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Theorem ostthlem1 23684
Description: Lemma for ostth 23696. 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 23676 . . . 4  |-  QQ  =  ( Base `  Q )
52, 4abvf 17346 . . 3  |-  ( F  e.  A  ->  F : QQ --> RR )
6 ffn 5721 . . 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 17346 . . 3  |-  ( G  e.  A  ->  G : QQ --> RR )
10 ffn 5721 . . 3  |-  ( G : QQ --> RR  ->  G  Fn  QQ )
118, 9, 103syl 20 . 2  |-  ( ph  ->  G  Fn  QQ )
12 elq 11193 . . . 4  |-  ( y  e.  QQ  <->  E. k  e.  ZZ  E. n  e.  NN  y  =  ( k  /  n ) )
133qdrng 23677 . . . . . . . . . 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 11197 . . . . . . . . . 10  |-  ( k  e.  ZZ  ->  k  e.  QQ )
1716ad2antrl 727 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
k  e.  QQ )
18 nnq 11204 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  e.  QQ )
1918ad2antll 728 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  ->  n  e.  QQ )
20 nnne0 10574 . . . . . . . . . 10  |-  ( n  e.  NN  ->  n  =/=  0 )
2120ad2antll 728 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  ->  n  =/=  0 )
223qrng0 23678 . . . . . . . . . 10  |-  0  =  ( 0g `  Q )
23 eqid 2443 . . . . . . . . . 10  |-  (/r `  Q
)  =  (/r `  Q
)
242, 4, 22, 23abvdiv 17360 . . . . . . . . 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 1236 . . . . . . . 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 17360 . . . . . . . . . 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 1236 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( G `  (
k (/r `  Q ) n ) )  =  ( ( G `  k
)  /  ( G `
 n ) ) )
292, 22abv0 17354 . . . . . . . . . . . . . . . . 17  |-  ( F  e.  A  ->  ( F `  0 )  =  0 )
301, 29syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( F `  0
)  =  0 )
312, 22abv0 17354 . . . . . . . . . . . . . . . . 17  |-  ( G  e.  A  ->  ( G `  0 )  =  0 )
328, 31syl 16 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( G `  0
)  =  0 )
3330, 32eqtr4d 2487 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( F `  0
)  =  ( G `
 0 ) )
34 fveq2 5856 . . . . . . . . . . . . . . . 16  |-  ( k  =  0  ->  ( F `  k )  =  ( F ` 
0 ) )
35 fveq2 5856 . . . . . . . . . . . . . . . 16  |-  ( k  =  0  ->  ( G `  k )  =  ( G ` 
0 ) )
3634, 35eqeq12d 2465 . . . . . . . . . . . . . . 15  |-  ( k  =  0  ->  (
( F `  k
)  =  ( G `
 k )  <->  ( F `  0 )  =  ( G `  0
) ) )
3733, 36syl5ibrcom 222 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( k  =  0  ->  ( F `  k )  =  ( G `  k ) ) )
3837adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( k  =  0  ->  ( F `  k )  =  ( G `  k ) ) )
3938imp 429 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  =  0 )  -> 
( F `  k
)  =  ( G `
 k ) )
40 elnn1uz2 11167 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  <->  ( n  =  1  \/  n  e.  ( ZZ>= `  2 )
) )
413qrng1 23679 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  =  ( 1r `  Q )
422, 41abv1 17356 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Q  e.  DivRing  /\  F  e.  A )  ->  ( F `  1 )  =  1 )
4313, 1, 42sylancr 663 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( F `  1
)  =  1 )
442, 41abv1 17356 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( Q  e.  DivRing  /\  G  e.  A )  ->  ( G `  1 )  =  1 )
4513, 8, 44sylancr 663 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( G `  1
)  =  1 )
4643, 45eqtr4d 2487 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
47 fveq2 5856 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  1  ->  ( F `  n )  =  ( F ` 
1 ) )
48 fveq2 5856 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  1  ->  ( G `  n )  =  ( G ` 
1 ) )
4947, 48eqeq12d 2465 . . . . . . . . . . . . . . . . . . 19  |-  ( n  =  1  ->  (
( F `  n
)  =  ( G `
 n )  <->  ( F `  1 )  =  ( G `  1
) ) )
5046, 49syl5ibrcom 222 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( n  =  1  ->  ( F `  n )  =  ( G `  n ) ) )
5150imp 429 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  = 
1 )  ->  ( F `  n )  =  ( G `  n ) )
52 ostthlem1.3 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( ZZ>= `  2 )
)  ->  ( F `  n )  =  ( G `  n ) )
5351, 52jaodan 785 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( n  =  1  \/  n  e.  ( ZZ>= `  2 )
) )  ->  ( F `  n )  =  ( G `  n ) )
5440, 53sylan2b 475 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  n  e.  NN )  ->  ( F `
 n )  =  ( G `  n
) )
5554ralrimiva 2857 . . . . . . . . . . . . . 14  |-  ( ph  ->  A. n  e.  NN  ( F `  n )  =  ( G `  n ) )
5655adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  k  e.  ZZ )  ->  A. n  e.  NN  ( F `  n )  =  ( G `  n ) )
57 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  ( F `  n )  =  ( F `  k ) )
58 fveq2 5856 . . . . . . . . . . . . . . 15  |-  ( n  =  k  ->  ( G `  n )  =  ( G `  k ) )
5957, 58eqeq12d 2465 . . . . . . . . . . . . . 14  |-  ( n  =  k  ->  (
( F `  n
)  =  ( G `
 n )  <->  ( F `  k )  =  ( G `  k ) ) )
6059rspccva 3195 . . . . . . . . . . . . 13  |-  ( ( A. n  e.  NN  ( F `  n )  =  ( G `  n )  /\  k  e.  NN )  ->  ( F `  k )  =  ( G `  k ) )
6156, 60sylan 471 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  k  e.  NN )  ->  ( F `  k )  =  ( G `  k ) )
6255ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  A. n  e.  NN  ( F `  n )  =  ( G `  n ) )
6316adantl 466 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  k  e.  ZZ )  ->  k  e.  QQ )
643qrngneg 23680 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  QQ  ->  (
( invg `  Q ) `  k
)  =  -u k
)
6563, 64syl 16 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( ( invg `  Q
) `  k )  =  -u k )
6665eleq1d 2512 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( ( ( invg `  Q ) `  k
)  e.  NN  <->  -u k  e.  NN ) )
6766biimpar 485 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  (
( invg `  Q ) `  k
)  e.  NN )
68 fveq2 5856 . . . . . . . . . . . . . . . 16  |-  ( n  =  ( ( invg `  Q ) `
 k )  -> 
( F `  n
)  =  ( F `
 ( ( invg `  Q ) `
 k ) ) )
69 fveq2 5856 . . . . . . . . . . . . . . . 16  |-  ( n  =  ( ( invg `  Q ) `
 k )  -> 
( G `  n
)  =  ( G `
 ( ( invg `  Q ) `
 k ) ) )
7068, 69eqeq12d 2465 . . . . . . . . . . . . . . 15  |-  ( n  =  ( ( invg `  Q ) `
 k )  -> 
( ( F `  n )  =  ( G `  n )  <-> 
( F `  (
( invg `  Q ) `  k
) )  =  ( G `  ( ( invg `  Q
) `  k )
) ) )
7170rspccva 3195 . . . . . . . . . . . . . 14  |-  ( ( A. n  e.  NN  ( F `  n )  =  ( G `  n )  /\  (
( invg `  Q ) `  k
)  e.  NN )  ->  ( F `  ( ( invg `  Q ) `  k
) )  =  ( G `  ( ( invg `  Q
) `  k )
) )
7262, 67, 71syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  ( F `  ( ( invg `  Q ) `
 k ) )  =  ( G `  ( ( invg `  Q ) `  k
) ) )
731ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  F  e.  A )
7416ad2antlr 726 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  k  e.  QQ )
75 eqid 2443 . . . . . . . . . . . . . . 15  |-  ( invg `  Q )  =  ( invg `  Q )
762, 4, 75abvneg 17357 . . . . . . . . . . . . . 14  |-  ( ( F  e.  A  /\  k  e.  QQ )  ->  ( F `  (
( invg `  Q ) `  k
) )  =  ( F `  k ) )
7773, 74, 76syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  ( F `  ( ( invg `  Q ) `
 k ) )  =  ( F `  k ) )
788ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  G  e.  A )
792, 4, 75abvneg 17357 . . . . . . . . . . . . . 14  |-  ( ( G  e.  A  /\  k  e.  QQ )  ->  ( G `  (
( invg `  Q ) `  k
) )  =  ( G `  k ) )
8078, 74, 79syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  ( G `  ( ( invg `  Q ) `
 k ) )  =  ( G `  k ) )
8172, 77, 803eqtr3d 2492 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  k  e.  ZZ )  /\  -u k  e.  NN )  ->  ( F `  k )  =  ( G `  k ) )
82 elz 10872 . . . . . . . . . . . . . 14  |-  ( k  e.  ZZ  <->  ( k  e.  RR  /\  ( k  =  0  \/  k  e.  NN  \/  -u k  e.  NN ) ) )
8382simprbi 464 . . . . . . . . . . . . 13  |-  ( k  e.  ZZ  ->  (
k  =  0  \/  k  e.  NN  \/  -u k  e.  NN ) )
8483adantl 466 . . . . . . . . . . . 12  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( k  =  0  \/  k  e.  NN  \/  -u k  e.  NN ) )
8539, 61, 81, 84mpjao3dan 1296 . . . . . . . . . . 11  |-  ( (
ph  /\  k  e.  ZZ )  ->  ( F `
 k )  =  ( G `  k
) )
8685adantrr 716 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  k
)  =  ( G `
 k ) )
8754adantrl 715 . . . . . . . . . 10  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  n
)  =  ( G `
 n ) )
8886, 87oveq12d 6299 . . . . . . . . 9  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( ( F `  k )  /  ( F `  n )
)  =  ( ( G `  k )  /  ( G `  n ) ) )
8928, 88eqtr4d 2487 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( G `  (
k (/r `  Q ) n ) )  =  ( ( F `  k
)  /  ( F `
 n ) ) )
9025, 89eqtr4d 2487 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  (
k (/r `  Q ) n ) )  =  ( G `  ( k (/r `  Q ) n ) ) )
913qrngdiv 23681 . . . . . . . . 9  |-  ( ( k  e.  QQ  /\  n  e.  QQ  /\  n  =/=  0 )  ->  (
k (/r `  Q ) n )  =  ( k  /  n ) )
9217, 19, 21, 91syl3anc 1229 . . . . . . . 8  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( k (/r `  Q
) n )  =  ( k  /  n
) )
9392fveq2d 5860 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  (
k (/r `  Q ) n ) )  =  ( F `  ( k  /  n ) ) )
9492fveq2d 5860 . . . . . . 7  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( G `  (
k (/r `  Q ) n ) )  =  ( G `  ( k  /  n ) ) )
9590, 93, 943eqtr3d 2492 . . . . . 6  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( F `  (
k  /  n ) )  =  ( G `
 ( k  /  n ) ) )
96 fveq2 5856 . . . . . . 7  |-  ( y  =  ( k  /  n )  ->  ( F `  y )  =  ( F `  ( k  /  n
) ) )
97 fveq2 5856 . . . . . . 7  |-  ( y  =  ( k  /  n )  ->  ( G `  y )  =  ( G `  ( k  /  n
) ) )
9896, 97eqeq12d 2465 . . . . . 6  |-  ( y  =  ( k  /  n )  ->  (
( F `  y
)  =  ( G `
 y )  <->  ( F `  ( k  /  n
) )  =  ( G `  ( k  /  n ) ) ) )
9995, 98syl5ibrcom 222 . . . . 5  |-  ( (
ph  /\  ( k  e.  ZZ  /\  n  e.  NN ) )  -> 
( y  =  ( k  /  n )  ->  ( F `  y )  =  ( G `  y ) ) )
10099rexlimdvva 2942 . . . 4  |-  ( ph  ->  ( E. k  e.  ZZ  E. n  e.  NN  y  =  ( k  /  n )  ->  ( F `  y )  =  ( G `  y ) ) )
10112, 100syl5bi 217 . . 3  |-  ( ph  ->  ( y  e.  QQ  ->  ( F `  y
)  =  ( G `
 y ) ) )
102101imp 429 . 2  |-  ( (
ph  /\  y  e.  QQ )  ->  ( F `
 y )  =  ( G `  y
) )
1037, 11, 102eqfnfvd 5969 1  |-  ( ph  ->  F  =  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 973    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794    Fn wfn 5573   -->wf 5574   ` cfv 5578  (class class class)co 6281   RRcr 9494   0cc0 9495   1c1 9496   -ucneg 9811    / cdiv 10212   NNcn 10542   2c2 10591   ZZcz 10870   ZZ>=cuz 11090   QQcq 11191   ↾s cress 14510   invgcminusg 15928  /rcdvr 17205   DivRingcdr 17270  AbsValcabv 17339  ℂfldccnfld 18294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-dec 10985  df-uz 11091  df-q 11192  df-ico 11544  df-fz 11682  df-seq 12087  df-exp 12146  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-starv 14589  df-tset 14593  df-ple 14594  df-ds 14596  df-unif 14597  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-subg 16072  df-cmn 16674  df-mgp 17016  df-ur 17028  df-ring 17074  df-cring 17075  df-oppr 17146  df-dvdsr 17164  df-unit 17165  df-invr 17195  df-dvr 17206  df-drng 17272  df-subrg 17301  df-abv 17340  df-cnfld 18295
This theorem is referenced by:  ostthlem2  23685  ostth2  23694
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