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Theorem abvres 17601
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 17551 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  C  =  ( Base `  S )
)
54adantl 464 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  =  ( Base `  S
) )
6 eqid 2382 . . . 4  |-  ( +g  `  R )  =  ( +g  `  R )
73, 6ressplusg 14748 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  ( +g  `  R )  =  ( +g  `  S ) )
87adantl 464 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( +g  `  R )  =  ( +g  `  S
) )
9 eqid 2382 . . . 4  |-  ( .r
`  R )  =  ( .r `  R
)
103, 9ressmulr 14759 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  ( .r `  R )  =  ( .r `  S ) )
1110adantl 464 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( .r `  R )  =  ( .r `  S
) )
12 subrgsubg 17548 . . . 4  |-  ( C  e.  (SubRing `  R
)  ->  C  e.  (SubGrp `  R ) )
1312adantl 464 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  e.  (SubGrp `  R )
)
14 eqid 2382 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  R
)
153, 14subg0 16324 . . 3  |-  ( C  e.  (SubGrp `  R
)  ->  ( 0g `  R )  =  ( 0g `  S ) )
1613, 15syl 16 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( 0g `  R )  =  ( 0g `  S
) )
173subrgring 17545 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  S  e.  Ring )
1817adantl 464 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  S  e.  Ring )
19 abvres.a . . . 4  |-  A  =  (AbsVal `  R )
20 eqid 2382 . . . 4  |-  ( Base `  R )  =  (
Base `  R )
2119, 20abvf 17585 . . 3  |-  ( F  e.  A  ->  F : ( Base `  R
) --> RR )
2220subrgss 17543 . . 3  |-  ( C  e.  (SubRing `  R
)  ->  C  C_  ( Base `  R ) )
23 fssres 5659 . . 3  |-  ( ( F : ( Base `  R ) --> RR  /\  C  C_  ( Base `  R
) )  ->  ( F  |`  C ) : C --> RR )
2421, 22, 23syl2an 475 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F  |`  C ) : C --> RR )
2514subg0cl 16326 . . . 4  |-  ( C  e.  (SubGrp `  R
)  ->  ( 0g `  R )  e.  C
)
26 fvres 5788 . . . 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 17593 . . . 4  |-  ( F  e.  A  ->  ( F `  ( 0g `  R ) )  =  0 )
2928adantr 463 . . 3  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F `  ( 0g `  R ) )  =  0 )
3027, 29eqtrd 2423 . 2  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  (
( F  |`  C ) `
 ( 0g `  R ) )  =  0 )
31 simp1l 1018 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  F  e.  A )
3222adantl 464 . . . . . 6  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  C  C_  ( Base `  R
) )
3332sselda 3417 . . . . 5  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C )  ->  x  e.  ( Base `  R ) )
34333adant3 1014 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  x  e.  ( Base `  R ) )
35 simp3 996 . . . 4  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  ->  x  =/=  ( 0g `  R ) )
3619, 20, 14abvgt0 17590 . . . 4  |-  ( ( F  e.  A  /\  x  e.  ( Base `  R )  /\  x  =/=  ( 0g `  R
) )  ->  0  <  ( F `  x
) )
3731, 34, 35, 36syl3anc 1226 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
0  <  ( F `  x ) )
38 fvres 5788 . . . 4  |-  ( x  e.  C  ->  (
( F  |`  C ) `
 x )  =  ( F `  x
) )
39383ad2ant2 1016 . . 3  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
( ( F  |`  C ) `  x
)  =  ( F `
 x ) )
4037, 39breqtrrd 4393 . 2  |-  ( ( ( F  e.  A  /\  C  e.  (SubRing `  R ) )  /\  x  e.  C  /\  x  =/=  ( 0g `  R ) )  -> 
0  <  ( ( F  |`  C ) `  x ) )
41 simp1l 1018 . . . 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 1019 . . . . . 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 1020 . . . . 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 3418 . . . 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 1022 . . . . 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 3418 . . . 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 17591 . . . 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 1226 . . 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 17554 . . . . 5  |-  ( ( C  e.  (SubRing `  R
)  /\  x  e.  C  /\  y  e.  C
)  ->  ( x
( .r `  R
) y )  e.  C )
5142, 44, 46, 50syl3anc 1226 . . . 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 5788 . . . 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 5788 . . . . 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 6214 . . 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 2433 . 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 17592 . . . 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 1226 . . 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 17553 . . . . 5  |-  ( ( C  e.  (SubRing `  R
)  /\  x  e.  C  /\  y  e.  C
)  ->  ( x
( +g  `  R ) y )  e.  C
)
6242, 44, 46, 61syl3anc 1226 . . . 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 5788 . . . 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 6214 . . 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 4397 . 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 17582 1  |-  ( ( F  e.  A  /\  C  e.  (SubRing `  R
) )  ->  ( F  |`  C )  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577    C_ wss 3389   class class class wbr 4367    |` cres 4915   -->wf 5492   ` cfv 5496  (class class class)co 6196   RRcr 9402   0cc0 9403    + caddc 9406    x. cmul 9408    < clt 9539    <_ cle 9540   Basecbs 14634   ↾s cress 14635   +g cplusg 14702   .rcmulr 14703   0gc0g 14847  SubGrpcsubg 16312   Ringcrg 17311  SubRingcsubrg 17538  AbsValcabv 17578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-recs 6960  df-rdg 6994  df-er 7229  df-map 7340  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-ico 11456  df-ndx 14637  df-slot 14638  df-base 14639  df-sets 14640  df-ress 14641  df-plusg 14715  df-mulr 14716  df-0g 14849  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-minusg 16175  df-subg 16315  df-mgp 17255  df-ring 17313  df-subrg 17540  df-abv 17579
This theorem is referenced by:  subrgnrg  21267  qabsabv  23931
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