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Theorem esumpfinvallem 28534
Description: Lemma for esumpfinval 28535 (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 6128 . . . 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 6308 . . . 4  |-  (flds  ( 0 [,) +oo ) )  e.  _V
43a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (flds  ( 0 [,) +oo )
)  e.  _V )
5 ovex 6308 . . . 4  |-  ( RR*ss  ( 0 [,) +oo ) )  e.  _V
65a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( RR*ss  (
0 [,) +oo )
)  e.  _V )
7 rge0ssre 11684 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
8 ax-resscn 9581 . . . . . . 7  |-  RR  C_  CC
97, 8sstri 3453 . . . . . 6  |-  ( 0 [,) +oo )  C_  CC
10 eqid 2404 . . . . . . 7  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
11 cnfldbas 18746 . . . . . . 7  |-  CC  =  ( Base ` fld )
1210, 11ressbas2 14901 . . . . . 6  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
139, 12ax-mp 5 . . . . 5  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
14 icossxr 11665 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR*
15 eqid 2404 . . . . . . 7  |-  ( RR*ss  ( 0 [,) +oo ) )  =  (
RR*ss  ( 0 [,) +oo ) )
16 xrsbas 18756 . . . . . . 7  |-  RR*  =  ( Base `  RR*s )
1715, 16ressbas2 14901 . . . . . 6  |-  ( ( 0 [,) +oo )  C_ 
RR*  ->  ( 0 [,) +oo )  =  ( Base `  ( RR*ss  (
0 [,) +oo )
) ) )
1814, 17ax-mp 5 . . . . 5  |-  ( 0 [,) +oo )  =  ( Base `  ( RR*ss  ( 0 [,) +oo ) ) )
1913, 18eqtr3i 2435 . . . 4  |-  ( Base `  (flds  ( 0 [,) +oo )
) )  =  (
Base `  ( RR*ss  ( 0 [,) +oo ) ) )
2019a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( Base `  (flds  ( 0 [,) +oo ) ) )  =  ( Base `  ( RR*ss  ( 0 [,) +oo ) ) ) )
21 simprl 758 . . . . 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 )
) ) )
2221, 13syl6eleqr 2503 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  /\  y  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  ->  x  e.  ( 0 [,) +oo ) )
23 simprr 760 . . . . 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 )
) ) )
2423, 13syl6eleqr 2503 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  ( x  e.  ( Base `  (flds  ( 0 [,) +oo ) ) )  /\  y  e.  ( Base `  (flds  ( 0 [,) +oo ) ) ) ) )  -> 
y  e.  ( 0 [,) +oo ) )
25 ge0addcl 11688 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  e.  ( 0 [,) +oo )
)
26 ovex 6308 . . . . . . 7  |-  ( 0 [,) +oo )  e. 
_V
27 cnfldadd 18747 . . . . . . . 8  |-  +  =  ( +g  ` fld )
2810, 27ressplusg 14957 . . . . . . 7  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
2926, 28ax-mp 5 . . . . . 6  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
3029oveqi 6293 . . . . 5  |-  ( x  +  y )  =  ( x ( +g  `  (flds  ( 0 [,) +oo )
) ) y )
3125, 30, 133eltr3g 2508 . . . 4  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x ( +g  `  (flds  ( 0 [,) +oo ) ) ) y )  e.  ( Base `  (flds  ( 0 [,) +oo )
) ) )
3222, 24, 31syl2anc 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 ) ) ) )
33 simpl 457 . . . . . . 7  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  x  e.  ( 0 [,) +oo )
)
347, 33sseldi 3442 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  x  e.  RR )
35 simpr 461 . . . . . . 7  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  y  e.  ( 0 [,) +oo )
)
367, 35sseldi 3442 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  y  e.  RR )
37 rexadd 11486 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x +e
y )  =  ( x  +  y ) )
3837eqcomd 2412 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  +  y )  =  ( x +e y ) )
3934, 36, 38syl2anc 661 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  =  ( x +e y ) )
40 xrsadd 18757 . . . . . . . 8  |-  +e 
=  ( +g  `  RR*s
)
4115, 40ressplusg 14957 . . . . . . 7  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) )
4226, 41ax-mp 5 . . . . . 6  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )
4342oveqi 6293 . . . . 5  |-  ( x +e y )  =  ( x ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) y )
4439, 30, 433eqtr3g 2468 . . . 4  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x ( +g  `  (flds  ( 0 [,) +oo ) ) ) y )  =  ( x ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) y ) )
4522, 24, 44syl2anc 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 ) )
46 simpr 461 . . . 4  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  F : A --> ( 0 [,) +oo ) )
47 ffun 5718 . . . 4  |-  ( F : A --> ( 0 [,) +oo )  ->  Fun  F )
4846, 47syl 17 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  Fun  F )
49 frn 5722 . . . . 5  |-  ( F : A --> ( 0 [,) +oo )  ->  ran  F  C_  ( 0 [,) +oo ) )
5046, 49syl 17 . . . 4  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ran  F  C_  (
0 [,) +oo )
)
5150, 13syl6sseq 3490 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ran  F  C_  ( Base `  (flds  ( 0 [,) +oo )
) ) )
522, 4, 6, 20, 32, 45, 48, 51gsumpropd2 16227 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( (flds  ( 0 [,) +oo ) )  gsumg  F )  =  ( ( RR*ss  ( 0 [,) +oo ) ) 
gsumg  F ) )
53 cnfldex 18745 . . . 4  |-fld  e.  _V
5453a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->fld 
e.  _V )
55 simpl 457 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  A  e.  V
)
569a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( 0 [,) +oo )  C_  CC )
57 0e0icopnf 11686 . . . 4  |-  0  e.  ( 0 [,) +oo )
5857a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  0  e.  ( 0 [,) +oo )
)
59 simpr 461 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  x  e.  CC )
6059addid2d 9817 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  ( 0  +  x )  =  x )
6159addid1d 9816 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  ( x  + 
0 )  =  x )
6260, 61jca 532 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  ( ( 0  +  x )  =  x  /\  ( x  +  0 )  =  x ) )
6311, 27, 10, 54, 55, 56, 46, 58, 62gsumress 16229 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (fld 
gsumg  F )  =  ( (flds  ( 0 [,) +oo )
)  gsumg  F ) )
64 xrge0base 28138 . . 3  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
65 xrge0plusg 28140 . . 3  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
66 ovex 6308 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
67 ressress 14908 . . . . 5  |-  ( ( ( 0 [,] +oo )  e.  _V  /\  (
0 [,) +oo )  e.  _V )  ->  (
( RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )  =  (
RR*ss  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) ) ) )
6866, 26, 67mp2an 672 . . . 4  |-  ( (
RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo ) )  =  (
RR*ss  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) ) )
69 incom 3634 . . . . . 6  |-  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) )  =  ( ( 0 [,) +oo )  i^i  ( 0 [,] +oo ) )
70 icossicc 11667 . . . . . . 7  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
71 dfss 3431 . . . . . . 7  |-  ( ( 0 [,) +oo )  C_  ( 0 [,] +oo ) 
<->  ( 0 [,) +oo )  =  ( (
0 [,) +oo )  i^i  ( 0 [,] +oo ) ) )
7270, 71mpbi 210 . . . . . 6  |-  ( 0 [,) +oo )  =  ( ( 0 [,) +oo )  i^i  (
0 [,] +oo )
)
7369, 72eqtr4i 2436 . . . . 5  |-  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) )  =  ( 0 [,) +oo )
7473oveq2i 6291 . . . 4  |-  ( RR*ss  ( ( 0 [,] +oo )  i^i  (
0 [,) +oo )
) )  =  (
RR*ss  ( 0 [,) +oo ) )
7568, 74eqtr2i 2434 . . 3  |-  ( RR*ss  ( 0 [,) +oo ) )  =  ( ( RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo )
)
76 ovex 6308 . . . 4  |-  ( RR*ss  ( 0 [,] +oo ) )  e.  _V
7776a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( RR*ss  (
0 [,] +oo )
)  e.  _V )
7870a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( 0 [,) +oo )  C_  ( 0 [,] +oo ) )
79 iccssxr 11663 . . . . . 6  |-  ( 0 [,] +oo )  C_  RR*
80 simpr 461 . . . . . 6  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  x  e.  ( 0 [,] +oo ) )
8179, 80sseldi 3442 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  x  e.  RR* )
82 xaddid2 11494 . . . . 5  |-  ( x  e.  RR*  ->  ( 0 +e x )  =  x )
8381, 82syl 17 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  ( 0 +e x )  =  x )
84 xaddid1 11493 . . . . 5  |-  ( x  e.  RR*  ->  ( x +e 0 )  =  x )
8581, 84syl 17 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  ( x +e 0 )  =  x )
8683, 85jca 532 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  ( (
0 +e x )  =  x  /\  ( x +e 0 )  =  x ) )
8764, 65, 75, 77, 55, 78, 46, 58, 86gsumress 16229 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  F )  =  ( ( RR*ss  ( 0 [,) +oo ) ) 
gsumg  F ) )
8852, 63, 873eqtr4d 2455 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 1407    e. wcel 1844   _Vcvv 3061    i^i cin 3415    C_ wss 3416   ran crn 4826   Fun wfun 5565   -->wf 5567   ` cfv 5571  (class class class)co 6280   CCcc 9522   RRcr 9523   0cc0 9524    + caddc 9527   +oocpnf 9657   RR*cxr 9659   +ecxad 11371   [,)cico 11586   [,]cicc 11587   Basecbs 14843   ↾s cress 14844   +g cplusg 14911    gsumg cgsu 15057   RR*scxrs 15116  ℂfldccnfld 18742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-1cn 9582  ax-icn 9583  ax-addcl 9584  ax-addrcl 9585  ax-mulcl 9586  ax-mulrcl 9587  ax-mulcom 9588  ax-addass 9589  ax-mulass 9590  ax-distr 9591  ax-i2m1 9592  ax-1ne0 9593  ax-1rid 9594  ax-rnegex 9595  ax-rrecex 9596  ax-cnre 9597  ax-pre-lttri 9598  ax-pre-lttrn 9599  ax-pre-ltadd 9600  ax-pre-mulgt0 9601  ax-addf 9603
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-oadd 7173  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-sub 9845  df-neg 9846  df-nn 10579  df-2 10637  df-3 10638  df-4 10639  df-5 10640  df-6 10641  df-7 10642  df-8 10643  df-9 10644  df-10 10645  df-n0 10839  df-z 10908  df-dec 11022  df-uz 11130  df-xadd 11374  df-ico 11590  df-icc 11591  df-fz 11729  df-seq 12154  df-struct 14845  df-ndx 14846  df-slot 14847  df-base 14848  df-sets 14849  df-ress 14850  df-plusg 14924  df-mulr 14925  df-starv 14926  df-tset 14930  df-ple 14931  df-ds 14933  df-unif 14934  df-0g 15058  df-gsum 15059  df-xrs 15118  df-cnfld 18743
This theorem is referenced by:  esumpfinval  28535  esumpfinvalf  28536  esumpcvgval  28538  esumcvg  28546
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