MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  telgsums Structured version   Visualization version   Unicode version

Theorem telgsums 17672
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 17485 . . . 4  |-  ( G  e.  Abel  ->  G  e. CMnd
)
53, 4syl 17 . . 3  |-  ( ph  ->  G  e. CMnd )
6 ablgrp 17484 . . . . . . 7  |-  ( G  e.  Abel  ->  G  e. 
Grp )
73, 6syl 17 . . . . . 6  |-  ( ph  ->  G  e.  Grp )
87adantr 471 . . . . 5  |-  ( (
ph  /\  i  e.  NN0 )  ->  G  e.  Grp )
9 simpr 467 . . . . . 6  |-  ( (
ph  /\  i  e.  NN0 )  ->  i  e.  NN0 )
10 telgsums.f . . . . . . 7  |-  ( ph  ->  A. k  e.  NN0  C  e.  B )
1110adantr 471 . . . . . 6  |-  ( (
ph  /\  i  e.  NN0 )  ->  A. k  e.  NN0  C  e.  B
)
12 rspcsbela 3807 . . . . . 6  |-  ( ( i  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ i  /  k ]_ C  e.  B )
139, 11, 12syl2anc 671 . . . . 5  |-  ( (
ph  /\  i  e.  NN0 )  ->  [_ i  / 
k ]_ C  e.  B
)
14 peano2nn0 10939 . . . . . 6  |-  ( i  e.  NN0  ->  ( i  +  1 )  e. 
NN0 )
15 rspcsbela 3807 . . . . . 6  |-  ( ( ( i  +  1 )  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ (
i  +  1 )  /  k ]_ C  e.  B )
1614, 10, 15syl2anr 485 . . . . 5  |-  ( (
ph  /\  i  e.  NN0 )  ->  [_ ( i  +  1 )  / 
k ]_ C  e.  B
)
17 telgsums.m . . . . . 6  |-  .-  =  ( -g `  G )
181, 17grpsubcl 16783 . . . . 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 1276 . . . 4  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( [_ i  /  k ]_ C  .- 
[_ ( i  +  1 )  /  k ]_ C )  e.  B
)
2019ralrimiva 2814 . . 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 3359 . . . . . . . . . . 11  |-  ( ( i  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  ->  [. i  /  k ]. ( S  <  k  ->  C  =  .0.  )
)
24 vex 3060 . . . . . . . . . . . 12  |-  i  e. 
_V
25 sbcimg 3321 . . . . . . . . . . . . 13  |-  ( i  e.  _V  ->  ( [. i  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  (
[. i  /  k ]. S  <  k  ->  [. i  /  k ]. C  =  .0.  ) ) )
26 sbcbr2g 4472 . . . . . . . . . . . . . . 15  |-  ( i  e.  _V  ->  ( [. i  /  k ]. S  <  k  <->  S  <  [_ i  /  k ]_ k ) )
27 csbvarg 3804 . . . . . . . . . . . . . . . 16  |-  ( i  e.  _V  ->  [_ i  /  k ]_ k  =  i )
2827breq2d 4428 . . . . . . . . . . . . . . 15  |-  ( i  e.  _V  ->  ( S  <  [_ i  /  k ]_ k  <->  S  <  i ) )
2926, 28bitrd 261 . . . . . . . . . . . . . 14  |-  ( i  e.  _V  ->  ( [. i  /  k ]. S  <  k  <->  S  <  i ) )
30 sbceq1g 3789 . . . . . . . . . . . . . 14  |-  ( i  e.  _V  ->  ( [. i  /  k ]. C  =  .0.  <->  [_ i  /  k ]_ C  =  .0.  ) )
3129, 30imbi12d 326 . . . . . . . . . . . . 13  |-  ( i  e.  _V  ->  (
( [. i  /  k ]. S  <  k  ->  [. i  /  k ]. C  =  .0.  ) 
<->  ( S  <  i  ->  [_ i  /  k ]_ C  =  .0.  ) ) )
3225, 31bitrd 261 . . . . . . . . . . . 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 201 . . . . . . . . . 10  |-  ( ( i  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  -> 
( S  <  i  ->  [_ i  /  k ]_ C  =  .0.  ) )
3534expcom 441 . . . . . . . . 9  |-  ( A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  )  ->  (
i  e.  NN0  ->  ( S  <  i  ->  [_ i  /  k ]_ C  =  .0.  ) ) )
3622, 35syl 17 . . . . . . . 8  |-  ( ph  ->  ( i  e.  NN0  ->  ( S  <  i  ->  [_ i  /  k ]_ C  =  .0.  ) ) )
3736imp31 438 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  [_ i  /  k ]_ C  =  .0.  )
3821nn0red 10955 . . . . . . . . . . . . 13  |-  ( ph  ->  S  e.  RR )
3938adantr 471 . . . . . . . . . . . 12  |-  ( (
ph  /\  i  e.  NN0 )  ->  S  e.  RR )
4039adantr 471 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  S  e.  RR )
41 nn0re 10907 . . . . . . . . . . . 12  |-  ( i  e.  NN0  ->  i  e.  RR )
4241ad2antlr 738 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  i  e.  RR )
4314ad2antlr 738 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  (
i  +  1 )  e.  NN0 )
4443nn0red 10955 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  (
i  +  1 )  e.  RR )
45 simpr 467 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  S  <  i )
4642ltp1d 10565 . . . . . . . . . . 11  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  i  <  ( i  +  1 ) )
4740, 42, 44, 45, 46lttrd 9822 . . . . . . . . . 10  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  S  <  ( i  +  1 ) )
4847ex 440 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( S  <  i  ->  S  <  ( i  +  1 ) ) )
49 rspsbca 3359 . . . . . . . . . . 11  |-  ( ( ( i  +  1 )  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  ->  [. ( i  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )
)
50 ovex 6343 . . . . . . . . . . . 12  |-  ( i  +  1 )  e. 
_V
51 sbcimg 3321 . . . . . . . . . . . . 13  |-  ( ( i  +  1 )  e.  _V  ->  ( [. ( i  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  (
[. ( i  +  1 )  /  k ]. S  <  k  ->  [. ( i  +  1 )  /  k ]. C  =  .0.  )
) )
52 sbcbr2g 4472 . . . . . . . . . . . . . . 15  |-  ( ( i  +  1 )  e.  _V  ->  ( [. ( i  +  1 )  /  k ]. S  <  k  <->  S  <  [_ ( i  +  1 )  /  k ]_ k ) )
53 csbvarg 3804 . . . . . . . . . . . . . . . 16  |-  ( ( i  +  1 )  e.  _V  ->  [_ (
i  +  1 )  /  k ]_ k  =  ( i  +  1 ) )
5453breq2d 4428 . . . . . . . . . . . . . . 15  |-  ( ( i  +  1 )  e.  _V  ->  ( S  <  [_ ( i  +  1 )  /  k ]_ k  <->  S  <  ( i  +  1 ) ) )
5552, 54bitrd 261 . . . . . . . . . . . . . 14  |-  ( ( i  +  1 )  e.  _V  ->  ( [. ( i  +  1 )  /  k ]. S  <  k  <->  S  <  ( i  +  1 ) ) )
56 sbceq1g 3789 . . . . . . . . . . . . . 14  |-  ( ( i  +  1 )  e.  _V  ->  ( [. ( i  +  1 )  /  k ]. C  =  .0.  <->  [_ ( i  +  1 )  / 
k ]_ C  =  .0.  ) )
5755, 56imbi12d 326 . . . . . . . . . . . . 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 261 . . . . . . . . . . . 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 201 . . . . . . . . . 10  |-  ( ( ( i  +  1 )  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  -> 
( S  <  (
i  +  1 )  ->  [_ ( i  +  1 )  /  k ]_ C  =  .0.  ) )
6114, 22, 60syl2anr 485 . . . . . . . . 9  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( S  <  ( i  +  1 )  ->  [_ ( i  +  1 )  / 
k ]_ C  =  .0.  ) )
6248, 61syld 45 . . . . . . . 8  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( S  <  i  ->  [_ ( i  +  1 )  / 
k ]_ C  =  .0.  ) )
6362imp 435 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  [_ (
i  +  1 )  /  k ]_ C  =  .0.  )
6437, 63oveq12d 6333 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  ( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
)  =  (  .0.  .-  .0.  ) )
658adantr 471 . . . . . . . 8  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  G  e.  Grp )
661, 2grpidcl 16743 . . . . . . . 8  |-  ( G  e.  Grp  ->  .0.  e.  B )
6765, 66jccir 546 . . . . . . 7  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  ( G  e.  Grp  /\  .0.  e.  B ) )
681, 2, 17grpsubid 16787 . . . . . . 7  |-  ( ( G  e.  Grp  /\  .0.  e.  B )  -> 
(  .0.  .-  .0.  )  =  .0.  )
6967, 68syl 17 . . . . . 6  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  (  .0.  .-  .0.  )  =  .0.  )
7064, 69eqtrd 2496 . . . . 5  |-  ( ( ( ph  /\  i  e.  NN0 )  /\  S  <  i )  ->  ( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
)  =  .0.  )
7170ex 440 . . . 4  |-  ( (
ph  /\  i  e.  NN0 )  ->  ( S  <  i  ->  ( [_ i  /  k ]_ C  .- 
[_ ( i  +  1 )  /  k ]_ C )  =  .0.  ) )
7271ralrimiva 2814 . . 3  |-  ( ph  ->  A. i  e.  NN0  ( S  <  i  -> 
( [_ i  /  k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
)  =  .0.  )
)
731, 2, 5, 20, 21, 72gsummptnn0fzv 17665 . 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 11868 . . . . . 6  |-  ( 0 ... ( S  + 
1 ) )  C_  ( ZZ>= `  0 )
7574a1i 11 . . . . 5  |-  ( ph  ->  ( 0 ... ( S  +  1 ) )  C_  ( ZZ>= ` 
0 ) )
76 nn0uz 11222 . . . . 5  |-  NN0  =  ( ZZ>= `  0 )
7775, 76syl6sseqr 3491 . . . 4  |-  ( ph  ->  ( 0 ... ( S  +  1 ) )  C_  NN0 )
78 ssralv 3505 . . . 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 62 . . 3  |-  ( ph  ->  A. k  e.  ( 0 ... ( S  +  1 ) ) C  e.  B )
801, 3, 17, 21, 79telgsumfz0s 17670 . 2  |-  ( ph  ->  ( G  gsumg  ( i  e.  ( 0 ... S ) 
|->  ( [_ i  / 
k ]_ C  .-  [_ (
i  +  1 )  /  k ]_ C
) ) )  =  ( [_ 0  / 
k ]_ C  .-  [_ ( S  +  1 )  /  k ]_ C
) )
81 peano2nn0 10939 . . . . . 6  |-  ( S  e.  NN0  ->  ( S  +  1 )  e. 
NN0 )
8221, 81syl 17 . . . . 5  |-  ( ph  ->  ( S  +  1 )  e.  NN0 )
8338ltp1d 10565 . . . . 5  |-  ( ph  ->  S  <  ( S  +  1 ) )
84 rspsbca 3359 . . . . . . 7  |-  ( ( ( S  +  1 )  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  ->  [. ( S  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )
)
85 ovex 6343 . . . . . . . 8  |-  ( S  +  1 )  e. 
_V
86 sbcimg 3321 . . . . . . . . 9  |-  ( ( S  +  1 )  e.  _V  ->  ( [. ( S  +  1 )  /  k ]. ( S  <  k  ->  C  =  .0.  )  <->  (
[. ( S  + 
1 )  /  k ]. S  <  k  ->  [. ( S  +  1 )  /  k ]. C  =  .0.  )
) )
87 sbcbr2g 4472 . . . . . . . . . . 11  |-  ( ( S  +  1 )  e.  _V  ->  ( [. ( S  +  1 )  /  k ]. S  <  k  <->  S  <  [_ ( S  +  1 )  /  k ]_ k ) )
88 csbvarg 3804 . . . . . . . . . . . 12  |-  ( ( S  +  1 )  e.  _V  ->  [_ ( S  +  1 )  /  k ]_ k  =  ( S  + 
1 ) )
8988breq2d 4428 . . . . . . . . . . 11  |-  ( ( S  +  1 )  e.  _V  ->  ( S  <  [_ ( S  + 
1 )  /  k ]_ k  <->  S  <  ( S  +  1 ) ) )
9087, 89bitrd 261 . . . . . . . . . 10  |-  ( ( S  +  1 )  e.  _V  ->  ( [. ( S  +  1 )  /  k ]. S  <  k  <->  S  <  ( S  +  1 ) ) )
91 sbceq1g 3789 . . . . . . . . . 10  |-  ( ( S  +  1 )  e.  _V  ->  ( [. ( S  +  1 )  /  k ]. C  =  .0.  <->  [_ ( S  +  1 )  / 
k ]_ C  =  .0.  ) )
9290, 91imbi12d 326 . . . . . . . . 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 261 . . . . . . . 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 201 . . . . . 6  |-  ( ( ( S  +  1 )  e.  NN0  /\  A. k  e.  NN0  ( S  <  k  ->  C  =  .0.  ) )  -> 
( S  <  ( S  +  1 )  ->  [_ ( S  + 
1 )  /  k ]_ C  =  .0.  ) )
9695ex 440 . . . . 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 63 . . . 4  |-  ( ph  ->  [_ ( S  + 
1 )  /  k ]_ C  =  .0.  )
9897oveq2d 6331 . . 3  |-  ( ph  ->  ( [_ 0  / 
k ]_ C  .-  [_ ( S  +  1 )  /  k ]_ C
)  =  ( [_
0  /  k ]_ C  .-  .0.  ) )
99 0nn0 10913 . . . . . 6  |-  0  e.  NN0
10099a1i 11 . . . . 5  |-  ( ph  ->  0  e.  NN0 )
101 rspcsbela 3807 . . . . 5  |-  ( ( 0  e.  NN0  /\  A. k  e.  NN0  C  e.  B )  ->  [_ 0  /  k ]_ C  e.  B )
102100, 10, 101syl2anc 671 . . . 4  |-  ( ph  ->  [_ 0  /  k ]_ C  e.  B
)
1031, 2, 17grpsubid1 16788 . . . 4  |-  ( ( G  e.  Grp  /\  [_ 0  /  k ]_ C  e.  B )  ->  ( [_ 0  / 
k ]_ C  .-  .0.  )  =  [_ 0  /  k ]_ C
)
1047, 102, 103syl2anc 671 . . 3  |-  ( ph  ->  ( [_ 0  / 
k ]_ C  .-  .0.  )  =  [_ 0  /  k ]_ C
)
10598, 104eqtrd 2496 . 2  |-  ( ph  ->  ( [_ 0  / 
k ]_ C  .-  [_ ( S  +  1 )  /  k ]_ C
)  =  [_ 0  /  k ]_ C
)
10673, 80, 1053eqtrd 2500 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 189    /\ wa 375    = wceq 1455    e. wcel 1898   A.wral 2749   _Vcvv 3057   [.wsbc 3279   [_csb 3375    C_ wss 3416   class class class wbr 4416    |-> cmpt 4475   ` cfv 5601  (class class class)co 6315   RRcr 9564   0cc0 9565   1c1 9566    + caddc 9568    < clt 9701   NN0cn0 10898   ZZ>=cuz 11188   ...cfz 11813   Basecbs 15170   0gc0g 15387    gsumg cgsu 15388   Grpcgrp 16718   -gcsg 16720  CMndccmn 17479   Abelcabl 17480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-om 6720  df-1st 6820  df-2nd 6821  df-supp 6942  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-map 7500  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fsupp 7910  df-oi 8051  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-2 10696  df-n0 10899  df-z 10967  df-uz 11189  df-fz 11814  df-fzo 11947  df-seq 12246  df-hash 12548  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-0g 15389  df-gsum 15390  df-mre 15541  df-mrc 15542  df-acs 15544  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-submnd 16632  df-grp 16722  df-minusg 16723  df-sbg 16724  df-mulg 16725  df-cntz 17020  df-cmn 17481  df-abl 17482
This theorem is referenced by:  telgsum  17673
  Copyright terms: Public domain W3C validator