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Theorem abvres 15932
Description: The restriction of an absolute value to a subring is an absolute value. (Contributed by Mario Carneiro, 4-Dec-2014.)
Hypotheses
Ref Expression
abvres.a  |-  A  =  (AbsVal `  R )
abvres.s  |-  S  =  ( Rs  C )
abvres.b  |-  B  =  (AbsVal `  S )
Assertion
Ref Expression
abvres  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F  |`  C )  e.  B )

Proof of Theorem abvres
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 abvres.b . . 3  |-  B  =  (AbsVal `  S )
21a1i 11 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  B  =  (AbsVal `  S )
)
3 abvres.s . . . 4  |-  S  =  ( Rs  C )
43subrgbas 15882 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  C  =  ( Base `  S )
)
54adantl 454 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  =  ( Base `  S
) )
6 eqid 2438 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
73, 6ressplusg 13576 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  ( +g  `  R )  =  ( +g  `  S ) )
87adantl 454 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( +g  `  R )  =  ( +g  `  S
) )
9 eqid 2438 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
103, 9ressmulr 13587 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1110adantl 454 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( .r `  R )  =  ( .r `  S
) )
12 subrgsubg 15879 . . . 4  |-  ( C  e.  (SubRing `  R
)  ->  C  e.  (SubGrp `  R ) )
1312adantl 454 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  e.  (SubGrp `  R )
)
14 eqid 2438 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
153, 14subg0 14955 . . 3  |-  ( C  e.  (SubGrp `  R
)  ->  ( 0g `  R )  =  ( 0g `  S ) )
1613, 15syl 16 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( 0g `  R )  =  ( 0g `  S
) )
173subrgrng 15876 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  S  e.  Ring )
1817adantl 454 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  S  e.  Ring )
19 abvres.a . . . 4  |-  A  =  (AbsVal `  R )
20 eqid 2438 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2119, 20abvf 15916 . . 3  |-  ( F  e.  A  ->  F : ( Base `  R
) --> RR )
2220subrgss 15874 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  C  C_  ( Base `  R ) )
23 fssres 5613 . . 3  |-  ( ( F : ( Base `  R ) --> RR  /\  C  C_  ( Base `  R
) )  ->  ( F  |`  C ) : C --> RR )
2421, 22, 23syl2an 465 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F  |`  C ) : C --> RR )
2514subg0cl 14957 . . . 4  |-  ( C  e.  (SubGrp `  R
)  ->  ( 0g `  R )  e.  C
)
26 fvres 5748 . . . 4  |-  ( ( 0g `  R )  e.  C  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  ( F `  ( 0g `  R ) ) )
2713, 25, 263syl 19 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  ( F `  ( 0g `  R ) ) )
2819, 14abv0 15924 . . . 4  |-  ( F  e.  A  ->  ( F `  ( 0g `  R ) )  =  0 )
2928adantr 453 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F `  ( 0g `  R ) )  =  0 )
3027, 29eqtrd 2470 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  0 )
31 simp1l 982 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  F  e.  A )
3222adantl 454 . . . . . 6  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  C_  ( Base `  R
) )
3332sselda 3350 . . . . 5  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C )  ->  x  e.  ( Base `  R ) )
34333adant3 978 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  x  e.  ( Base `  R ) )
35 simp3 960 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  x  =/=  ( 0g `  R ) )
3619, 20, 14abvgt0 15921 . . . 4  |-  ( ( F  e.  A  /\  x  e.  ( Base `  R )  /\  x  =/=  ( 0g `  R
) )  ->  0  <  ( F `  x
) )
3731, 34, 35, 36syl3anc 1185 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
0  <  ( F `  x ) )
38 fvres 5748 . . . 4  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
39383ad2ant2 980 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
( ( F  |`  C ) `  x
)  =  ( F `
 x ) )
4037, 39breqtrrd 4241 . 2  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
0  <  ( ( F  |`  C ) `  x ) )
41 simp1l 982 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  ->  F  e.  A )
42 simp1r 983 . . . . . 6  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  ->  C  e.  (SubRing `  R
) )
4342, 22syl 16 . . . . 5  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  ->  C  C_  ( Base `  R
) )
44 simp2l 984 . . . . 5  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  ->  x  e.  C )
4543, 44sseldd 3351 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  ->  x  e.  ( Base `  R ) )
46 simp3l 986 . . . . 5  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
y  e.  C )
4743, 46sseldd 3351 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
y  e.  ( Base `  R ) )
4819, 20, 9abvmul 15922 . . . 4  |-  ( ( F  e.  A  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  ( F `  ( x
( .r `  R
) y ) )  =  ( ( F `
 x )  x.  ( F `  y
) ) )
4941, 45, 47, 48syl3anc 1185 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( F `  (
x ( .r `  R ) y ) )  =  ( ( F `  x )  x.  ( F `  y ) ) )
509subrgmcl 15885 . . . . 5  |-  ( ( C  e.  (SubRing `  R
)  /\  x  e.  C  /\  y  e.  C
)  ->  ( x
( .r `  R
) y )  e.  C )
5142, 44, 46, 50syl3anc 1185 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( x ( .r
`  R ) y )  e.  C )
52 fvres 5748 . . . 4  |-  ( ( x ( .r `  R ) y )  e.  C  ->  (
( F  |`  C ) `
 ( x ( .r `  R ) y ) )  =  ( F `  (
x ( .r `  R ) y ) ) )
5351, 52syl 16 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( ( F  |`  C ) `  (
x ( .r `  R ) y ) )  =  ( F `
 ( x ( .r `  R ) y ) ) )
5444, 38syl 16 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( ( F  |`  C ) `  x
)  =  ( F `
 x ) )
55 fvres 5748 . . . . 5  |-  ( y  e.  C  ->  (
( F  |`  C ) `
 y )  =  ( F `  y
) )
5646, 55syl 16 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( ( F  |`  C ) `  y
)  =  ( F `
 y ) )
5754, 56oveq12d 6102 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( ( ( F  |`  C ) `  x
)  x.  ( ( F  |`  C ) `  y ) )  =  ( ( F `  x )  x.  ( F `  y )
) )
5849, 53, 573eqtr4d 2480 . 2  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( ( F  |`  C ) `  (
x ( .r `  R ) y ) )  =  ( ( ( F  |`  C ) `
 x )  x.  ( ( F  |`  C ) `  y
) ) )
5919, 20, 6abvtri 15923 . . . 4  |-  ( ( F  e.  A  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R
) )  ->  ( F `  ( x
( +g  `  R ) y ) )  <_ 
( ( F `  x )  +  ( F `  y ) ) )
6041, 45, 47, 59syl3anc 1185 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( F `  (
x ( +g  `  R
) y ) )  <_  ( ( F `
 x )  +  ( F `  y
) ) )
616subrgacl 15884 . . . . 5  |-  ( ( C  e.  (SubRing `  R
)  /\  x  e.  C  /\  y  e.  C
)  ->  ( x
( +g  `  R ) y )  e.  C
)
6242, 44, 46, 61syl3anc 1185 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( x ( +g  `  R ) y )  e.  C )
63 fvres 5748 . . . 4  |-  ( ( x ( +g  `  R
) y )  e.  C  ->  ( ( F  |`  C ) `  ( x ( +g  `  R ) y ) )  =  ( F `
 ( x ( +g  `  R ) y ) ) )
6462, 63syl 16 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( ( F  |`  C ) `  (
x ( +g  `  R
) y ) )  =  ( F `  ( x ( +g  `  R ) y ) ) )
6554, 56oveq12d 6102 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( ( ( F  |`  C ) `  x
)  +  ( ( F  |`  C ) `  y ) )  =  ( ( F `  x )  +  ( F `  y ) ) )
6660, 64, 653brtr4d 4245 . 2  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  ( x  e.  C  /\  x  =/=  ( 0g `  R ) )  /\  ( y  e.  C  /\  y  =/=  ( 0g `  R
) ) )  -> 
( ( F  |`  C ) `  (
x ( +g  `  R
) y ) )  <_  ( ( ( F  |`  C ) `  x )  +  ( ( F  |`  C ) `
 y ) ) )
672, 5, 8, 11, 16, 18, 24, 30, 40, 58, 66isabvd 15913 1  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F  |`  C )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601    C_ wss 3322   class class class wbr 4215    |` cres 4883   -->wf 5453   ` cfv 5457  (class class class)co 6084   RRcr 8994   0cc0 8995    + caddc 8998    x. cmul 9000    < clt 9125    <_ cle 9126   Basecbs 13474   ↾s cress 13475   +g cplusg 13534   .rcmulr 13535   0gc0g 13728  SubGrpcsubg 14943   Ringcrg 15665  SubRingcsubrg 15869  AbsValcabv 15909
This theorem is referenced by:  subrgnrg  18714  qabsabv  21328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-riota 6552  df-recs 6636  df-rdg 6671  df-er 6908  df-map 7023  df-en 7113  df-dom 7114  df-sdom 7115  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-nn 10006  df-2 10063  df-3 10064  df-ico 10927  df-ndx 13477  df-slot 13478  df-base 13479  df-sets 13480  df-ress 13481  df-plusg 13547  df-mulr 13548  df-0g 13732  df-mnd 14695  df-grp 14817  df-minusg 14818  df-subg 14946  df-mgp 15654  df-rng 15668  df-subrg 15871  df-abv 15910
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