Theorem isabvd 21111

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 6721	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
4	1, 3	mpbid 235	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶ℝ)
5	4	ffnd 6738	. . . 4 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑅))
6	4	ffvelcdmda 7114	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
7		0le0 12482	. . . . . . . . . 10 ⊢ 0 ≤ 0
8		isabvd.z	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 = (0g‘𝑅))
9	8	fveq2d 6918	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = (𝐹‘(0g‘𝑅)))
10		isabvd.3	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = 0)
11	9, 10	eqtr3d 2800	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = 0)
12	7, 11	breqtrrid 5147	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐹‘(0g‘𝑅)))
13	12	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g‘𝑅)))
14		fveq2 6914	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝑅) → (𝐹‘𝑥) = (𝐹‘(0g‘𝑅)))
15	14	breq2d 5118	. . . . . . . 8 ⊢ (𝑥 = (0g‘𝑅) → (0 ≤ (𝐹‘𝑥) ↔ 0 ≤ (𝐹‘(0g‘𝑅))))
16	13, 15	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g‘𝑅) → 0 ≤ (𝐹‘𝑥)))
17		simp1 1154	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝜑)
18		simp2 1155	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
19	2	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝐵 = (Base‘𝑅))
20	18, 19	eleqtrrd 2867	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ∈ 𝐵)
21		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ≠ (0g‘𝑅))
22	8	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 = (0g‘𝑅))
23	21, 22	neeqtrrd 3033	. . . . . . . . . 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 1399	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 < (𝐹‘𝑥))
26		0re 11347	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
27	6	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
28		ltle 11435	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ) → (0 < (𝐹‘𝑥) → 0 ≤ (𝐹‘𝑥)))
29	26, 27, 28	sylancr 599	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (0 < (𝐹‘𝑥) → 0 ≤ (𝐹‘𝑥)))
30	25, 29	mpd 16	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 ≤ (𝐹‘𝑥))
31	30	3expia 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g‘𝑅) → 0 ≤ (𝐹‘𝑥)))
32	16, 31	pm2.61dne 3045	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘𝑥))
33		elrege0 13623	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
34	6, 32, 33	sylanbrc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (0[,)+∞))
35	34	ralrimiva 3158	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞))
36		ffnfv 7149	. . . 4 ⊢ (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞)))
37	5, 35, 36	sylanbrc 595	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶(0[,)+∞))
38	25	gt0ne0d 11918	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (𝐹‘𝑥) ≠ 0)
39	38	3expia 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g‘𝑅) → (𝐹‘𝑥) ≠ 0))
40	39	necon4d 2983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐹‘𝑥) = 0 → 𝑥 = (0g‘𝑅)))
41	11	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
42		fveqeq2 6923	. . . . . . 7 ⊢ (𝑥 = (0g‘𝑅) → ((𝐹‘𝑥) = 0 ↔ (𝐹‘(0g‘𝑅)) = 0))
43	41, 42	syl5ibrcom 250	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g‘𝑅) → (𝐹‘𝑥) = 0))
44	40, 43	impbid 215	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)))
45	11	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
46	45	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
47		oveq1 7456	. . . . . . . . . . . 12 ⊢ (𝑥 = (0g‘𝑅) → (𝑥(.r‘𝑅)𝑦) = ((0g‘𝑅)(.r‘𝑅)𝑦))
48		isabvd.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
49	48	3ad2ant1 1151	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
50		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
51		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
52		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
53		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
54	51, 52, 53	ringlz 20566	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
55	49, 50, 54	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
56	47, 55	sylan9eqr 2820	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
57	56	fveq2d 6918	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
58	14, 45	sylan9eqr 2820	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑥) = 0)
59	58	oveq1d 7464	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = (0 · (𝐹‘𝑦)))
60	4	3ad2ant1 1151	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹:(Base‘𝑅)⟶ℝ)
61	60, 50	ffvelcdmd 7115	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℝ)
62	61	recnd 11374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
63	62	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
64	63	mul02d 11545	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (0 · (𝐹‘𝑦)) = 0)
65	59, 64	eqtrd 2798	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
66	46, 57, 65	3eqtr4d 2808	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
67	45	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
68		oveq2 7457	. . . . . . . . . . . 12 ⊢ (𝑦 = (0g‘𝑅) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑅)(0g‘𝑅)))
69		simp2 1155	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
70	51, 52, 53	ringrz 20567	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
71	49, 69, 70	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
72	68, 71	sylan9eqr 2820	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
73	72	fveq2d 6918	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
74		fveq2 6914	. . . . . . . . . . . . 13 ⊢ (𝑦 = (0g‘𝑅) → (𝐹‘𝑦) = (𝐹‘(0g‘𝑅)))
75	74, 45	sylan9eqr 2820	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑦) = 0)
76	75	oveq2d 7465	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = ((𝐹‘𝑥) · 0))
77	60, 69	ffvelcdmd 7115	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
78	77	recnd 11374	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
79	78	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
80	79	mul01d 11546	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · 0) = 0)
81	76, 80	eqtrd 2798	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
82	67, 73, 81	3eqtr4d 2808	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
83		simpl1 1210	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → · = (.r‘𝑅))
86	85	oveqd 7466	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 · 𝑦) = (𝑥(.r‘𝑅)𝑦))
87	86	fveq2d 6918	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑥(.r‘𝑅)𝑦)))
88		simpl2 1211	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ∈ (Base‘𝑅))
89	83, 2	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝐵 = (Base‘𝑅))
90	88, 89	eleqtrrd 2867	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ∈ 𝐵)
91		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ≠ (0g‘𝑅))
92	83, 8	syl 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 0 = (0g‘𝑅))
93	91, 92	neeqtrrd 3033	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ≠ 0 )
94		simpl3 1212	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
95	94, 89	eleqtrrd 2867	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ∈ 𝐵)
96		simprr 785	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ≠ (0g‘𝑅))
97	96, 92	neeqtrrd 3033	. . . . . . . . . . 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 1407	. . . . . . . . . 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 2800	. . . . . . . . 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 3047	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
102		oveq1 7456	. . . . . . . . . . . 12 ⊢ (𝑥 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = ((0g‘𝑅)(+g‘𝑅)𝑦))
103		ringgrp 20506	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
104	49, 103	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
105		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
106	51, 105, 53	grplid 19219	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
107	104, 50, 106	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
108	102, 107	sylan9eqr 2820	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑦)
109	108	fveq2d 6918	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑦))
110	7, 58	breqtrrid 5147	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑥))
111	61, 77	addge02d 11943	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
112	111	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
113	110, 112	mpbid 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
114	109, 113	eqbrtrd 5131	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
115		oveq2 7457	. . . . . . . . . . . 12 ⊢ (𝑦 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑅)(0g‘𝑅)))
116	51, 105, 53	grprid 19220	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
117	104, 69, 116	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
118	115, 117	sylan9eqr 2820	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑥)
119	118	fveq2d 6918	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑥))
120	7, 75	breqtrrid 5147	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑦))
121	77, 61	addge01d 11942	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
122	121	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (0 ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
123	120, 122	mpbid 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
124	119, 123	eqbrtrd 5131	. . . . . . . . 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 18	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → + = (+g‘𝑅))
127	126	oveqd 7466	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦))
128	127	fveq2d 6918	. . . . . . . . . 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 1407	. . . . . . . . . 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 5133	. . . . . . . . 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 3047	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
133	101, 132	jca 521	. . . . . . 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 1139	. . . . . 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 3157	. . . . 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 521	. . . 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 3158	. . 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 2763	. . . . 5 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
139	138, 51, 105, 52, 53	isabv 21110	. . . 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 18	. . 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 726	. 2 ⊢ (𝜑 → 𝐹 ∈ (AbsVal‘𝑅))
142		isabvd.a	. 2 ⊢ (𝜑 → 𝐴 = (AbsVal‘𝑅))
143	141, 142	eleqtrrd 2867	1 ⊢ (𝜑 → 𝐹 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   class class class wbr 5106   Fn wfn 6559  ⟶wf 6560  ‘cfv 6564  (class class class)co 7449  ℂcc 11235  ℝcr 11236  0cc0 11237   + caddc 11240   · cmul 11242  +∞cpnf 11377   < clt 11380   ≤ cle 11381  [,)cico 13516  Basecbs 17428  +_gcplusg 17469  ._rcmulr 17470  0_gc0g 17651  Grpcgrp 19185  Ringcrg 20501  AbsValcabv 21107

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-er 8753  df-map 8885  df-en 9010  df-dom 9011  df-sdom 9012  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-nn 12374  df-2 12443  df-ico 13520  df-sets 17383  df-slot 17401  df-ndx 17413  df-base 17429  df-plusg 17482  df-0g 17653  df-mgm 18857  df-sgrp 18949  df-mnd 18965  df-grp 19188  df-minusg 19189  df-cmn 20038  df-abl 20039  df-mgp 20403  df-rng 20417  df-ur 20450  df-ring 20503  df-abv 21108

This theorem is used by:  abvres  21130  abvtrivd  21131  absabv  21774  abvcxp  27988  padicabv  28003