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Theorem esumpfinvallem 26538
Description: Lemma for esumpfinval 26539 (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 5965 . . . 4  |-  ( ( F : A --> ( 0 [,) +oo )  /\  A  e.  V )  ->  F  e.  _V )
21ancoms 453 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  F  e.  _V )
3 ovex 6131 . . . 4  |-  (flds  ( 0 [,) +oo ) )  e.  _V
43a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (flds  ( 0 [,) +oo )
)  e.  _V )
5 ovex 6131 . . . 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 11109 . . . . . . . . 9  |- -oo  e.  RR*
8 pnfxr 11107 . . . . . . . . 9  |- +oo  e.  RR*
9 mnflt0 11120 . . . . . . . . 9  |- -oo  <  0
10 pnfge 11125 . . . . . . . . . 10  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
118, 10ax-mp 5 . . . . . . . . 9  |- +oo  <_ +oo
12 icossioo 26077 . . . . . . . . 9  |-  ( ( ( -oo  e.  RR*  /\ +oo  e.  RR* )  /\  ( -oo  <  0  /\ +oo  <_ +oo ) )  -> 
( 0 [,) +oo )  C_  ( -oo (,) +oo ) )
137, 8, 9, 11, 12mp4an 673 . . . . . . . 8  |-  ( 0 [,) +oo )  C_  ( -oo (,) +oo )
14 ioomax 11385 . . . . . . . 8  |-  ( -oo (,) +oo )  =  RR
1513, 14sseqtri 3403 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
16 ax-resscn 9354 . . . . . . 7  |-  RR  C_  CC
1715, 16sstri 3380 . . . . . 6  |-  ( 0 [,) +oo )  C_  CC
18 eqid 2443 . . . . . . 7  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
19 cnfldbas 17837 . . . . . . 7  |-  CC  =  ( Base ` fld )
2018, 19ressbas2 14244 . . . . . 6  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
2117, 20ax-mp 5 . . . . 5  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
22 icossxr 11395 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR*
23 eqid 2443 . . . . . . 7  |-  ( RR*ss  ( 0 [,) +oo ) )  =  (
RR*ss  ( 0 [,) +oo ) )
24 xrsbas 17847 . . . . . . 7  |-  RR*  =  ( Base `  RR*s )
2523, 24ressbas2 14244 . . . . . 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 2465 . . . 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 755 . . . . 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 2534 . . . 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 756 . . . . 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 2534 . . . 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 11412 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  e.  ( 0 [,) +oo )
)
34 ovex 6131 . . . . . . 7  |-  ( 0 [,) +oo )  e. 
_V
35 cnfldadd 17838 . . . . . . . 8  |-  +  =  ( +g  ` fld )
3618, 35ressplusg 14295 . . . . . . 7  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
3734, 36ax-mp 5 . . . . . 6  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
3837oveqi 6119 . . . . 5  |-  ( x  +  y )  =  ( x ( +g  `  (flds  ( 0 [,) +oo )
) ) y )
3933, 38, 213eltr3g 2525 . . . 4  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x ( +g  `  (flds  ( 0 [,) +oo ) ) ) y )  e.  ( Base `  (flds  ( 0 [,) +oo )
) ) )
4030, 32, 39syl2anc 661 . . 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 457 . . . . . . 7  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  x  e.  ( 0 [,) +oo )
)
4215, 41sseldi 3369 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  x  e.  RR )
43 simpr 461 . . . . . . 7  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  y  e.  ( 0 [,) +oo )
)
4415, 43sseldi 3369 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  y  e.  RR )
45 rexadd 11217 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x +e
y )  =  ( x  +  y ) )
4645eqcomd 2448 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  =  ( x +e y ) )
4742, 44, 46syl2anc 661 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  =  ( x +e y ) )
48 xrsadd 17848 . . . . . . . 8  |-  +e 
=  ( +g  `  RR*s
)
4923, 48ressplusg 14295 . . . . . . 7  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) )
5034, 49ax-mp 5 . . . . . 6  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )
5150oveqi 6119 . . . . 5  |-  ( x +e y )  =  ( x ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) y )
5247, 38, 513eqtr3g 2498 . . . 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 661 . . 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 461 . . . 4  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  F : A --> ( 0 [,) +oo ) )
55 ffun 5576 . . . 4  |-  ( F : A --> ( 0 [,) +oo )  ->  Fun  F )
5654, 55syl 16 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  Fun  F )
57 frn 5580 . . . . 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 3417 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ran  F  C_  ( Base `  (flds  ( 0 [,) +oo )
) ) )
602, 4, 6, 28, 40, 53, 56, 59gsumpropd2 15521 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( (flds  ( 0 [,) +oo ) )  gsumg  F )  =  ( ( RR*ss  ( 0 [,) +oo ) ) 
gsumg  F ) )
61 cnfldex 17836 . . . 4  |-fld  e.  _V
6261a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->fld 
e.  _V )
63 simpl 457 . . 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 11410 . . . 4  |-  0  e.  ( 0 [,) +oo )
6665a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  0  e.  ( 0 [,) +oo )
)
67 simpr 461 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  x  e.  CC )
6867addid2d 9585 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  ( 0  +  x )  =  x )
6967addid1d 9584 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  ( x  + 
0 )  =  x )
7068, 69jca 532 . . 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 15522 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (fld 
gsumg  F )  =  ( (flds  ( 0 [,) +oo )
)  gsumg  F ) )
72 xrge0base 26161 . . 3  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
73 xrge0plusg 26163 . . 3  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
74 ovex 6131 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
75 ressress 14250 . . . . 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 672 . . . 4  |-  ( (
RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )  =  (
RR*ss  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) ) )
77 incom 3558 . . . . . 6  |-  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) )  =  ( ( 0 [,) +oo )  i^i  ( 0 [,] +oo ) )
78 icossicc 26073 . . . . . . 7  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
79 dfss 3358 . . . . . . 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 2466 . . . . 5  |-  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) )  =  ( 0 [,) +oo )
8281oveq2i 6117 . . . 4  |-  ( RR*ss  ( ( 0 [,] +oo )  i^i  (
0 [,) +oo )
) )  =  (
RR*ss  ( 0 [,) +oo ) )
8376, 82eqtr2i 2464 . . 3  |-  ( RR*ss  ( 0 [,) +oo ) )  =  ( ( RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo )
)
84 ovex 6131 . . . 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 11393 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
88 simpr 461 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  x  e.  ( 0 [,] +oo ) )
8987, 88sseldi 3369 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  x  e.  RR* )
90 xaddid2 11225 . . . . 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 11224 . . . . 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 532 . . 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 15522 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  F )  =  ( ( RR*ss  ( 0 [,) +oo ) ) 
gsumg  F ) )
9660, 71, 953eqtr4d 2485 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 1369    e. wcel 1756   _Vcvv 2987    i^i cin 3342    C_ wss 3343   class class class wbr 4307   ran crn 4856   Fun wfun 5427   -->wf 5429   ` cfv 5433  (class class class)co 6106   CCcc 9295   RRcr 9296   0cc0 9297    + caddc 9300   +oocpnf 9430   -oocmnf 9431   RR*cxr 9432    < clt 9433    <_ cle 9434   +ecxad 11102   (,)cioo 11315   [,)cico 11317   [,]cicc 11318   Basecbs 14189   ↾s cress 14190   +g cplusg 14253    gsumg cgsu 14394   RR*scxrs 14453  ℂfldccnfld 17833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-addf 9376
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-1o 6935  df-oadd 6939  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-fin 7329  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-2 10395  df-3 10396  df-4 10397  df-5 10398  df-6 10399  df-7 10400  df-8 10401  df-9 10402  df-10 10403  df-n0 10595  df-z 10662  df-dec 10771  df-uz 10877  df-xadd 11105  df-ioo 11319  df-ico 11321  df-icc 11322  df-fz 11453  df-seq 11822  df-struct 14191  df-ndx 14192  df-slot 14193  df-base 14194  df-sets 14195  df-ress 14196  df-plusg 14266  df-mulr 14267  df-starv 14268  df-tset 14272  df-ple 14273  df-ds 14275  df-unif 14276  df-0g 14395  df-gsum 14396  df-xrs 14455  df-cnfld 17834
This theorem is referenced by:  esumpfinval  26539  esumpfinvalf  26540  esumpcvgval  26542  esumcvg  26550
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