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Theorem telgsums 17135
Description: Telescoping finitely supported group sum ranging over nonnegative integers, using explicit substitution. (Contributed by AV, 24-Oct-2019.)
Hypotheses
Ref Expression
telgsums.b  |-  B  =  ( Base `  G
)
telgsums.g  |-  ( ph  ->  G  e.  Abel )
telgsums.m  |-  .-  =  ( -g `  G )
telgsums.0  |-  .0.  =  ( 0g `  G )
telgsums.f  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
telgsums.s  |-  ( ph  ->  S  e.  NN0 )
telgsums.u  |-  ( ph  ->  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )
)
Assertion
Ref Expression
telgsums  |-  ( ph  ->  ( G  gsumg  ( i  e.  NN0  |->  ( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
) ) )  = 
[_ 0  /  k ]_ C )
Distinct variable groups:    B, i,
k    C, i    i, G    S, i, k    .0. , i,
k    ph, i    .- , i
Allowed substitution hints:    ph( k)    C( k)    G( k)    .- ( k)

Proof of Theorem telgsums
StepHypRef Expression
1 telgsums.b . . 3  |-  B  =  ( Base `  G
)
2 telgsums.0 . . 3  |-  .0.  =  ( 0g `  G )
3 telgsums.g . . . 4  |-  ( ph  ->  G  e.  Abel )
4 ablcmn 16921 . . . 4  |-  ( G  e.  Abel  ->  G  e. CMnd
)
53, 4syl 16 . . 3  |-  ( ph  ->  G  e. CMnd )
6 ablgrp 16920 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
73, 6syl 16 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
87adantr 463 . . . . 5  |-  ( (
ph  /\  i  e.  NN0 )  ->  G  e.  Grp )
9 simpr 459 . . . . . 6  |-  ( (
ph  /\  i  e.  NN0 )  ->  i  e.  NN0 )
10 telgsums.f . . . . . . 7  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
1110adantr 463 . . . . . 6  |-  ( (
ph  /\  i  e.  NN0 )  ->  A. k  e.  NN0  C  e.  B
)
12 rspcsbela 3773 . . . . . 6  |-  ( ( i  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ i  /  k ]_ C  e.  B )
139, 11, 12syl2anc 659 . . . . 5  |-  ( (
ph  /\  i  e.  NN0 )  ->  [_ i  / 
k ]_ C  e.  B
)
14 peano2nn0 10753 . . . . . 6  |-  ( i  e.  NN0  ->  ( i  +  1 )  e. 
NN0 )
15 rspcsbela 3773 . . . . . 6  |-  ( ( ( i  +  1 )  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ (
i  +  1 )  /  k ]_ C  e.  B )
1614, 10, 15syl2anr 476 . . . . 5  |-  ( (
ph  /\  i  e.  NN0 )  ->  [_ ( i  +  1 )  / 
k ]_ C  e.  B
)
17 telgsums.m . . . . . 6  |-  .-  =  ( -g `  G )
181, 17grpsubcl 16235 . . . . 5  |-  ( ( G  e.  Grp  /\  [_ i  /  k ]_ C  e.  B  /\  [_ ( i  +  1 )  /  k ]_ C  e.  B )  ->  ( [_ i  / 
k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
)  e.  B )
198, 13, 16, 18syl3anc 1226 . . . 4  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( [_ i  /  k ]_ C  .- 
[_ ( i  +  1 )  /  k ]_ C )  e.  B
)
2019ralrimiva 2796 . . 3  |-  ( ph  ->  A. i  e.  NN0  ( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
)  e.  B )
21 telgsums.s . . 3  |-  ( ph  ->  S  e.  NN0 )
22 telgsums.u . . . . . . . . 9  |-  ( ph  ->  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )
)
23 rspsbca 3332 . . . . . . . . . . 11  |-  ( ( i  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  ->  [. i  /  k ]. ( S  <  k  ->  C  =  .0.  )
)
24 vex 3037 . . . . . . . . . . . 12  |-  i  e. 
_V
25 sbcimg 3294 . . . . . . . . . . . . 13  |-  ( i  e.  _V  ->  ( [. i  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  (
[. i  /  k ]. S  <  k  ->  [. i  /  k ]. C  =  .0.  ) ) )
26 sbcbr2g 4422 . . . . . . . . . . . . . . 15  |-  ( i  e.  _V  ->  ( [. i  /  k ]. S  <  k  <->  S  <  [_ i  /  k ]_ k ) )
27 csbvarg 3770 . . . . . . . . . . . . . . . 16  |-  ( i  e.  _V  ->  [_ i  /  k ]_ k  =  i )
2827breq2d 4379 . . . . . . . . . . . . . . 15  |-  ( i  e.  _V  ->  ( S  <  [_ i  /  k ]_ k  <->  S  <  i ) )
2926, 28bitrd 253 . . . . . . . . . . . . . 14  |-  ( i  e.  _V  ->  ( [. i  /  k ]. S  <  k  <->  S  <  i ) )
30 sbceq1g 3755 . . . . . . . . . . . . . 14  |-  ( i  e.  _V  ->  ( [. i  /  k ]. C  =  .0.  <->  [_ i  /  k ]_ C  =  .0.  ) )
3129, 30imbi12d 318 . . . . . . . . . . . . 13  |-  ( i  e.  _V  ->  (
( [. i  /  k ]. S  <  k  ->  [. i  /  k ]. C  =  .0.  ) 
<->  ( S  <  i  ->  [_ i  /  k ]_ C  =  .0.  ) ) )
3225, 31bitrd 253 . . . . . . . . . . . 12  |-  ( i  e.  _V  ->  ( [. i  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  ( S  <  i  ->  [_ i  /  k ]_ C  =  .0.  ) ) )
3324, 32ax-mp 5 . . . . . . . . . . 11  |-  ( [. i  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  ( S  <  i  ->  [_ i  / 
k ]_ C  =  .0.  ) )
3423, 33sylib 196 . . . . . . . . . 10  |-  ( ( i  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  -> 
( S  <  i  ->  [_ i  /  k ]_ C  =  .0.  ) )
3534expcom 433 . . . . . . . . 9  |-  ( A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )  ->  (
i  e.  NN0  ->  ( S  <  i  ->  [_ i  /  k ]_ C  =  .0.  ) ) )
3622, 35syl 16 . . . . . . . 8  |-  ( ph  ->  ( i  e.  NN0  ->  ( S  <  i  ->  [_ i  /  k ]_ C  =  .0.  ) ) )
3736imp31 430 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  [_ i  /  k ]_ C  =  .0.  )
3821nn0red 10770 . . . . . . . . . . . . 13  |-  ( ph  ->  S  e.  RR )
3938adantr 463 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  NN0 )  ->  S  e.  RR )
4039adantr 463 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  S  e.  RR )
41 nn0re 10721 . . . . . . . . . . . 12  |-  ( i  e.  NN0  ->  i  e.  RR )
4241ad2antlr 724 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  i  e.  RR )
4314ad2antlr 724 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  (
i  +  1 )  e.  NN0 )
4443nn0red 10770 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  (
i  +  1 )  e.  RR )
45 simpr 459 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  S  <  i )
4642ltp1d 10392 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  i  <  ( i  +  1 ) )
4740, 42, 44, 45, 46lttrd 9654 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  S  <  ( i  +  1 ) )
4847ex 432 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( S  <  i  ->  S  <  ( i  +  1 ) ) )
49 rspsbca 3332 . . . . . . . . . . 11  |-  ( ( ( i  +  1 )  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  ->  [. ( i  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )
)
50 ovex 6224 . . . . . . . . . . . 12  |-  ( i  +  1 )  e. 
_V
51 sbcimg 3294 . . . . . . . . . . . . 13  |-  ( ( i  +  1 )  e.  _V  ->  ( [. ( i  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  (
[. ( i  +  1 )  /  k ]. S  <  k  ->  [. ( i  +  1 )  /  k ]. C  =  .0.  )
) )
52 sbcbr2g 4422 . . . . . . . . . . . . . . 15  |-  ( ( i  +  1 )  e.  _V  ->  ( [. ( i  +  1 )  /  k ]. S  <  k  <->  S  <  [_ ( i  +  1 )  /  k ]_ k ) )
53 csbvarg 3770 . . . . . . . . . . . . . . . 16  |-  ( ( i  +  1 )  e.  _V  ->  [_ (
i  +  1 )  /  k ]_ k  =  ( i  +  1 ) )
5453breq2d 4379 . . . . . . . . . . . . . . 15  |-  ( ( i  +  1 )  e.  _V  ->  ( S  <  [_ ( i  +  1 )  /  k ]_ k  <->  S  <  ( i  +  1 ) ) )
5552, 54bitrd 253 . . . . . . . . . . . . . 14  |-  ( ( i  +  1 )  e.  _V  ->  ( [. ( i  +  1 )  /  k ]. S  <  k  <->  S  <  ( i  +  1 ) ) )
56 sbceq1g 3755 . . . . . . . . . . . . . 14  |-  ( ( i  +  1 )  e.  _V  ->  ( [. ( i  +  1 )  /  k ]. C  =  .0.  <->  [_ ( i  +  1 )  / 
k ]_ C  =  .0.  ) )
5755, 56imbi12d 318 . . . . . . . . . . . . 13  |-  ( ( i  +  1 )  e.  _V  ->  (
( [. ( i  +  1 )  /  k ]. S  <  k  ->  [. ( i  +  1 )  /  k ]. C  =  .0.  )  <->  ( S  <  ( i  +  1 )  ->  [_ ( i  +  1 )  /  k ]_ C  =  .0.  )
) )
5851, 57bitrd 253 . . . . . . . . . . . 12  |-  ( ( i  +  1 )  e.  _V  ->  ( [. ( i  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  ( S  <  ( i  +  1 )  ->  [_ ( i  +  1 )  /  k ]_ C  =  .0.  )
) )
5950, 58ax-mp 5 . . . . . . . . . . 11  |-  ( [. ( i  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  ( S  <  ( i  +  1 )  ->  [_ ( i  +  1 )  /  k ]_ C  =  .0.  )
)
6049, 59sylib 196 . . . . . . . . . 10  |-  ( ( ( i  +  1 )  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  -> 
( S  <  (
i  +  1 )  ->  [_ ( i  +  1 )  /  k ]_ C  =  .0.  ) )
6114, 22, 60syl2anr 476 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( S  <  ( i  +  1 )  ->  [_ ( i  +  1 )  / 
k ]_ C  =  .0.  ) )
6248, 61syld 44 . . . . . . . 8  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( S  <  i  ->  [_ ( i  +  1 )  / 
k ]_ C  =  .0.  ) )
6362imp 427 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  [_ (
i  +  1 )  /  k ]_ C  =  .0.  )
6437, 63oveq12d 6214 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  ( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
)  =  (  .0.  .-  .0.  ) )
658adantr 463 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  G  e.  Grp )
661, 2grpidcl 16195 . . . . . . . 8  |-  ( G  e.  Grp  ->  .0.  e.  B )
6765, 66jccir 537 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  ( G  e.  Grp  /\  .0.  e.  B ) )
681, 2, 17grpsubid 16239 . . . . . . 7  |-  ( ( G  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  .-  .0.  )  =  .0.  )
6967, 68syl 16 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  (  .0.  .-  .0.  )  =  .0.  )
7064, 69eqtrd 2423 . . . . 5  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  ( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
)  =  .0.  )
7170ex 432 . . . 4  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( S  <  i  ->  ( [_ i  /  k ]_ C  .- 
[_ ( i  +  1 )  /  k ]_ C )  =  .0.  ) )
7271ralrimiva 2796 . . 3  |-  ( ph  ->  A. i  e.  NN0  ( S  <  i  -> 
( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
)  =  .0.  )
)
731, 2, 5, 20, 21, 72gsummptnn0fzv 17128 . 2  |-  ( ph  ->  ( G  gsumg  ( i  e.  NN0  |->  ( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
) ) )  =  ( G  gsumg  ( i  e.  ( 0 ... S ) 
|->  ( [_ i  / 
k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
) ) ) )
74 fzssuz 11646 . . . . . 6  |-  ( 0 ... ( S  + 
1 ) )  C_  ( ZZ>= `  0 )
7574a1i 11 . . . . 5  |-  ( ph  ->  ( 0 ... ( S  +  1 ) )  C_  ( ZZ>= ` 
0 ) )
76 nn0uz 11035 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
7775, 76syl6sseqr 3464 . . . 4  |-  ( ph  ->  ( 0 ... ( S  +  1 ) )  C_  NN0 )
78 ssralv 3478 . . . 4  |-  ( ( 0 ... ( S  +  1 ) ) 
C_  NN0  ->  ( A. k  e.  NN0  C  e.  B  ->  A. k  e.  ( 0 ... ( S  +  1 ) ) C  e.  B
) )
7977, 10, 78sylc 60 . . 3  |-  ( ph  ->  A. k  e.  ( 0 ... ( S  +  1 ) ) C  e.  B )
801, 3, 17, 21, 79telgsumfz0s 17133 . 2  |-  ( ph  ->  ( G  gsumg  ( i  e.  ( 0 ... S ) 
|->  ( [_ i  / 
k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
) ) )  =  ( [_ 0  / 
k ]_ C  .-  [_ ( S  +  1 )  /  k ]_ C
) )
81 peano2nn0 10753 . . . . . 6  |-  ( S  e.  NN0  ->  ( S  +  1 )  e. 
NN0 )
8221, 81syl 16 . . . . 5  |-  ( ph  ->  ( S  +  1 )  e.  NN0 )
8338ltp1d 10392 . . . . 5  |-  ( ph  ->  S  <  ( S  +  1 ) )
84 rspsbca 3332 . . . . . . 7  |-  ( ( ( S  +  1 )  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  ->  [. ( S  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )
)
85 ovex 6224 . . . . . . . 8  |-  ( S  +  1 )  e. 
_V
86 sbcimg 3294 . . . . . . . . 9  |-  ( ( S  +  1 )  e.  _V  ->  ( [. ( S  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  (
[. ( S  + 
1 )  /  k ]. S  <  k  ->  [. ( S  +  1 )  /  k ]. C  =  .0.  )
) )
87 sbcbr2g 4422 . . . . . . . . . . 11  |-  ( ( S  +  1 )  e.  _V  ->  ( [. ( S  +  1 )  /  k ]. S  <  k  <->  S  <  [_ ( S  +  1 )  /  k ]_ k ) )
88 csbvarg 3770 . . . . . . . . . . . 12  |-  ( ( S  +  1 )  e.  _V  ->  [_ ( S  +  1 )  /  k ]_ k  =  ( S  + 
1 ) )
8988breq2d 4379 . . . . . . . . . . 11  |-  ( ( S  +  1 )  e.  _V  ->  ( S  <  [_ ( S  + 
1 )  /  k ]_ k  <->  S  <  ( S  +  1 ) ) )
9087, 89bitrd 253 . . . . . . . . . 10  |-  ( ( S  +  1 )  e.  _V  ->  ( [. ( S  +  1 )  /  k ]. S  <  k  <->  S  <  ( S  +  1 ) ) )
91 sbceq1g 3755 . . . . . . . . . 10  |-  ( ( S  +  1 )  e.  _V  ->  ( [. ( S  +  1 )  /  k ]. C  =  .0.  <->  [_ ( S  +  1 )  / 
k ]_ C  =  .0.  ) )
9290, 91imbi12d 318 . . . . . . . . 9  |-  ( ( S  +  1 )  e.  _V  ->  (
( [. ( S  + 
1 )  /  k ]. S  <  k  ->  [. ( S  +  1 )  /  k ]. C  =  .0.  )  <->  ( S  <  ( S  +  1 )  ->  [_ ( S  +  1 )  /  k ]_ C  =  .0.  )
) )
9386, 92bitrd 253 . . . . . . . 8  |-  ( ( S  +  1 )  e.  _V  ->  ( [. ( S  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  ( S  <  ( S  +  1 )  ->  [_ ( S  +  1 )  /  k ]_ C  =  .0.  )
) )
9485, 93ax-mp 5 . . . . . . 7  |-  ( [. ( S  +  1
)  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  ( S  <  ( S  +  1 )  ->  [_ ( S  +  1 )  /  k ]_ C  =  .0.  )
)
9584, 94sylib 196 . . . . . 6  |-  ( ( ( S  +  1 )  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  -> 
( S  <  ( S  +  1 )  ->  [_ ( S  + 
1 )  /  k ]_ C  =  .0.  ) )
9695ex 432 . . . . 5  |-  ( ( S  +  1 )  e.  NN0  ->  ( A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )  ->  ( S  <  ( S  + 
1 )  ->  [_ ( S  +  1 )  /  k ]_ C  =  .0.  ) ) )
9782, 22, 83, 96syl3c 61 . . . 4  |-  ( ph  ->  [_ ( S  + 
1 )  /  k ]_ C  =  .0.  )
9897oveq2d 6212 . . 3  |-  ( ph  ->  ( [_ 0  / 
k ]_ C  .-  [_ ( S  +  1 )  /  k ]_ C
)  =  ( [_
0  /  k ]_ C  .-  .0.  ) )
99 0nn0 10727 . . . . . 6  |-  0  e.  NN0
10099a1i 11 . . . . 5  |-  ( ph  ->  0  e.  NN0 )
101 rspcsbela 3773 . . . . 5  |-  ( ( 0  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ 0  /  k ]_ C  e.  B )
102100, 10, 101syl2anc 659 . . . 4  |-  ( ph  ->  [_ 0  /  k ]_ C  e.  B
)
1031, 2, 17grpsubid1 16240 . . . 4  |-  ( ( G  e.  Grp  /\  [_ 0  /  k ]_ C  e.  B )  ->  ( [_ 0  / 
k ]_ C  .-  .0.  )  =  [_ 0  /  k ]_ C
)
1047, 102, 103syl2anc 659 . . 3  |-  ( ph  ->  ( [_ 0  / 
k ]_ C  .-  .0.  )  =  [_ 0  /  k ]_ C
)
10598, 104eqtrd 2423 . 2  |-  ( ph  ->  ( [_ 0  / 
k ]_ C  .-  [_ ( S  +  1 )  /  k ]_ C
)  =  [_ 0  /  k ]_ C
)
10673, 80, 1053eqtrd 2427 1  |-  ( ph  ->  ( G  gsumg  ( i  e.  NN0  |->  ( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
) ) )  = 
[_ 0  /  k ]_ C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1399    e. wcel 1826   A.wral 2732   _Vcvv 3034   [.wsbc 3252   [_csb 3348    C_ wss 3389   class class class wbr 4367    |-> cmpt 4425   ` cfv 5496  (class class class)co 6196   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    < clt 9539   NN0cn0 10712   ZZ>=cuz 11001   ...cfz 11593   Basecbs 14634   0gc0g 14847    gsumg cgsu 14848   Grpcgrp 16170   -gcsg 16172  CMndccmn 16915   Abelcabl 16916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-inf2 7972  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-fal 1405  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-se 4753  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-isom 5505  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-of 6439  df-om 6600  df-1st 6699  df-2nd 6700  df-supp 6818  df-recs 6960  df-rdg 6994  df-1o 7048  df-oadd 7052  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-fin 7439  df-fsupp 7745  df-oi 7850  df-card 8233  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594  df-fzo 11718  df-seq 12011  df-hash 12308  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-0g 14849  df-gsum 14850  df-mre 14993  df-mrc 14994  df-acs 14996  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-submnd 16084  df-grp 16174  df-minusg 16175  df-sbg 16176  df-mulg 16177  df-cntz 16472  df-cmn 16917  df-abl 16918
This theorem is referenced by:  telgsum  17136
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