Theorem isabvd 19703

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 6484	. . . . . 6 ⊢ (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
4	1, 3	mpbid 235	. . . . 5 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶ℝ)
5	4	ffnd 6499	. . . 4 ⊢ (𝜑 → 𝐹 Fn (Base‘𝑅))
6	4	ffvelrnda 6855	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
7		0le0 11810	. . . . . . . . . 10 ⊢ 0 ≤ 0
8		isabvd.z	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 = (0g‘𝑅))
9	8	fveq2d 6672	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = (𝐹‘(0g‘𝑅)))
10		isabvd.3	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐹‘ 0 ) = 0)
11	9, 10	eqtr3d 2775	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹‘(0g‘𝑅)) = 0)
12	7, 11	breqtrrid 5065	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝐹‘(0g‘𝑅)))
13	12	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g‘𝑅)))
14		fveq2 6668	. . . . . . . . 9 ⊢ (𝑥 = (0g‘𝑅) → (𝐹‘𝑥) = (𝐹‘(0g‘𝑅)))
15	14	breq2d 5039	. . . . . . . 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 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 2836	. . . . . . . . . 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 3008	. . . . . . . . . 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 1372	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → 0 < (𝐹‘𝑥))
26		0re 10714	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
27	6	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
28		ltle 10800	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ ∧ (𝐹‘𝑥) ∈ ℝ) → (0 < (𝐹‘𝑥) → 0 ≤ (𝐹‘𝑥)))
29	26, 27, 28	sylancr 590	. . . . . . . . 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 3020	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘𝑥))
33		elrege0 12921	. . . . . 6 ⊢ ((𝐹‘𝑥) ∈ (0[,)+∞) ↔ ((𝐹‘𝑥) ∈ ℝ ∧ 0 ≤ (𝐹‘𝑥)))
34	6, 32, 33	sylanbrc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ (0[,)+∞))
35	34	ralrimiva 3096	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞))
36		ffnfv 6886	. . . 4 ⊢ (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹‘𝑥) ∈ (0[,)+∞)))
37	5, 35, 36	sylanbrc 586	. . 3 ⊢ (𝜑 → 𝐹:(Base‘𝑅)⟶(0[,)+∞))
38	25	gt0ne0d 11275	. . . . . . . 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 2958	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → ((𝐹‘𝑥) = 0 → 𝑥 = (0g‘𝑅)))
41	11	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
42		fveqeq2 6677	. . . . . . 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 1134	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
46	45	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
47		oveq1 7171	. . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑅) = (Base‘𝑅)
52		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (.r‘𝑅) = (.r‘𝑅)
53		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (0g‘𝑅) = (0g‘𝑅)
54	51, 52, 53	ringlz 19452	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
55	49, 50, 54	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(.r‘𝑅)𝑦) = (0g‘𝑅))
56	47, 55	sylan9eqr 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
57	56	fveq2d 6672	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
58	14, 45	sylan9eqr 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑥) = 0)
59	58	oveq1d 7179	. . . . . . . . . . 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	ffvelrnd 6856	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℝ)
62	61	recnd 10740	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
63	62	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘𝑦) ∈ ℂ)
64	63	mul02d 10909	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (0 · (𝐹‘𝑦)) = 0)
65	59, 64	eqtrd 2773	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
66	46, 57, 65	3eqtr4d 2783	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
67	45	adantr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(0g‘𝑅)) = 0)
68		oveq2 7172	. . . . . . . . . . . 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 19453	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
71	49, 69, 70	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))
72	68, 71	sylan9eqr 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(.r‘𝑅)𝑦) = (0g‘𝑅))
73	72	fveq2d 6672	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = (𝐹‘(0g‘𝑅)))
74		fveq2 6668	. . . . . . . . . . . . 13 ⊢ (𝑦 = (0g‘𝑅) → (𝐹‘𝑦) = (𝐹‘(0g‘𝑅)))
75	74, 45	sylan9eqr 2795	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑦) = 0)
76	75	oveq2d 7180	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = ((𝐹‘𝑥) · 0))
77	60, 69	ffvelrnd 6856	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℝ)
78	77	recnd 10740	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
79	78	adantr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘𝑥) ∈ ℂ)
80	79	mul01d 10910	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · 0) = 0)
81	76, 80	eqtrd 2773	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → ((𝐹‘𝑥) · (𝐹‘𝑦)) = 0)
82	67, 73, 81	3eqtr4d 2783	. . . . . . . . 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 7181	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 · 𝑦) = (𝑥(.r‘𝑅)𝑦))
87	86	fveq2d 6672	. . . . . . . . . 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 2836	. . . . . . . . . . 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 3008	. . . . . . . . . . 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 2836	. . . . . . . . . . 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 3008	. . . . . . . . . . 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 1380	. . . . . . . . . 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 2775	. . . . . . . . 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 3022	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)))
102		oveq1 7171	. . . . . . . . . . . 12 ⊢ (𝑥 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = ((0g‘𝑅)(+g‘𝑅)𝑦))
103		ringgrp 19414	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
104	49, 103	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
105		eqid 2738	. . . . . . . . . . . . . 14 ⊢ (+g‘𝑅) = (+g‘𝑅)
106	51, 105, 53	grplid 18244	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
107	104, 50, 106	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g‘𝑅)(+g‘𝑅)𝑦) = 𝑦)
108	102, 107	sylan9eqr 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑦)
109	108	fveq2d 6672	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑦))
110	7, 58	breqtrrid 5065	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑥))
111	61, 77	addge02d 11300	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑥) ↔ (𝐹‘𝑦) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
112	111	adantr 484	. . . . . . . . . . 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 5049	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
115		oveq2 7172	. . . . . . . . . . . 12 ⊢ (𝑦 = (0g‘𝑅) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑅)(0g‘𝑅)))
116	51, 105, 53	grprid 18245	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
117	104, 69, 116	syl2anc 587	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g‘𝑅)(0g‘𝑅)) = 𝑥)
118	115, 117	sylan9eqr 2795	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝑥(+g‘𝑅)𝑦) = 𝑥)
119	118	fveq2d 6672	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) = (𝐹‘𝑥))
120	7, 75	breqtrrid 5065	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g‘𝑅)) → 0 ≤ (𝐹‘𝑦))
121	77, 61	addge01d 11299	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹‘𝑦) ↔ (𝐹‘𝑥) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))
122	121	adantr 484	. . . . . . . . . . 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 5049	. . . . . . . . 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 7181	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅))) → (𝑥 + 𝑦) = (𝑥(+g‘𝑅)𝑦))
128	127	fveq2d 6672	. . . . . . . . . 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 1380	. . . . . . . . . 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 5051	. . . . . . . . 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 3022	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))
133	101, 132	jca 515	. . . . . . 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 3095	. . . . 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 515	. . . 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 3096	. . 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 2738	. . . . 5 ⊢ (AbsVal‘𝑅) = (AbsVal‘𝑅)
139	138, 51, 105, 52, 53	isabv 19702	. . . 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 2836	1 ⊢ (𝜑 → 𝐹 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2113   ≠ wne 2934  ∀wral 3053   class class class wbr 5027   Fn wfn 6328  ⟶wf 6329  ‘cfv 6333  (class class class)co 7164  ℂcc 10606  ℝcr 10607  0cc0 10608   + caddc 10611   · cmul 10613  +∞cpnf 10743   < clt 10746   ≤ cle 10747  [,)cico 12816  Basecbs 16579  +_gcplusg 16661  ._rcmulr 16662  0_gc0g 16809  Grpcgrp 18212  Ringcrg 19409  AbsValcabv 19699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-er 8313  df-map 8432  df-en 8549  df-dom 8550  df-sdom 8551  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-2 11772  df-ico 12820  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-plusg 16674  df-0g 16811  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-grp 18215  df-minusg 18216  df-mgp 19352  df-ring 19411  df-abv 19700

This theorem is referenced by:  abvres  19722  abvtrivd  19723  absabv  20267  abvcxp  26343  padicabv  26358