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Theorem abvres 15606
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 10 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  B  =  (AbsVal `  S )
)
3 abvres.s . . . 4  |-  S  =  ( Rs  C )
43subrgbas 15556 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  C  =  ( Base `  S )
)
54adantl 452 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  =  ( Base `  S
) )
6 eqid 2285 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
73, 6ressplusg 13252 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  ( +g  `  R )  =  ( +g  `  S ) )
87adantl 452 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( +g  `  R )  =  ( +g  `  S
) )
9 eqid 2285 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
103, 9ressmulr 13263 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1110adantl 452 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( .r `  R )  =  ( .r `  S
) )
12 subrgsubg 15553 . . . 4  |-  ( C  e.  (SubRing `  R
)  ->  C  e.  (SubGrp `  R ) )
1312adantl 452 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  e.  (SubGrp `  R )
)
14 eqid 2285 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
153, 14subg0 14629 . . 3  |-  ( C  e.  (SubGrp `  R
)  ->  ( 0g `  R )  =  ( 0g `  S ) )
1613, 15syl 15 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( 0g `  R )  =  ( 0g `  S
) )
173subrgrng 15550 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  S  e.  Ring )
1817adantl 452 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  S  e.  Ring )
19 abvres.a . . . 4  |-  A  =  (AbsVal `  R )
20 eqid 2285 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2119, 20abvf 15590 . . 3  |-  ( F  e.  A  ->  F : ( Base `  R
) --> RR )
2220subrgss 15548 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  C  C_  ( Base `  R ) )
23 fssres 5410 . . 3  |-  ( ( F : ( Base `  R ) --> RR  /\  C  C_  ( Base `  R
) )  ->  ( F  |`  C ) : C --> RR )
2421, 22, 23syl2an 463 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F  |`  C ) : C --> RR )
2514subg0cl 14631 . . . 4  |-  ( C  e.  (SubGrp `  R
)  ->  ( 0g `  R )  e.  C
)
26 fvres 5544 . . . 4  |-  ( ( 0g `  R )  e.  C  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  ( F `  ( 0g `  R ) ) )
2713, 25, 263syl 18 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  ( F `  ( 0g `  R ) ) )
2819, 14abv0 15598 . . . 4  |-  ( F  e.  A  ->  ( F `  ( 0g `  R ) )  =  0 )
2928adantr 451 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F `  ( 0g `  R ) )  =  0 )
3027, 29eqtrd 2317 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  0 )
31 simp1l 979 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  F  e.  A )
3222adantl 452 . . . . . 6  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  C_  ( Base `  R
) )
3332sselda 3182 . . . . 5  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C )  ->  x  e.  ( Base `  R ) )
34333adant3 975 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  x  e.  ( Base `  R ) )
35 simp3 957 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  x  =/=  ( 0g `  R ) )
3619, 20, 14abvgt0 15595 . . . 4  |-  ( ( F  e.  A  /\  x  e.  ( Base `  R )  /\  x  =/=  ( 0g `  R
) )  ->  0  <  ( F `  x
) )
3731, 34, 35, 36syl3anc 1182 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
0  <  ( F `  x ) )
38 fvres 5544 . . . 4  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
39383ad2ant2 977 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
( ( F  |`  C ) `  x
)  =  ( F `
 x ) )
4037, 39breqtrrd 4051 . 2  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
0  <  ( ( F  |`  C ) `  x ) )
41 simp1l 979 . . . 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 980 . . . . . 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 15 . . . . 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 981 . . . . 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 3183 . . . 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 983 . . . . 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 3183 . . . 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 15596 . . . 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 1182 . . 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 15559 . . . . 5  |-  ( ( C  e.  (SubRing `  R
)  /\  x  e.  C  /\  y  e.  C
)  ->  ( x
( .r `  R
) y )  e.  C )
5142, 44, 46, 50syl3anc 1182 . . . 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 5544 . . . 4  |-  ( ( x ( .r `  R ) y )  e.  C  ->  (
( F  |`  C ) `
 ( x ( .r `  R ) y ) )  =  ( F `  (
x ( .r `  R ) y ) ) )
5351, 52syl 15 . . 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 15 . . . 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 5544 . . . . 5  |-  ( y  e.  C  ->  (
( F  |`  C ) `
 y )  =  ( F `  y
) )
5646, 55syl 15 . . . 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 5878 . . 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 2327 . 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 15597 . . . 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 1182 . . 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 15558 . . . . 5  |-  ( ( C  e.  (SubRing `  R
)  /\  x  e.  C  /\  y  e.  C
)  ->  ( x
( +g  `  R ) y )  e.  C
)
6242, 44, 46, 61syl3anc 1182 . . . 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 5544 . . . 4  |-  ( ( x ( +g  `  R
) y )  e.  C  ->  ( ( F  |`  C ) `  ( x ( +g  `  R ) y ) )  =  ( F `
 ( x ( +g  `  R ) y ) ) )
6462, 63syl 15 . . 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 5878 . . 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 4055 . 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 15587 1  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F  |`  C )  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448    C_ wss 3154   class class class wbr 4025    |` cres 4693   -->wf 5253   ` cfv 5257  (class class class)co 5860   RRcr 8738   0cc0 8739    + caddc 8742    x. cmul 8744    < clt 8869    <_ cle 8870   Basecbs 13150   ↾s cress 13151   +g cplusg 13210   .rcmulr 13211   0gc0g 13402  SubGrpcsubg 14617   Ringcrg 15339  SubRingcsubrg 15543  AbsValcabv 15583
This theorem is referenced by:  subrgnrg  18186  qabsabv  20780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-ico 10664  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-0g 13406  df-mnd 14369  df-grp 14491  df-minusg 14492  df-subg 14620  df-mgp 15328  df-rng 15342  df-subrg 15545  df-abv 15584
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