Theorem abvres 18839

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 = ( R ↾s 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.

Step	Hyp	Ref	Expression
1		abvres.b	. . 3 |- B = (AbsVal ` S )
2	1	a1i 11	. 2 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> B = (AbsVal ` S ) )
3		abvres.s	. . . 4 |- S = ( R ↾s C )
4	3	subrgbas 18789	. . 3 |- ( C e. (SubRing ` R ) -> C = ( Base ` S ) )
5	4	adantl 482	. 2 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> C = ( Base ` S ) )
6		eqid 2622	. . . 4 |- ( +g ` R ) = ( +g ` R )
7	3, 6	ressplusg 15993	. . 3 |- ( C e. (SubRing ` R ) -> ( +g ` R ) = ( +g ` S ) )
8	7	adantl 482	. 2 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> ( +g ` R ) = ( +g ` S ) )
9		eqid 2622	. . . 4 |- ( .r ` R ) = ( .r ` R )
10	3, 9	ressmulr 16006	. . 3 |- ( C e. (SubRing ` R ) -> ( .r ` R ) = ( .r ` S ) )
11	10	adantl 482	. 2 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> ( .r ` R ) = ( .r ` S ) )
12		subrgsubg 18786	. . . 4 |- ( C e. (SubRing ` R ) -> C e. (SubGrp ` R ) )
13	12	adantl 482	. . 3 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> C e. (SubGrp ` R ) )
14		eqid 2622	. . . 4 |- ( 0g ` R ) = ( 0g ` R )
15	3, 14	subg0 17600	. . 3 |- ( C e. (SubGrp ` R ) -> ( 0g ` R ) = ( 0g ` S ) )
16	13, 15	syl 17	. 2 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> ( 0g ` R ) = ( 0g ` S ) )
17	3	subrgring 18783	. . 3 |- ( C e. (SubRing ` R ) -> S e. Ring )
18	17	adantl 482	. 2 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> S e. Ring )
19		abvres.a	. . . 4 |- A = (AbsVal ` R )
20		eqid 2622	. . . 4 |- ( Base ` R ) = ( Base ` R )
21	19, 20	abvf 18823	. . 3 |- ( F e. A -> F : ( Base ` R ) --> RR )
22	20	subrgss 18781	. . 3 |- ( C e. (SubRing ` R ) -> C C_ ( Base ` R ) )
23		fssres 6070	. . 3 |- ( ( F : ( Base ` R ) --> RR /\ C C_ ( Base ` R ) ) -> ( F |` C ) : C --> RR )
24	21, 22, 23	syl2an 494	. 2 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> ( F |` C ) : C --> RR )
25	14	subg0cl 17602	. . . 4 |- ( C e. (SubGrp ` R ) -> ( 0g ` R ) e. C )
26		fvres 6207	. . . 4 |- ( ( 0g ` R ) e. C -> ( ( F |` C ) ` ( 0g ` R ) ) = ( F ` ( 0g ` R ) ) )
27	13, 25, 26	3syl 18	. . 3 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> ( ( F |` C ) ` ( 0g ` R ) ) = ( F ` ( 0g ` R ) ) )
28	19, 14	abv0 18831	. . . 4 |- ( F e. A -> ( F ` ( 0g ` R ) ) = 0 )
29	28	adantr 481	. . 3 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> ( F ` ( 0g ` R ) ) = 0 )
30	27, 29	eqtrd 2656	. 2 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> ( ( F |` C ) ` ( 0g ` R ) ) = 0 )
31		simp1l 1085	. . . 4 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ x e. C /\ x =/= ( 0g ` R ) ) -> F e. A )
32	22	adantl 482	. . . . . 6 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> C C_ ( Base ` R ) )
33	32	sselda 3603	. . . . 5 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ x e. C ) -> x e. ( Base ` R ) )
34	33	3adant3 1081	. . . 4 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ x e. C /\ x =/= ( 0g ` R ) ) -> x e. ( Base ` R ) )
35		simp3 1063	. . . 4 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ x e. C /\ x =/= ( 0g ` R ) ) -> x =/= ( 0g ` R ) )
36	19, 20, 14	abvgt0 18828	. . . 4 |- ( ( F e. A /\ x e. ( Base ` R ) /\ x =/= ( 0g ` R ) ) -> 0 < ( F ` x ) )
37	31, 34, 35, 36	syl3anc 1326	. . 3 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ x e. C /\ x =/= ( 0g ` R ) ) -> 0 < ( F ` x ) )
38		fvres 6207	. . . 4 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
39	38	3ad2ant2 1083	. . 3 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ x e. C /\ x =/= ( 0g ` R ) ) -> ( ( F |` C ) ` x ) = ( F ` x ) )
40	37, 39	breqtrrd 4681	. 2 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ x e. C /\ x =/= ( 0g ` R ) ) -> 0 < ( ( F |` C ) ` x ) )
41		simp1l 1085	. . . 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 1086	. . . . . 6 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ ( x e. C /\ x =/= ( 0g ` R ) ) /\ ( y e. C /\ y =/= ( 0g ` R ) ) ) -> C e. (SubRing ` R ) )
43	42, 22	syl 17	. . . . 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 1087	. . . . 5 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ ( x e. C /\ x =/= ( 0g ` R ) ) /\ ( y e. C /\ y =/= ( 0g ` R ) ) ) -> x e. C )
45	43, 44	sseldd 3604	. . . 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 1089	. . . . 5 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ ( x e. C /\ x =/= ( 0g ` R ) ) /\ ( y e. C /\ y =/= ( 0g ` R ) ) ) -> y e. C )
47	43, 46	sseldd 3604	. . . 4 |- ( ( ( F e. A /\ C e. (SubRing ` R ) ) /\ ( x e. C /\ x =/= ( 0g ` R ) ) /\ ( y e. C /\ y =/= ( 0g ` R ) ) ) -> y e. ( Base ` R ) )
48	19, 20, 9	abvmul 18829	. . . 4 |- ( ( F e. A /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( F ` ( x ( .r ` R ) y ) ) = ( ( F ` x ) x. ( F ` y ) ) )
49	41, 45, 47, 48	syl3anc 1326	. . 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 ) ) )
50	9	subrgmcl 18792	. . . . 5 |- ( ( C e. (SubRing ` R ) /\ x e. C /\ y e. C ) -> ( x ( .r ` R ) y ) e. C )
51	42, 44, 46, 50	syl3anc 1326	. . . 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 6207	. . . 4 |- ( ( x ( .r ` R ) y ) e. C -> ( ( F |` C ) ` ( x ( .r ` R ) y ) ) = ( F ` ( x ( .r ` R ) y ) ) )
53	51, 52	syl 17	. . 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 ) ) )
54	44, 38	syl 17	. . . 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 6207	. . . . 5 |- ( y e. C -> ( ( F |` C ) ` y ) = ( F ` y ) )
56	46, 55	syl 17	. . . 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 ) )
57	54, 56	oveq12d 6668	. . 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 ) ) )
58	49, 53, 57	3eqtr4d 2666	. 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 ) ) )
59	19, 20, 6	abvtri 18830	. . . 4 |- ( ( F e. A /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( F ` ( x ( +g ` R ) y ) ) <_ ( ( F ` x ) + ( F ` y ) ) )
60	41, 45, 47, 59	syl3anc 1326	. . 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 ) ) )
61	6	subrgacl 18791	. . . . 5 |- ( ( C e. (SubRing ` R ) /\ x e. C /\ y e. C ) -> ( x ( +g ` R ) y ) e. C )
62	42, 44, 46, 61	syl3anc 1326	. . . 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 6207	. . . 4 |- ( ( x ( +g ` R ) y ) e. C -> ( ( F |` C ) ` ( x ( +g ` R ) y ) ) = ( F ` ( x ( +g ` R ) y ) ) )
64	62, 63	syl 17	. . 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 ) ) )
65	54, 56	oveq12d 6668	. . 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 ) ) )
66	60, 64, 65	3brtr4d 4685	. 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 ) ) )
67	2, 5, 8, 11, 16, 18, 24, 30, 40, 58, 66	isabvd 18820	1 |- ( ( F e. A /\ C e. (SubRing ` R ) ) -> ( F |` C ) e. B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   class class class wbr 4653   |` cres 5116   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   Basecbs 15857   ↾_s cress 15858   +g cplusg 15941   .rcmulr 15942   0gc0g 16100  SubGrpcsubg 17588   Ringcrg 18547  SubRingcsubrg 18776  AbsValcabv 18816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-ico 12181  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-subg 17591  df-mgp 18490  df-ring 18549  df-subrg 18778  df-abv 18817

This theorem is referenced by:  subrgnrg  22477  qabsabv  25318