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Theorem ostthlem1 23655
Description: Lemma for ostth 23667. If two absolute values agree on the positive integers greater than one, then they agree for all rational numbers and thus are equal as functions. (Contributed by Mario Carneiro, 9-Sep-2014.)
Hypotheses
Ref Expression
qrng.q  |-  Q  =  (flds  QQ )
qabsabv.a  |-  A  =  (AbsVal `  Q )
ostthlem1.1  |-  ( ph  ->  F  e.  A )
ostthlem1.2  |-  ( ph  ->  G  e.  A )
ostthlem1.3  |-  ( (
ph  /\  n  e.  ( ZZ>= `  2 )
)  ->  ( F `  n )  =  ( G `  n ) )
Assertion
Ref Expression
ostthlem1  |-  ( ph  ->  F  =  G )
Distinct variable groups:    n, G    ph, n    A, n    Q, n   
n, F

Proof of Theorem ostthlem1
Dummy variables  k 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ostthlem1.1 . . 3  |-  ( ph  ->  F  e.  A )
2 qabsabv.a . . . 4  |-  A  =  (AbsVal `  Q )
3 qrng.q . . . . 5  |-  Q  =  (flds  QQ )
43qrngbas 23647 . . . 4  |-  QQ  =  ( Base `  Q )
52, 4abvf 17320 . . 3  |-  ( F  e.  A  ->  F : QQ --> RR )
6 ffn 5736 . . 3  |-  ( F : QQ --> RR  ->  F  Fn  QQ )
71, 5, 63syl 20 . 2  |-  ( ph  ->  F  Fn  QQ )
8 ostthlem1.2 . . 3  |-  ( ph  ->  G  e.  A )
92, 4abvf 17320 . . 3  |-  ( G  e.  A  ->  G : QQ --> RR )
10 ffn 5736 . . 3  |-  ( G : QQ --> RR  ->  G  Fn  QQ )
118, 9, 103syl 20 . 2  |-  ( ph  ->  G  Fn  QQ )
12 elq 11194 . . . 4  |-  ( y  e.  QQ  <->  E. k  e.  ZZ  E. n  e.  NN  y  =  ( k  /  n ) )
133qdrng 23648 . . . . . . . . . 10  |-  Q  e.  DivRing
1413a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  ->  Q  e.  DivRing )
151adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  ->  F  e.  A )
16 zq 11198 . . . . . . . . . 10  |-  ( k  e.  ZZ  ->  k  e.  QQ )
1716ad2antrl 727 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
k  e.  QQ )
18 nnq 11205 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  QQ )
1918ad2antll 728 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  ->  n  e.  QQ )
20 nnne0 10578 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  =/=  0 )
2120ad2antll 728 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  ->  n  =/=  0 )
223qrng0 23649 . . . . . . . . . 10  |-  0  =  ( 0g `  Q )
23 eqid 2467 . . . . . . . . . 10  |-  (/r `  Q
)  =  (/r `  Q
)
242, 4, 22, 23abvdiv 17334 . . . . . . . . 9  |-  ( ( ( Q  e.  DivRing  /\  F  e.  A )  /\  ( k  e.  QQ  /\  n  e.  QQ  /\  n  =/=  0 ) )  ->  ( F `  ( k (/r `  Q
) n ) )  =  ( ( F `
 k )  / 
( F `  n
) ) )
2514, 15, 17, 19, 21, 24syl23anc 1235 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  (
k (/r `  Q ) n ) )  =  ( ( F `  k
)  /  ( F `
 n ) ) )
268adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  ->  G  e.  A )
272, 4, 22, 23abvdiv 17334 . . . . . . . . . 10  |-  ( ( ( Q  e.  DivRing  /\  G  e.  A )  /\  ( k  e.  QQ  /\  n  e.  QQ  /\  n  =/=  0 ) )  ->  ( G `  ( k (/r `  Q
) n ) )  =  ( ( G `
 k )  / 
( G `  n
) ) )
2814, 26, 17, 19, 21, 27syl23anc 1235 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( G `  (
k (/r `  Q ) n ) )  =  ( ( G `  k
)  /  ( G `
 n ) ) )
29 elz 10876 . . . . . . . . . . . . . 14  |-  ( k  e.  ZZ  <->  ( k  e.  RR  /\  ( k  =  0  \/  k  e.  NN  \/  -u k  e.  NN ) ) )
3029simprbi 464 . . . . . . . . . . . . 13  |-  ( k  e.  ZZ  ->  (
k  =  0  \/  k  e.  NN  \/  -u k  e.  NN ) )
3130adantl 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( k  =  0  \/  k  e.  NN  \/  -u k  e.  NN ) )
322, 22abv0 17328 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  A  ->  ( F `  0 )  =  0 )
331, 32syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( F `  0
)  =  0 )
342, 22abv0 17328 . . . . . . . . . . . . . . . . . 18  |-  ( G  e.  A  ->  ( G `  0 )  =  0 )
358, 34syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( G `  0
)  =  0 )
3633, 35eqtr4d 2511 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( F `  0
)  =  ( G `
 0 ) )
37 fveq2 5871 . . . . . . . . . . . . . . . . 17  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
38 fveq2 5871 . . . . . . . . . . . . . . . . 17  |-  ( k  =  0  ->  ( G `  k )  =  ( G ` 
0 ) )
3937, 38eqeq12d 2489 . . . . . . . . . . . . . . . 16  |-  ( k  =  0  ->  (
( F `  k
)  =  ( G `
 k )  <->  ( F `  0 )  =  ( G `  0
) ) )
4036, 39syl5ibrcom 222 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( k  =  0  ->  ( F `  k )  =  ( G `  k ) ) )
4140adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( k  =  0  ->  ( F `  k )  =  ( G `  k ) ) )
4241imp 429 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  =  0 )  -> 
( F `  k
)  =  ( G `
 k ) )
43 elnn1uz2 11168 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  <->  ( n  =  1  \/  n  e.  ( ZZ>= `  2 )
) )
443qrng1 23650 . . . . . . . . . . . . . . . . . . . . . . 23  |-  1  =  ( 1r `  Q )
452, 44abv1 17330 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Q  e.  DivRing  /\  F  e.  A )  ->  ( F `  1 )  =  1 )
4613, 1, 45sylancr 663 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( F `  1
)  =  1 )
472, 44abv1 17330 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Q  e.  DivRing  /\  G  e.  A )  ->  ( G `  1 )  =  1 )
4813, 8, 47sylancr 663 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( G `  1
)  =  1 )
4946, 48eqtr4d 2511 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
50 fveq2 5871 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  1  ->  ( F `  n )  =  ( F ` 
1 ) )
51 fveq2 5871 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =  1  ->  ( G `  n )  =  ( G ` 
1 ) )
5250, 51eqeq12d 2489 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  1  ->  (
( F `  n
)  =  ( G `
 n )  <->  ( F `  1 )  =  ( G `  1
) ) )
5349, 52syl5ibrcom 222 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( n  =  1  ->  ( F `  n )  =  ( G `  n ) ) )
5453imp 429 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  n  = 
1 )  ->  ( F `  n )  =  ( G `  n ) )
55 ostthlem1.3 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  n  e.  ( ZZ>= `  2 )
)  ->  ( F `  n )  =  ( G `  n ) )
5654, 55jaodan 783 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( n  =  1  \/  n  e.  ( ZZ>= `  2 )
) )  ->  ( F `  n )  =  ( G `  n ) )
5743, 56sylan2b 475 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =  ( G `  n
) )
5857ralrimiva 2881 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A. n  e.  NN  ( F `  n )  =  ( G `  n ) )
5958adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  k  e.  ZZ )  ->  A. n  e.  NN  ( F `  n )  =  ( G `  n ) )
60 fveq2 5871 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
61 fveq2 5871 . . . . . . . . . . . . . . . 16  |-  ( n  =  k  ->  ( G `  n )  =  ( G `  k ) )
6260, 61eqeq12d 2489 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  (
( F `  n
)  =  ( G `
 n )  <->  ( F `  k )  =  ( G `  k ) ) )
6362rspccva 3218 . . . . . . . . . . . . . 14  |-  ( ( A. n  e.  NN  ( F `  n )  =  ( G `  n )  /\  k  e.  NN )  ->  ( F `  k )  =  ( G `  k ) )
6459, 63sylan 471 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  NN )  ->  ( F `  k )  =  ( G `  k ) )
6558ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  A. n  e.  NN  ( F `  n )  =  ( G `  n ) )
6616adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  k  e.  ZZ )  ->  k  e.  QQ )
673qrngneg 23651 . . . . . . . . . . . . . . . . . 18  |-  ( k  e.  QQ  ->  (
( invg `  Q ) `  k
)  =  -u k
)
6866, 67syl 16 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( ( invg `  Q
) `  k )  =  -u k )
6968eleq1d 2536 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( ( ( invg `  Q ) `  k
)  e.  NN  <->  -u k  e.  NN ) )
7069biimpar 485 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  (
( invg `  Q ) `  k
)  e.  NN )
71 fveq2 5871 . . . . . . . . . . . . . . . . 17  |-  ( n  =  ( ( invg `  Q ) `
 k )  -> 
( F `  n
)  =  ( F `
 ( ( invg `  Q ) `
 k ) ) )
72 fveq2 5871 . . . . . . . . . . . . . . . . 17  |-  ( n  =  ( ( invg `  Q ) `
 k )  -> 
( G `  n
)  =  ( G `
 ( ( invg `  Q ) `
 k ) ) )
7371, 72eqeq12d 2489 . . . . . . . . . . . . . . . 16  |-  ( n  =  ( ( invg `  Q ) `
 k )  -> 
( ( F `  n )  =  ( G `  n )  <-> 
( F `  (
( invg `  Q ) `  k
) )  =  ( G `  ( ( invg `  Q
) `  k )
) ) )
7473rspccva 3218 . . . . . . . . . . . . . . 15  |-  ( ( A. n  e.  NN  ( F `  n )  =  ( G `  n )  /\  (
( invg `  Q ) `  k
)  e.  NN )  ->  ( F `  ( ( invg `  Q ) `  k
) )  =  ( G `  ( ( invg `  Q
) `  k )
) )
7565, 70, 74syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  ( F `  ( ( invg `  Q ) `
 k ) )  =  ( G `  ( ( invg `  Q ) `  k
) ) )
761ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  F  e.  A )
7716ad2antlr 726 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  k  e.  QQ )
78 eqid 2467 . . . . . . . . . . . . . . . 16  |-  ( invg `  Q )  =  ( invg `  Q )
792, 4, 78abvneg 17331 . . . . . . . . . . . . . . 15  |-  ( ( F  e.  A  /\  k  e.  QQ )  ->  ( F `  (
( invg `  Q ) `  k
) )  =  ( F `  k ) )
8076, 77, 79syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  ( F `  ( ( invg `  Q ) `
 k ) )  =  ( F `  k ) )
818ad2antrr 725 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  G  e.  A )
822, 4, 78abvneg 17331 . . . . . . . . . . . . . . 15  |-  ( ( G  e.  A  /\  k  e.  QQ )  ->  ( G `  (
( invg `  Q ) `  k
) )  =  ( G `  k ) )
8381, 77, 82syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  ( G `  ( ( invg `  Q ) `
 k ) )  =  ( G `  k ) )
8475, 80, 833eqtr3d 2516 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  ( F `  k )  =  ( G `  k ) )
8542, 64, 843jaodan 1294 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  (
k  =  0  \/  k  e.  NN  \/  -u k  e.  NN ) )  ->  ( F `  k )  =  ( G `  k ) )
8631, 85mpdan 668 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( F `
 k )  =  ( G `  k
) )
8786adantrr 716 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  k
)  =  ( G `
 k ) )
8857adantrl 715 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  n
)  =  ( G `
 n ) )
8987, 88oveq12d 6312 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( ( F `  k )  /  ( F `  n )
)  =  ( ( G `  k )  /  ( G `  n ) ) )
9028, 89eqtr4d 2511 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( G `  (
k (/r `  Q ) n ) )  =  ( ( F `  k
)  /  ( F `
 n ) ) )
9125, 90eqtr4d 2511 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  (
k (/r `  Q ) n ) )  =  ( G `  ( k (/r `  Q ) n ) ) )
923qrngdiv 23652 . . . . . . . . 9  |-  ( ( k  e.  QQ  /\  n  e.  QQ  /\  n  =/=  0 )  ->  (
k (/r `  Q ) n )  =  ( k  /  n ) )
9317, 19, 21, 92syl3anc 1228 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( k (/r `  Q
) n )  =  ( k  /  n
) )
9493fveq2d 5875 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  (
k (/r `  Q ) n ) )  =  ( F `  ( k  /  n ) ) )
9593fveq2d 5875 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( G `  (
k (/r `  Q ) n ) )  =  ( G `  ( k  /  n ) ) )
9691, 94, 953eqtr3d 2516 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  (
k  /  n ) )  =  ( G `
 ( k  /  n ) ) )
97 fveq2 5871 . . . . . . 7  |-  ( y  =  ( k  /  n )  ->  ( F `  y )  =  ( F `  ( k  /  n
) ) )
98 fveq2 5871 . . . . . . 7  |-  ( y  =  ( k  /  n )  ->  ( G `  y )  =  ( G `  ( k  /  n
) ) )
9997, 98eqeq12d 2489 . . . . . 6  |-  ( y  =  ( k  /  n )  ->  (
( F `  y
)  =  ( G `
 y )  <->  ( F `  ( k  /  n
) )  =  ( G `  ( k  /  n ) ) ) )
10096, 99syl5ibrcom 222 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( y  =  ( k  /  n )  ->  ( F `  y )  =  ( G `  y ) ) )
101100rexlimdvva 2966 . . . 4  |-  ( ph  ->  ( E. k  e.  ZZ  E. n  e.  NN  y  =  ( k  /  n )  ->  ( F `  y )  =  ( G `  y ) ) )
10212, 101syl5bi 217 . . 3  |-  ( ph  ->  ( y  e.  QQ  ->  ( F `  y
)  =  ( G `
 y ) ) )
103102imp 429 . 2  |-  ( (
ph  /\  y  e.  QQ )  ->  ( F `
 y )  =  ( G `  y
) )
1047, 11, 103eqfnfvd 5984 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    Fn wfn 5588   -->wf 5589   ` cfv 5593  (class class class)co 6294   RRcr 9501   0cc0 9502   1c1 9503   -ucneg 9816    / cdiv 10216   NNcn 10546   2c2 10595   ZZcz 10874   ZZ>=cuz 11092   QQcq 11192   ↾s cress 14503   invgcminusg 15903  /rcdvr 17180   DivRingcdr 17244  AbsValcabv 17313  ℂfldccnfld 18267
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 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-addf 9581  ax-mulf 9582
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 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-tpos 6965  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-7 10609  df-8 10610  df-9 10611  df-10 10612  df-n0 10806  df-z 10875  df-dec 10987  df-uz 11093  df-q 11193  df-ico 11545  df-fz 11683  df-seq 12086  df-exp 12145  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-starv 14582  df-tset 14586  df-ple 14587  df-ds 14589  df-unif 14590  df-0g 14709  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-grp 15906  df-minusg 15907  df-subg 16047  df-cmn 16650  df-mgp 16991  df-ur 17003  df-ring 17049  df-cring 17050  df-oppr 17121  df-dvdsr 17139  df-unit 17140  df-invr 17170  df-dvr 17181  df-drng 17246  df-subrg 17275  df-abv 17314  df-cnfld 18268
This theorem is referenced by:  ostthlem2  23656  ostth2  23665
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