Theorem isabvd 20813

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 6722	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
4	1, 3	mpbid 232	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶ℝ)
5	4	ffnd 6737	. . . 4 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑅))
6	4	ffvelcdmda 7104	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
7		0le0 12367	. . . . . . . . . 10 ⊢ 0 ≤ 0
8		isabvd.z	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 = (0g‘𝑅))
9	8	fveq2d 6910	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = (𝐹‘(0g‘𝑅)))
10		isabvd.3	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = 0)
11	9, 10	eqtr3d 2779	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = 0)
12	7, 11	breqtrrid 5181	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐹‘(0g‘𝑅)))
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g‘𝑅)))
14		fveq2 6906	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝑅) → (𝐹‘𝑥) = (𝐹‘(0g‘𝑅)))
15	14	breq2d 5155	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝑅) → (0 ≤ (𝐹‘𝑥) ↔ 0 ≤ (𝐹‘(0g‘𝑅))))
16	13, 15	syl5ibrcom 247	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g‘𝑅) → 0 ≤ (𝐹‘𝑥)))
17		simp1 1137	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝜑)
18		simp2 1138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
19	2	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝐵 = (Base‘𝑅))
20	18, 19	eleqtrrd 2844	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ∈ 𝐵)
21		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ≠ (0g‘𝑅))
22	8	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 = (0g‘𝑅))
23	21, 22	neeqtrrd 3015	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 < (𝐹‘𝑥))
26		0re 11263	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
27	6	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
28		ltle 11349	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ) → (0 < (𝐹‘𝑥) → 0 ≤ (𝐹‘𝑥)))
29	26, 27, 28	sylancr 587	. . . . . . . . 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 1122	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g‘𝑅) → 0 ≤ (𝐹‘𝑥)))
32	16, 31	pm2.61dne 3028	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘𝑥))
33		elrege0 13494	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
34	6, 32, 33	sylanbrc 583	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (0[,)+∞))
35	34	ralrimiva 3146	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞))
36		ffnfv 7139	. . . 4 ⊢ (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞)))
37	5, 35, 36	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶(0[,)+∞))
38	25	gt0ne0d 11827	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (𝐹‘𝑥) ≠ 0)
39	38	3expia 1122	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g‘𝑅) → (𝐹‘𝑥) ≠ 0))
40	39	necon4d 2964	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐹‘𝑥) = 0 → 𝑥 = (0g‘𝑅)))
41	11	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
42		fveqeq2 6915	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑅) → ((𝐹‘𝑥) = 0 ↔ (𝐹‘(0g‘𝑅)) = 0))
43	41, 42	syl5ibrcom 247	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g‘𝑅) → (𝐹‘𝑥) = 0))
44	40, 43	impbid 212	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)))
45	11	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
46	45	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
47		oveq1 7438	. . . . . . . . . . . 12 ⊢ (𝑥 = (0g‘𝑅) → (𝑥(.r‘𝑅)𝑦) = ((0g‘𝑅)(.r‘𝑅)𝑦))
48		isabvd.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
49	48	3ad2ant1 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
50		simp3 1139	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
51		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
52		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
53		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
54	51, 52, 53	ringlz 20290	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
55	49, 50, 54	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
56	47, 55	sylan9eqr 2799	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
57	56	fveq2d 6910	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
58	14, 45	sylan9eqr 2799	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑥) = 0)
59	58	oveq1d 7446	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = (0 · (𝐹‘𝑦)))
60	4	3ad2ant1 1134	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹:(Base‘𝑅)⟶ℝ)
61	60, 50	ffvelcdmd 7105	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℝ)
62	61	recnd 11289	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
63	62	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
64	63	mul02d 11459	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (0 · (𝐹‘𝑦)) = 0)
65	59, 64	eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
66	46, 57, 65	3eqtr4d 2787	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
67	45	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
68		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑦 = (0g‘𝑅) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑅)(0g‘𝑅)))
69		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
70	51, 52, 53	ringrz 20291	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
71	49, 69, 70	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
72	68, 71	sylan9eqr 2799	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
73	72	fveq2d 6910	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
74		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑦 = (0g‘𝑅) → (𝐹‘𝑦) = (𝐹‘(0g‘𝑅)))
75	74, 45	sylan9eqr 2799	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑦) = 0)
76	75	oveq2d 7447	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = ((𝐹‘𝑥) · 0))
77	60, 69	ffvelcdmd 7105	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
78	77	recnd 11289	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
79	78	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
80	79	mul01d 11460	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · 0) = 0)
81	76, 80	eqtrd 2777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
82	67, 73, 81	3eqtr4d 2787	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
83		simpl1 1192	. . . . . . . . . . . . 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 7448	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 · 𝑦) = (𝑥(.r‘𝑅)𝑦))
87	86	fveq2d 6910	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑥(.r‘𝑅)𝑦)))
88		simpl2 1193	. . . . . . . . . . . 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 2844	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ∈ 𝐵)
91		simprl 771	. . . . . . . . . . . 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 3015	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ≠ 0 )
94		simpl3 1194	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
95	94, 89	eleqtrrd 2844	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ∈ 𝐵)
96		simprr 773	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ≠ (0g‘𝑅))
97	96, 92	neeqtrrd 3015	. . . . . . . . . . 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 1381	. . . . . . . . . 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 2779	. . . . . . . . 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 3030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
102		oveq1 7438	. . . . . . . . . . . 12 ⊢ (𝑥 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = ((0g‘𝑅)(+g‘𝑅)𝑦))
103		ringgrp 20235	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
104	49, 103	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
105		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
106	51, 105, 53	grplid 18985	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
107	104, 50, 106	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
108	102, 107	sylan9eqr 2799	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑦)
109	108	fveq2d 6910	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑦))
110	7, 58	breqtrrid 5181	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑥))
111	61, 77	addge02d 11852	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
112	111	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
113	110, 112	mpbid 232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
114	109, 113	eqbrtrd 5165	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
115		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑦 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑅)(0g‘𝑅)))
116	51, 105, 53	grprid 18986	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
117	104, 69, 116	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
118	115, 117	sylan9eqr 2799	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑥)
119	118	fveq2d 6910	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑥))
120	7, 75	breqtrrid 5181	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑦))
121	77, 61	addge01d 11851	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
122	121	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (0 ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
123	120, 122	mpbid 232	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
124	119, 123	eqbrtrd 5165	. . . . . . . . 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 7448	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦))
128	127	fveq2d 6910	. . . . . . . . . 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 1381	. . . . . . . . . 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 5167	. . . . . . . . 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 3030	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
133	101, 132	jca 511	. . . . . . 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 1122	. . . . . 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 3145	. . . . 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 511	. . . 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 3146	. . 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 2737	. . . . 5 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
139	138, 51, 105, 52, 53	isabv 20812	. . . 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 713	. 2 ⊢ (𝜑 → 𝐹 ∈ (AbsVal‘𝑅))
142		isabvd.a	. 2 ⊢ (𝜑 → 𝐴 = (AbsVal‘𝑅))
143	141, 142	eleqtrrd 2844	1 ⊢ (𝜑 → 𝐹 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061   class class class wbr 5143   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℂcc 11153  ℝcr 11154  0cc0 11155   + caddc 11158   · cmul 11160  +∞cpnf 11292   < clt 11295   ≤ cle 11296  [,)cico 13389  Basecbs 17247  +_gcplusg 17297  ._rcmulr 17298  0_gc0g 17484  Grpcgrp 18951  Ringcrg 20230  AbsValcabv 20809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-ico 13393  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-abv 20810

This theorem is referenced by:  abvres  20832  abvtrivd  20833  absabv  21442  abvcxp  27659  padicabv  27674