . . 5  |-  ( x  e.  RR*  ->  ( x +e 0 )  =  x )
9389, 92syl 16 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  ( x +e 0 )  =  x )
9491, 93jca 529 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  ( (
0 +e x )  =  x  /\  ( x +e 0 )  =  x ) )
9572, 73, 83, 85, 63, 86, 54, 66, 94gsumress 15500 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  F )  =  ( ( RR*ss  ( 0 [,) +oo ) ) 
gsumg  F ) )
9660, 71, 953eqtr4d 2483 1  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (fld 
gsumg  F )  =  ( ( RR*ss  ( 0 [,] +oo ) ) 
gsumg  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761   _Vcvv 2970    i^i cin 3324    C_ wss 3325   class class class wbr 4289   ran crn 4837   Fun wfun 5409   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278    + caddc 9281   +oocpnf 9411   -oocmnf 9412   RR*cxr 9413    < clt 9414    <_ cle 9415   +ecxad 11083   (,)cioo 11296   [,)cico 11298   [,]cicc 11299   Basecbs 14170   ↾s cress 14171   +g cplusg 14234    gsumg cgsu 14375   RR*scxrs 14434  ℂfldccnfld 17777
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-addf 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-oadd 6920  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-xadd 11086  df-ioo 11300  df-ico 11302  df-icc 11303  df-fz 11434  df-seq 11803  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-0g 14376  df-gsum 14377  df-xrs 14436  df-cnfld 17778
This theorem is referenced by:  esumpfinval  26460  esumpfinvalf  26461  esumpcvgval  26463  esumcvg  26471
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