Theorem isabvd 20884

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 6679	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
4	1, 3	mpbid 235	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶ℝ)
5	4	ffnd 6696	. . . 4 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑅))
6	4	ffvelcdmda 7069	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
7		0le0 12333	. . . . . . . . . 10 ⊢ 0 ≤ 0
8		isabvd.z	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 = (0g‘𝑅))
9	8	fveq2d 6875	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = (𝐹‘(0g‘𝑅)))
10		isabvd.3	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = 0)
11	9, 10	eqtr3d 2802	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = 0)
12	7, 11	breqtrrid 5143	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐹‘(0g‘𝑅)))
13	12	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g‘𝑅)))
14		fveq2 6871	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝑅) → (𝐹‘𝑥) = (𝐹‘(0g‘𝑅)))
15	14	breq2d 5117	. . . . . . . 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 1152	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝜑)
18		simp2 1153	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
19	2	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝐵 = (Base‘𝑅))
20	18, 19	eleqtrrd 2868	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ∈ 𝐵)
21		simp3 1154	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 𝑥 ≠ (0g‘𝑅))
22	8	3ad2ant1 1149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 = (0g‘𝑅))
23	21, 22	neeqtrrd 3034	. . . . . . . . . 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 1394	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 < (𝐹‘𝑥))
26		0re 11198	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
27	6	3adant3 1148	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
28		ltle 11286	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ) → (0 < (𝐹‘𝑥) → 0 ≤ (𝐹‘𝑥)))
29	26, 27, 28	sylancr 598	. . . . . . . . 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 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g‘𝑅) → 0 ≤ (𝐹‘𝑥)))
32	16, 31	pm2.61dne 3046	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘𝑥))
33		elrege0 13472	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
34	6, 32, 33	sylanbrc 594	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (0[,)+∞))
35	34	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞))
36		ffnfv 7104	. . . 4 ⊢ (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞)))
37	5, 35, 36	sylanbrc 594	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶(0[,)+∞))
38	25	gt0ne0d 11766	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (𝐹‘𝑥) ≠ 0)
39	38	3expia 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g‘𝑅) → (𝐹‘𝑥) ≠ 0))
40	39	necon4d 2984	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐹‘𝑥) = 0 → 𝑥 = (0g‘𝑅)))
41	11	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
42		fveqeq2 6880	. . . . . . 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 1149	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
46	45	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
47		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑥 = (0g‘𝑅) → (𝑥(.r‘𝑅)𝑦) = ((0g‘𝑅)(.r‘𝑅)𝑦))
48		isabvd.1	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ Ring)
49	48	3ad2ant1 1149	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
50		simp3 1154	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
51		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
52		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
53		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
54	51, 52, 53	ringlz 20367	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
55	49, 50, 54	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
56	47, 55	sylan9eqr 2822	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
57	56	fveq2d 6875	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
58	14, 45	sylan9eqr 2822	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑥) = 0)
59	58	oveq1d 7415	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = (0 · (𝐹‘𝑦)))
60	4	3ad2ant1 1149	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹:(Base‘𝑅)⟶ℝ)
61	60, 50	ffvelcdmd 7070	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℝ)
62	61	recnd 11225	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
63	62	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
64	63	mul02d 11396	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (0 · (𝐹‘𝑦)) = 0)
65	59, 64	eqtrd 2800	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
66	46, 57, 65	3eqtr4d 2810	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
67	45	adantr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
68		oveq2 7408	. . . . . . . . . . . 12 ⊢ (𝑦 = (0g‘𝑅) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑅)(0g‘𝑅)))
69		simp2 1153	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
70	51, 52, 53	ringrz 20368	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
71	49, 69, 70	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
72	68, 71	sylan9eqr 2822	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
73	72	fveq2d 6875	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
74		fveq2 6871	. . . . . . . . . . . . 13 ⊢ (𝑦 = (0g‘𝑅) → (𝐹‘𝑦) = (𝐹‘(0g‘𝑅)))
75	74, 45	sylan9eqr 2822	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑦) = 0)
76	75	oveq2d 7416	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = ((𝐹‘𝑥) · 0))
77	60, 69	ffvelcdmd 7070	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
78	77	recnd 11225	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
79	78	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
80	79	mul01d 11397	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · 0) = 0)
81	76, 80	eqtrd 2800	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
82	67, 73, 81	3eqtr4d 2810	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
83		simpl1 1208	. . . . . . . . . . . . 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 7417	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 · 𝑦) = (𝑥(.r‘𝑅)𝑦))
87	86	fveq2d 6875	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑥(.r‘𝑅)𝑦)))
88		simpl2 1209	. . . . . . . . . . . 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 2868	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ∈ 𝐵)
91		simprl 782	. . . . . . . . . . . 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 3034	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑥 ≠ 0 )
94		simpl3 1210	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ∈ (Base‘𝑅))
95	94, 89	eleqtrrd 2868	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ∈ 𝐵)
96		simprr 784	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → 𝑦 ≠ (0g‘𝑅))
97	96, 92	neeqtrrd 3034	. . . . . . . . . . 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 1402	. . . . . . . . . 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 2802	. . . . . . . . 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 3048	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
102		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑥 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = ((0g‘𝑅)(+g‘𝑅)𝑦))
103		ringgrp 20311	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
104	49, 103	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
105		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
106	51, 105, 53	grplid 19024	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
107	104, 50, 106	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
108	102, 107	sylan9eqr 2822	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑦)
109	108	fveq2d 6875	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑦))
110	7, 58	breqtrrid 5143	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑥))
111	61, 77	addge02d 11791	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
112	111	adantr 485	. . . . . . . . . . 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 5127	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
115		oveq2 7408	. . . . . . . . . . . 12 ⊢ (𝑦 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑅)(0g‘𝑅)))
116	51, 105, 53	grprid 19025	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
117	104, 69, 116	syl2anc 595	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
118	115, 117	sylan9eqr 2822	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑥)
119	118	fveq2d 6875	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑥))
120	7, 75	breqtrrid 5143	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑦))
121	77, 61	addge01d 11790	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
122	121	adantr 485	. . . . . . . . . . 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 5127	. . . . . . . . 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 7417	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦))
128	127	fveq2d 6875	. . . . . . . . . 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 1402	. . . . . . . . . 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 5129	. . . . . . . . 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 3048	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
133	101, 132	jca 520	. . . . . . 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 1137	. . . . . 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 3156	. . . . 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 520	. . . 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 3157	. . 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 2765	. . . . 5 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
139	138, 51, 105, 52, 53	isabv 20883	. . . 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 725	. 2 ⊢ (𝜑 → 𝐹 ∈ (AbsVal‘𝑅))
142		isabvd.a	. 2 ⊢ (𝜑 → 𝐴 = (AbsVal‘𝑅))
143	141, 142	eleqtrrd 2868	1 ⊢ (𝜑 → 𝐹 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079   class class class wbr 5105   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  ℂcc 11086  ℝcr 11087  0cc0 11088   + caddc 11091   · cmul 11093  +∞cpnf 11228   < clt 11231   ≤ cle 11232  [,)cico 13365  Basecbs 17259  +_gcplusg 17300  ._rcmulr 17301  0_gc0g 17482  Grpcgrp 18990  Ringcrg 20306  AbsValcabv 20880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-ico 13369  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-plusg 17313  df-0g 17484  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-grp 18993  df-minusg 18994  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-abv 20881

This theorem is referenced by:  abvres  20903  abvtrivd  20904  absabv  21534  abvcxp  27737  padicabv  27752