Theorem isabvd 19585

Description: Properties that determine an absolute value. (Contributed by Mario Carneiro, 8-Sep-2014.) (Revised by Mario Carneiro, 4-Dec-2014.)

Hypotheses

Ref	Expression
isabvd.a	⊢ (𝜑 → 𝐴 = (AbsVal‘𝑅))
isabvd.b	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
isabvd.p	⊢ (𝜑 → + = (+g‘𝑅))
isabvd.t	⊢ (𝜑 → · = (.r‘𝑅))
isabvd.z	⊢ (𝜑 → 0 = (0g‘𝑅))
isabvd.1	⊢ (𝜑 → 𝑅 ∈ Ring)
isabvd.2	⊢ (𝜑 → 𝐹:𝐵⟶ℝ)
isabvd.3	⊢ (𝜑 → (𝐹‘ 0 ) = 0)
isabvd.4	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 0 < (𝐹‘𝑥))
isabvd.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
isabvd.6	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))

Assertion

Ref	Expression
isabvd	⊢ (𝜑 → 𝐹 ∈ 𝐴)

Distinct variable groups:   𝑥,𝑦,𝐹   𝜑,𝑥,𝑦   𝑥,𝑅,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   + (𝑥,𝑦)   · (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem isabvd

Step	Hyp	Ref	Expression
1		isabvd.2	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ℝ)
2		isabvd.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
3	2	feq2d 6494	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
4	1, 3	mpbid 234	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶ℝ)
5	4	ffnd 6509	. . . 4 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑅))
6	4	ffvelrnda 6845	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
7		0le0 11732	. . . . . . . . . 10 ⊢ 0 ≤ 0
8		isabvd.z	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 = (0g‘𝑅))
9	8	fveq2d 6668	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = (𝐹‘(0g‘𝑅)))
10		isabvd.3	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = 0)
11	9, 10	eqtr3d 2858	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = 0)
12	7, 11	breqtrrid 5096	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐹‘(0g‘𝑅)))
13	12	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g‘𝑅)))
14		fveq2 6664	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝑅) → (𝐹‘𝑥) = (𝐹‘(0g‘𝑅)))
15	14	breq2d 5070	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝑅) → (0 ≤ (𝐹‘𝑥) ↔ 0 ≤ (𝐹‘(0g‘𝑅))))
16	13, 15	syl5ibrcom 249	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g‘𝑅) → 0 ≤ (𝐹‘𝑥)))
17		simp1 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝜑)
18		simp2 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
19	2	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝐵 = (Base‘𝑅))
20	18, 19	eleqtrrd 2916	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ∈ 𝐵)
21		simp3 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ≠ (0g‘𝑅))
22	8	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 = (0g‘𝑅))
23	21, 22	neeqtrrd 3090	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ≠ 0 )
24		isabvd.4	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) → 0 < (𝐹‘𝑥))
25	17, 20, 23, 24	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 < (𝐹‘𝑥))
26		0re 10637	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
27	6	3adant3 1128	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
28		ltle 10723	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ) → (0 < (𝐹‘𝑥) → 0 ≤ (𝐹‘𝑥)))
29	26, 27, 28	sylancr 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (0 < (𝐹‘𝑥) → 0 ≤ (𝐹‘𝑥)))
30	25, 29	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 ≤ (𝐹‘𝑥))
31	30	3expia 1117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g‘𝑅) → 0 ≤ (𝐹‘𝑥)))
32	16, 31	pm2.61dne 3103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘𝑥))
33		elrege0 12836	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
34	6, 32, 33	sylanbrc 585	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (0[,)+∞))
35	34	ralrimiva 3182	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞))
36		ffnfv 6876	. . . 4 ⊢ (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞)))
37	5, 35, 36	sylanbrc 585	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶(0[,)+∞))
38	25	gt0ne0d 11198	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (𝐹‘𝑥) ≠ 0)
39	38	3expia 1117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g‘𝑅) → (𝐹‘𝑥) ≠ 0))
40	39	necon4d 3040	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐹‘𝑥) = 0 → 𝑥 = (0g‘𝑅)))
41	11	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
42		fveqeq2 6673	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑅) → ((𝐹‘𝑥) = 0 ↔ (𝐹‘(0g‘𝑅)) = 0))
43	41, 42	syl5ibrcom 249	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g‘𝑅) → (𝐹‘𝑥) = 0))
44	40, 43	impbid 214	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)))
45	11	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
46	45	adantr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
47		oveq1 7157	. . . . . . . . . . . 12 ⊢ (𝑥 = (0g‘𝑅) → (𝑥(.r‘𝑅)𝑦) = ((0g‘𝑅)(.r‘𝑅)𝑦))
48		isabvd.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
49	48	3ad2ant1 1129	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
50		simp3 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
51		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
52		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
53		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
54	51, 52, 53	ringlz 19331	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
55	49, 50, 54	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
56	47, 55	sylan9eqr 2878	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
57	56	fveq2d 6668	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
58	14, 45	sylan9eqr 2878	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑥) = 0)
59	58	oveq1d 7165	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = (0 · (𝐹‘𝑦)))
60	4	3ad2ant1 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹:(Base‘𝑅)⟶ℝ)
61	60, 50	ffvelrnd 6846	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℝ)
62	61	recnd 10663	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
63	62	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
64	63	mul02d 10832	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (0 · (𝐹‘𝑦)) = 0)
65	59, 64	eqtrd 2856	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
66	46, 57, 65	3eqtr4d 2866	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
67	45	adantr 483	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
68		oveq2 7158	. . . . . . . . . . . 12 ⊢ (𝑦 = (0g‘𝑅) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑅)(0g‘𝑅)))
69		simp2 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
70	51, 52, 53	ringrz 19332	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
71	49, 69, 70	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
72	68, 71	sylan9eqr 2878	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
73	72	fveq2d 6668	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
74		fveq2 6664	. . . . . . . . . . . . 13 ⊢ (𝑦 = (0g‘𝑅) → (𝐹‘𝑦) = (𝐹‘(0g‘𝑅)))
75	74, 45	sylan9eqr 2878	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑦) = 0)
76	75	oveq2d 7166	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = ((𝐹‘𝑥) · 0))
77	60, 69	ffvelrnd 6846	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
78	77	recnd 10663	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
79	78	adantr 483	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
80	79	mul01d 10833	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · 0) = 0)
81	76, 80	eqtrd 2856	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
82	67, 73, 81	3eqtr4d 2866	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
83		simpl1 1187	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝜑)
84		isabvd.t	. . . . . . . . . . . . 13 ⊢ (𝜑 → · = (.r‘𝑅))
85	83, 84	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → · = (.r‘𝑅))
86	85	oveqd 7167	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 · 𝑦) = (𝑥(.r‘𝑅)𝑦))
87	86	fveq2d 6668	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑥(.r‘𝑅)𝑦)))
88		simpl2 1188	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
89	83, 2	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝐵 = (Base‘𝑅))
90	88, 89	eleqtrrd 2916	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ∈ 𝐵)
91		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ≠ (0g‘𝑅))
92	83, 8	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 0 = (0g‘𝑅))
93	91, 92	neeqtrrd 3090	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ≠ 0 )
94		simpl3 1189	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
95	94, 89	eleqtrrd 2916	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ∈ 𝐵)
96		simprr 771	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ≠ (0g‘𝑅))
97	96, 92	neeqtrrd 3090	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ≠ 0 )
98		isabvd.5	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
99	83, 90, 93, 95, 97, 98	syl122anc 1375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
100	87, 99	eqtr3d 2858	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
101	66, 82, 100	pm2.61da2ne 3105	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
102		oveq1 7157	. . . . . . . . . . . 12 ⊢ (𝑥 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = ((0g‘𝑅)(+g‘𝑅)𝑦))
103		ringgrp 19296	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
104	49, 103	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
105		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
106	51, 105, 53	grplid 18127	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
107	104, 50, 106	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
108	102, 107	sylan9eqr 2878	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑦)
109	108	fveq2d 6668	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑦))
110	7, 58	breqtrrid 5096	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑥))
111	61, 77	addge02d 11223	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
112	111	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
113	110, 112	mpbid 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
114	109, 113	eqbrtrd 5080	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
115		oveq2 7158	. . . . . . . . . . . 12 ⊢ (𝑦 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑅)(0g‘𝑅)))
116	51, 105, 53	grprid 18128	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
117	104, 69, 116	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
118	115, 117	sylan9eqr 2878	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑥)
119	118	fveq2d 6668	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑥))
120	7, 75	breqtrrid 5096	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑦))
121	77, 61	addge01d 11222	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
122	121	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (0 ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
123	120, 122	mpbid 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
124	119, 123	eqbrtrd 5080	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
125		isabvd.p	. . . . . . . . . . . . 13 ⊢ (𝜑 → + = (+g‘𝑅))
126	83, 125	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → + = (+g‘𝑅))
127	126	oveqd 7167	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦))
128	127	fveq2d 6668	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑥(+g‘𝑅)𝑦)))
129		isabvd.6	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑥 ≠ 0 ) ∧ (𝑦 ∈ 𝐵 ∧ 𝑦 ≠ 0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
130	83, 90, 93, 95, 97, 129	syl122anc 1375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
131	128, 130	eqbrtrrd 5082	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
132	114, 124, 131	pm2.61da2ne 3105	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
133	101, 132	jca 514	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
134	133	3expia 1117	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦 ∈ (Base‘𝑅) → ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))
135	134	ralrimiv 3181	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
136	44, 135	jca 514	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))
137	136	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))
138		eqid 2821	. . . . 5 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
139	138, 51, 105, 52, 53	isabv 19584	. . . 4 ⊢ (𝑅 ∈ Ring → (𝐹 ∈ (AbsVal‘𝑅) ↔ (𝐹:(Base‘𝑅)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝑅)(((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))))
140	48, 139	syl 17	. . 3 ⊢ (𝜑 → (𝐹 ∈ (AbsVal‘𝑅) ↔ (𝐹:(Base‘𝑅)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝑅)(((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))))
141	37, 137, 140	mpbir2and 711	. 2 ⊢ (𝜑 → 𝐹 ∈ (AbsVal‘𝑅))
142		isabvd.a	. 2 ⊢ (𝜑 → 𝐴 = (AbsVal‘𝑅))
143	141, 142	eleqtrrd 2916	1 ⊢ (𝜑 → 𝐹 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138   class class class wbr 5058   Fn wfn 6344  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150  ℂcc 10529  ℝcr 10530  0cc0 10531   + caddc 10534   · cmul 10536  +∞cpnf 10666   < clt 10669   ≤ cle 10670  [,)cico 12734  Basecbs 16477  +_gcplusg 16559  ._rcmulr 16560  0_gc0g 16707  Grpcgrp 18097  Ringcrg 19291  AbsValcabv 19581

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-ico 12738  df-ndx 16480  df-slot 16481  df-base 16483  df-sets 16484  df-plusg 16572  df-0g 16709  df-mgm 17846  df-sgrp 17895  df-mnd 17906  df-grp 18100  df-minusg 18101  df-mgp 19234  df-ring 19293  df-abv 19582

This theorem is referenced by:  abvres  19604  abvtrivd  19605  absabv  20596  abvcxp  26185  padicabv  26200