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Theorem abvres 20439

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 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → (𝐹 ↾ 𝐶) ∈ 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 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → 𝐵 = (AbsVal‘𝑆))
3		abvres.s	. . . 4 ⊢ 𝑆 = (𝑅 ↾_s 𝐶)
4	3	subrgbas 20364	. . 3 ⊢ (𝐶 ∈ (SubRing‘𝑅) → 𝐶 = (Base‘𝑆))
5	4	adantl 482	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → 𝐶 = (Base‘𝑆))
6		eqid 2732	. . . 4 ⊢ (+_g‘𝑅) = (+_g‘𝑅)
7	3, 6	ressplusg 17231	. . 3 ⊢ (𝐶 ∈ (SubRing‘𝑅) → (+_g‘𝑅) = (+_g‘𝑆))
8	7	adantl 482	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → (+_g‘𝑅) = (+_g‘𝑆))
9		eqid 2732	. . . 4 ⊢ (._r‘𝑅) = (._r‘𝑅)
10	3, 9	ressmulr 17248	. . 3 ⊢ (𝐶 ∈ (SubRing‘𝑅) → (._r‘𝑅) = (._r‘𝑆))
11	10	adantl 482	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → (._r‘𝑅) = (._r‘𝑆))
12		subrgsubg 20361	. . . 4 ⊢ (𝐶 ∈ (SubRing‘𝑅) → 𝐶 ∈ (SubGrp‘𝑅))
13	12	adantl 482	. . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → 𝐶 ∈ (SubGrp‘𝑅))
14		eqid 2732	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
15	3, 14	subg0 19006	. . 3 ⊢ (𝐶 ∈ (SubGrp‘𝑅) → (0_g‘𝑅) = (0_g‘𝑆))
16	13, 15	syl 17	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → (0_g‘𝑅) = (0_g‘𝑆))
17	3	subrgring 20358	. . 3 ⊢ (𝐶 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
18	17	adantl 482	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → 𝑆 ∈ Ring)
19		abvres.a	. . . 4 ⊢ 𝐴 = (AbsVal‘𝑅)
20		eqid 2732	. . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅)
21	19, 20	abvf 20423	. . 3 ⊢ (𝐹 ∈ 𝐴 → 𝐹:(Base‘𝑅)⟶ℝ)
22	20	subrgss 20356	. . 3 ⊢ (𝐶 ∈ (SubRing‘𝑅) → 𝐶 ⊆ (Base‘𝑅))
23		fssres 6754	. . 3 ⊢ ((𝐹:(Base‘𝑅)⟶ℝ ∧ 𝐶 ⊆ (Base‘𝑅)) → (𝐹 ↾ 𝐶):𝐶⟶ℝ)
24	21, 22, 23	syl2an 596	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → (𝐹 ↾ 𝐶):𝐶⟶ℝ)
25	14	subg0cl 19008	. . . 4 ⊢ (𝐶 ∈ (SubGrp‘𝑅) → (0_g‘𝑅) ∈ 𝐶)
26		fvres 6907	. . . 4 ⊢ ((0_g‘𝑅) ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘(0_g‘𝑅)) = (𝐹‘(0_g‘𝑅)))
27	13, 25, 26	3syl 18	. . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → ((𝐹 ↾ 𝐶)‘(0_g‘𝑅)) = (𝐹‘(0_g‘𝑅)))
28	19, 14	abv0 20431	. . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹‘(0_g‘𝑅)) = 0)
29	28	adantr 481	. . 3 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → (𝐹‘(0_g‘𝑅)) = 0)
30	27, 29	eqtrd 2772	. 2 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → ((𝐹 ↾ 𝐶)‘(0_g‘𝑅)) = 0)
31		simp1l 1197	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) → 𝐹 ∈ 𝐴)
32	22	adantl 482	. . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → 𝐶 ⊆ (Base‘𝑅))
33	32	sselda 3981	. . . . 5 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ (Base‘𝑅))
34	33	3adant3 1132	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
35		simp3 1138	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) → 𝑥 ≠ (0_g‘𝑅))
36	19, 20, 14	abvgt0 20428	. . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0_g‘𝑅)) → 0 < (𝐹‘𝑥))
37	31, 34, 35, 36	syl3anc 1371	. . 3 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) → 0 < (𝐹‘𝑥))
38		fvres 6907	. . . 4 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
39	38	3ad2ant2 1134	. . 3 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
40	37, 39	breqtrrd 5175	. 2 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ 𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) → 0 < ((𝐹 ↾ 𝐶)‘𝑥))
41		simp1l 1197	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → 𝐹 ∈ 𝐴)
42		simp1r 1198	. . . . . 6 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → 𝐶 ∈ (SubRing‘𝑅))
43	42, 22	syl 17	. . . . 5 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → 𝐶 ⊆ (Base‘𝑅))
44		simp2l 1199	. . . . 5 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → 𝑥 ∈ 𝐶)
45	43, 44	sseldd 3982	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
46		simp3l 1201	. . . . 5 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → 𝑦 ∈ 𝐶)
47	43, 46	sseldd 3982	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
48	19, 20, 9	abvmul 20429	. . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(._r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
49	41, 45, 47, 48	syl3anc 1371	. . 3 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → (𝐹‘(𝑥(._r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
50	9	subrgmcl 20367	. . . . 5 ⊢ ((𝐶 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥(._r‘𝑅)𝑦) ∈ 𝐶)
51	42, 44, 46, 50	syl3anc 1371	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → (𝑥(._r‘𝑅)𝑦) ∈ 𝐶)
52	51	fvresd 6908	. . 3 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → ((𝐹 ↾ 𝐶)‘(𝑥(._r‘𝑅)𝑦)) = (𝐹‘(𝑥(._r‘𝑅)𝑦)))
53	44	fvresd 6908	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
54	46	fvresd 6908	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → ((𝐹 ↾ 𝐶)‘𝑦) = (𝐹‘𝑦))
55	53, 54	oveq12d 7423	. . 3 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → (((𝐹 ↾ 𝐶)‘𝑥) · ((𝐹 ↾ 𝐶)‘𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
56	49, 52, 55	3eqtr4d 2782	. 2 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → ((𝐹 ↾ 𝐶)‘(𝑥(._r‘𝑅)𝑦)) = (((𝐹 ↾ 𝐶)‘𝑥) · ((𝐹 ↾ 𝐶)‘𝑦)))
57	19, 20, 6	abvtri 20430	. . . 4 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+_g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
58	41, 45, 47, 57	syl3anc 1371	. . 3 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → (𝐹‘(𝑥(+_g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
59	6	subrgacl 20366	. . . . 5 ⊢ ((𝐶 ∈ (SubRing‘𝑅) ∧ 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (𝑥(+_g‘𝑅)𝑦) ∈ 𝐶)
60	42, 44, 46, 59	syl3anc 1371	. . . 4 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → (𝑥(+_g‘𝑅)𝑦) ∈ 𝐶)
61	60	fvresd 6908	. . 3 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → ((𝐹 ↾ 𝐶)‘(𝑥(+_g‘𝑅)𝑦)) = (𝐹‘(𝑥(+_g‘𝑅)𝑦)))
62	53, 54	oveq12d 7423	. . 3 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → (((𝐹 ↾ 𝐶)‘𝑥) + ((𝐹 ↾ 𝐶)‘𝑦)) = ((𝐹‘𝑥) + (𝐹‘𝑦)))
63	58, 61, 62	3brtr4d 5179	. 2 ⊢ (((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) ∧ (𝑥 ∈ 𝐶 ∧ 𝑥 ≠ (0_g‘𝑅)) ∧ (𝑦 ∈ 𝐶 ∧ 𝑦 ≠ (0_g‘𝑅))) → ((𝐹 ↾ 𝐶)‘(𝑥(+_g‘𝑅)𝑦)) ≤ (((𝐹 ↾ 𝐶)‘𝑥) + ((𝐹 ↾ 𝐶)‘𝑦)))
64	2, 5, 8, 11, 16, 18, 24, 30, 40, 56, 63	isabvd 20420	1 ⊢ ((𝐹 ∈ 𝐴 ∧ 𝐶 ∈ (SubRing‘𝑅)) → (𝐹 ↾ 𝐶) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ⊆ wss 3947 class class class wbr 5147 ↾ cres 5677 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℝcr 11105 0cc0 11106 + caddc 11109 · cmul 11111 < clt 11244 ≤ cle 11245 Basecbs 17140 ↾_s cress 17169 +_gcplusg 17193 ._rcmulr 17194 0_gc0g 17381 SubGrpcsubg 18994 Ringcrg 20049 SubRingcsubrg 20351 AbsValcabv 20416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-ico 13326 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-subg 18997 df-mgp 19982 df-ring 20051 df-subrg 20353 df-abv 20417

This theorem is referenced by: subrgnrg 24181 qabsabv 27121

W3C validator