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Theorem esumpfinvallem 27905
Description: Lemma for esumpfinval 27906 (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 6144 . . . 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 6320 . . . 4  |-  (flds  ( 0 [,) +oo ) )  e.  _V
43a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (flds  ( 0 [,) +oo )
)  e.  _V )
5 ovex 6320 . . . 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 11335 . . . . . . . . 9  |- -oo  e.  RR*
8 pnfxr 11333 . . . . . . . . 9  |- +oo  e.  RR*
9 mnflt0 11346 . . . . . . . . 9  |- -oo  <  0
10 pnfge 11351 . . . . . . . . . 10  |-  ( +oo  e.  RR*  -> +oo  <_ +oo )
118, 10ax-mp 5 . . . . . . . . 9  |- +oo  <_ +oo
12 icossioo 11627 . . . . . . . . 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 11611 . . . . . . . 8  |-  ( -oo (,) +oo )  =  RR
1513, 14sseqtri 3541 . . . . . . 7  |-  ( 0 [,) +oo )  C_  RR
16 ax-resscn 9561 . . . . . . 7  |-  RR  C_  CC
1715, 16sstri 3518 . . . . . 6  |-  ( 0 [,) +oo )  C_  CC
18 eqid 2467 . . . . . . 7  |-  (flds  ( 0 [,) +oo ) )  =  (flds  ( 0 [,) +oo ) )
19 cnfldbas 18294 . . . . . . 7  |-  CC  =  ( Base ` fld )
2018, 19ressbas2 14563 . . . . . 6  |-  ( ( 0 [,) +oo )  C_  CC  ->  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) ) )
2117, 20ax-mp 5 . . . . 5  |-  ( 0 [,) +oo )  =  ( Base `  (flds  ( 0 [,) +oo ) ) )
22 icossxr 11621 . . . . . 6  |-  ( 0 [,) +oo )  C_  RR*
23 eqid 2467 . . . . . . 7  |-  ( RR*ss  ( 0 [,) +oo ) )  =  (
RR*ss  ( 0 [,) +oo ) )
24 xrsbas 18304 . . . . . . 7  |-  RR*  =  ( Base `  RR*s )
2523, 24ressbas2 14563 . . . . . 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 2498 . . . 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 2566 . . . 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 2566 . . . 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 11644 . . . . 5  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  ( x  +  y )  e.  ( 0 [,) +oo )
)
34 ovex 6320 . . . . . . 7  |-  ( 0 [,) +oo )  e. 
_V
35 cnfldadd 18295 . . . . . . . 8  |-  +  =  ( +g  ` fld )
3618, 35ressplusg 14614 . . . . . . 7  |-  ( ( 0 [,) +oo )  e.  _V  ->  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) ) )
3734, 36ax-mp 5 . . . . . 6  |-  +  =  ( +g  `  (flds  ( 0 [,) +oo ) ) )
3837oveqi 6308 . . . . 5  |-  ( x  +  y )  =  ( x ( +g  `  (flds  ( 0 [,) +oo )
) ) y )
3933, 38, 213eltr3g 2571 . . . 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 3507 . . . . . 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 3507 . . . . . 6  |-  ( ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo ) )  ->  y  e.  RR )
45 rexadd 11443 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x +e
y )  =  ( x  +  y ) )
4645eqcomd 2475 . . . . . 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 18305 . . . . . . . 8  |-  +e 
=  ( +g  `  RR*s
)
4923, 48ressplusg 14614 . . . . . . 7  |-  ( ( 0 [,) +oo )  e.  _V  ->  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) )
5034, 49ax-mp 5 . . . . . 6  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) )
5150oveqi 6308 . . . . 5  |-  ( x +e y )  =  ( x ( +g  `  ( RR*ss  ( 0 [,) +oo ) ) ) y )
5247, 38, 513eqtr3g 2531 . . . 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 5739 . . . 4  |-  ( F : A --> ( 0 [,) +oo )  ->  Fun  F )
5654, 55syl 16 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  Fun  F )
57 frn 5743 . . . . 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 3555 . . 3  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ran  F  C_  ( Base `  (flds  ( 0 [,) +oo )
) ) )
602, 4, 6, 28, 40, 53, 56, 59gsumpropd2 15775 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( (flds  ( 0 [,) +oo ) )  gsumg  F )  =  ( ( RR*ss  ( 0 [,) +oo ) ) 
gsumg  F ) )
61 cnfldex 18293 . . . 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 11642 . . . 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 9792 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  CC )  ->  ( 0  +  x )  =  x )
6967addid1d 9791 . . . 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 15777 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  (fld 
gsumg  F )  =  ( (flds  ( 0 [,) +oo )
)  gsumg  F ) )
72 xrge0base 27497 . . 3  |-  ( 0 [,] +oo )  =  ( Base `  ( RR*ss  ( 0 [,] +oo ) ) )
73 xrge0plusg 27499 . . 3  |-  +e 
=  ( +g  `  ( RR*ss  ( 0 [,] +oo ) ) )
74 ovex 6320 . . . . 5  |-  ( 0 [,] +oo )  e. 
_V
75 ressress 14569 . . . . 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 3696 . . . . . 6  |-  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) )  =  ( ( 0 [,) +oo )  i^i  ( 0 [,] +oo ) )
78 icossicc 11623 . . . . . . 7  |-  ( 0 [,) +oo )  C_  ( 0 [,] +oo )
79 dfss 3496 . . . . . . 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 2499 . . . . 5  |-  ( ( 0 [,] +oo )  i^i  ( 0 [,) +oo ) )  =  ( 0 [,) +oo )
8281oveq2i 6306 . . . 4  |-  ( RR*ss  ( ( 0 [,] +oo )  i^i  (
0 [,) +oo )
) )  =  (
RR*ss  ( 0 [,) +oo ) )
8376, 82eqtr2i 2497 . . 3  |-  ( RR*ss  ( 0 [,) +oo ) )  =  ( ( RR*ss  ( 0 [,] +oo ) )s  ( 0 [,) +oo )
)
84 ovex 6320 . . . 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 11619 . . . . . 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 3507 . . . . 5  |-  ( ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  /\  x  e.  ( 0 [,] +oo )
)  ->  x  e.  RR* )
90 xaddid2 11451 . . . . 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 11450 . . . . 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 15777 . 2  |-  ( ( A  e.  V  /\  F : A --> ( 0 [,) +oo ) )  ->  ( ( RR*ss  ( 0 [,] +oo ) )  gsumg  F )  =  ( ( RR*ss  ( 0 [,) +oo ) ) 
gsumg  F ) )
9660, 71, 953eqtr4d 2518 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 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480    C_ wss 3481   class class class wbr 4453   ran crn 5006   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    + caddc 9507   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    < clt 9640    <_ cle 9641   +ecxad 11328   (,)cioo 11541   [,)cico 11543   [,]cicc 11544   Basecbs 14507   ↾s cress 14508   +g cplusg 14572    gsumg cgsu 14713   RR*scxrs 14772  ℂfldccnfld 18290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-addf 9583
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-xadd 11331  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-seq 12088  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-0g 14714  df-gsum 14715  df-xrs 14774  df-cnfld 18291
This theorem is referenced by:  esumpfinval  27906  esumpfinvalf  27907  esumpcvgval  27909  esumcvg  27917
  Copyright terms: Public domain W3C validator