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Theorem abvres 17271
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 17221 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  C  =  ( Base `  S )
)
54adantl 466 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  =  ( Base `  S
) )
6 eqid 2467 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
73, 6ressplusg 14593 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  ( +g  `  R )  =  ( +g  `  S ) )
87adantl 466 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( +g  `  R )  =  ( +g  `  S
) )
9 eqid 2467 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
103, 9ressmulr 14604 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1110adantl 466 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( .r `  R )  =  ( .r `  S
) )
12 subrgsubg 17218 . . . 4  |-  ( C  e.  (SubRing `  R
)  ->  C  e.  (SubGrp `  R ) )
1312adantl 466 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  e.  (SubGrp `  R )
)
14 eqid 2467 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
153, 14subg0 16002 . . 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 17215 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  S  e.  Ring )
1817adantl 466 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  S  e.  Ring )
19 abvres.a . . . 4  |-  A  =  (AbsVal `  R )
20 eqid 2467 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2119, 20abvf 17255 . . 3  |-  ( F  e.  A  ->  F : ( Base `  R
) --> RR )
2220subrgss 17213 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  C  C_  ( Base `  R ) )
23 fssres 5749 . . 3  |-  ( ( F : ( Base `  R ) --> RR  /\  C  C_  ( Base `  R
) )  ->  ( F  |`  C ) : C --> RR )
2421, 22, 23syl2an 477 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F  |`  C ) : C --> RR )
2514subg0cl 16004 . . . 4  |-  ( C  e.  (SubGrp `  R
)  ->  ( 0g `  R )  e.  C
)
26 fvres 5878 . . . 4  |-  ( ( 0g `  R )  e.  C  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  ( F `  ( 0g `  R ) ) )
2713, 25, 263syl 20 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  ( F `  ( 0g `  R ) ) )
2819, 14abv0 17263 . . . 4  |-  ( F  e.  A  ->  ( F `  ( 0g `  R ) )  =  0 )
2928adantr 465 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F `  ( 0g `  R ) )  =  0 )
3027, 29eqtrd 2508 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  0 )
31 simp1l 1020 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  F  e.  A )
3222adantl 466 . . . . . 6  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  C_  ( Base `  R
) )
3332sselda 3504 . . . . 5  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C )  ->  x  e.  ( Base `  R ) )
34333adant3 1016 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  x  e.  ( Base `  R ) )
35 simp3 998 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  x  =/=  ( 0g `  R ) )
3619, 20, 14abvgt0 17260 . . . 4  |-  ( ( F  e.  A  /\  x  e.  ( Base `  R )  /\  x  =/=  ( 0g `  R
) )  ->  0  <  ( F `  x
) )
3731, 34, 35, 36syl3anc 1228 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
0  <  ( F `  x ) )
38 fvres 5878 . . . 4  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
39383ad2ant2 1018 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
( ( F  |`  C ) `  x
)  =  ( F `
 x ) )
4037, 39breqtrrd 4473 . 2  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
0  <  ( ( F  |`  C ) `  x ) )
41 simp1l 1020 . . . 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 1021 . . . . . 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 1022 . . . . 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 3505 . . . 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 1024 . . . . 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 3505 . . . 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 17261 . . . 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 1228 . . 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 17224 . . . . 5  |-  ( ( C  e.  (SubRing `  R
)  /\  x  e.  C  /\  y  e.  C
)  ->  ( x
( .r `  R
) y )  e.  C )
5142, 44, 46, 50syl3anc 1228 . . . 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 5878 . . . 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 5878 . . . . 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 6300 . . 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 2518 . 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 17262 . . . 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 1228 . . 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 17223 . . . . 5  |-  ( ( C  e.  (SubRing `  R
)  /\  x  e.  C  /\  y  e.  C
)  ->  ( x
( +g  `  R ) y )  e.  C
)
6242, 44, 46, 61syl3anc 1228 . . . 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 5878 . . . 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 6300 . . 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 4477 . 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 17252 1  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F  |`  C )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   class class class wbr 4447    |` cres 5001   -->wf 5582   ` cfv 5586  (class class class)co 6282   RRcr 9487   0cc0 9488    + caddc 9491    x. cmul 9493    < clt 9624    <_ cle 9625   Basecbs 14486   ↾s cress 14487   +g cplusg 14551   .rcmulr 14552   0gc0g 14691  SubGrpcsubg 15990   Ringcrg 16986  SubRingcsubrg 17208  AbsValcabv 17248
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
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 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-map 7419  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-ico 11531  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-0g 14693  df-mnd 15728  df-grp 15858  df-minusg 15859  df-subg 15993  df-mgp 16932  df-rng 16988  df-subrg 17210  df-abv 17249
This theorem is referenced by:  subrgnrg  20917  qabsabv  23542
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