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Theorem isabvd 19514
 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 (𝜑𝐴 = (AbsVal‘𝑅))
isabvd.b (𝜑𝐵 = (Base‘𝑅))
isabvd.p (𝜑+ = (+g𝑅))
isabvd.t (𝜑· = (.r𝑅))
isabvd.z (𝜑0 = (0g𝑅))
isabvd.1 (𝜑𝑅 ∈ Ring)
isabvd.2 (𝜑𝐹:𝐵⟶ℝ)
isabvd.3 (𝜑 → (𝐹0 ) = 0)
isabvd.4 ((𝜑𝑥𝐵𝑥0 ) → 0 < (𝐹𝑥))
isabvd.5 ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
isabvd.6 ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
Assertion
Ref Expression
isabvd (𝜑𝐹𝐴)
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
StepHypRef Expression
1 isabvd.2 . . . . . 6 (𝜑𝐹:𝐵⟶ℝ)
2 isabvd.b . . . . . . 7 (𝜑𝐵 = (Base‘𝑅))
32feq2d 6496 . . . . . 6 (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
41, 3mpbid 233 . . . . 5 (𝜑𝐹:(Base‘𝑅)⟶ℝ)
54ffnd 6511 . . . 4 (𝜑𝐹 Fn (Base‘𝑅))
64ffvelrnda 6846 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
7 0le0 11730 . . . . . . . . . 10 0 ≤ 0
8 isabvd.z . . . . . . . . . . . 12 (𝜑0 = (0g𝑅))
98fveq2d 6670 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = (𝐹‘(0g𝑅)))
10 isabvd.3 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = 0)
119, 10eqtr3d 2862 . . . . . . . . . 10 (𝜑 → (𝐹‘(0g𝑅)) = 0)
127, 11breqtrrid 5100 . . . . . . . . 9 (𝜑 → 0 ≤ (𝐹‘(0g𝑅)))
1312adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g𝑅)))
14 fveq2 6666 . . . . . . . . 9 (𝑥 = (0g𝑅) → (𝐹𝑥) = (𝐹‘(0g𝑅)))
1514breq2d 5074 . . . . . . . 8 (𝑥 = (0g𝑅) → (0 ≤ (𝐹𝑥) ↔ 0 ≤ (𝐹‘(0g𝑅))))
1613, 15syl5ibrcom 248 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g𝑅) → 0 ≤ (𝐹𝑥)))
17 simp1 1130 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝜑)
18 simp2 1131 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ∈ (Base‘𝑅))
1923ad2ant1 1127 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝐵 = (Base‘𝑅))
2018, 19eleqtrrd 2920 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥𝐵)
21 simp3 1132 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ≠ (0g𝑅))
2283ad2ant1 1127 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 = (0g𝑅))
2321, 22neeqtrrd 3094 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥0 )
24 isabvd.4 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑥0 ) → 0 < (𝐹𝑥))
2517, 20, 23, 24syl3anc 1365 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 < (𝐹𝑥))
26 0re 10635 . . . . . . . . . 10 0 ∈ ℝ
2763adant3 1126 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ∈ ℝ)
28 ltle 10721 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (𝐹𝑥) ∈ ℝ) → (0 < (𝐹𝑥) → 0 ≤ (𝐹𝑥)))
2926, 27, 28sylancr 587 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (0 < (𝐹𝑥) → 0 ≤ (𝐹𝑥)))
3025, 29mpd 15 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 ≤ (𝐹𝑥))
31303expia 1115 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g𝑅) → 0 ≤ (𝐹𝑥)))
3216, 31pm2.61dne 3107 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹𝑥))
33 elrege0 12835 . . . . . 6 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
346, 32, 33sylanbrc 583 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ (0[,)+∞))
3534ralrimiva 3186 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞))
36 ffnfv 6877 . . . 4 (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞)))
375, 35, 36sylanbrc 583 . . 3 (𝜑𝐹:(Base‘𝑅)⟶(0[,)+∞))
3825gt0ne0d 11196 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ≠ 0)
39383expia 1115 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g𝑅) → (𝐹𝑥) ≠ 0))
4039necon4d 3044 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((𝐹𝑥) = 0 → 𝑥 = (0g𝑅)))
4111adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹‘(0g𝑅)) = 0)
42 fveqeq2 6675 . . . . . . 7 (𝑥 = (0g𝑅) → ((𝐹𝑥) = 0 ↔ (𝐹‘(0g𝑅)) = 0))
4341, 42syl5ibrcom 248 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g𝑅) → (𝐹𝑥) = 0))
4440, 43impbid 213 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)))
45113ad2ant1 1127 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(0g𝑅)) = 0)
4645adantr 481 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(0g𝑅)) = 0)
47 oveq1 7158 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = ((0g𝑅)(.r𝑅)𝑦))
48 isabvd.1 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
49483ad2ant1 1127 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
50 simp3 1132 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
51 eqid 2825 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
52 eqid 2825 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
53 eqid 2825 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5451, 52, 53ringlz 19260 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)𝑦) = (0g𝑅))
5549, 50, 54syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)𝑦) = (0g𝑅))
5647, 55sylan9eqr 2882 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
5756fveq2d 6670 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
5814, 45sylan9eqr 2882 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑥) = 0)
5958oveq1d 7166 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = (0 · (𝐹𝑦)))
6043ad2ant1 1127 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹:(Base‘𝑅)⟶ℝ)
6160, 50ffvelrnd 6847 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℝ)
6261recnd 10661 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℂ)
6362adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑦) ∈ ℂ)
6463mul02d 10830 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (0 · (𝐹𝑦)) = 0)
6559, 64eqtrd 2860 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
6646, 57, 653eqtr4d 2870 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
6745adantr 481 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(0g𝑅)) = 0)
68 oveq2 7159 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = (𝑥(.r𝑅)(0g𝑅)))
69 simp2 1131 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
7051, 52, 53ringrz 19261 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)(0g𝑅)) = (0g𝑅))
7149, 69, 70syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)(0g𝑅)) = (0g𝑅))
7268, 71sylan9eqr 2882 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
7372fveq2d 6670 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
74 fveq2 6666 . . . . . . . . . . . . 13 (𝑦 = (0g𝑅) → (𝐹𝑦) = (𝐹‘(0g𝑅)))
7574, 45sylan9eqr 2882 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑦) = 0)
7675oveq2d 7167 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = ((𝐹𝑥) · 0))
7760, 69ffvelrnd 6847 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
7877recnd 10661 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℂ)
7978adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑥) ∈ ℂ)
8079mul01d 10831 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · 0) = 0)
8176, 80eqtrd 2860 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
8267, 73, 813eqtr4d 2870 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
83 simpl1 1185 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝜑)
84 isabvd.t . . . . . . . . . . . . 13 (𝜑· = (.r𝑅))
8583, 84syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → · = (.r𝑅))
8685oveqd 7168 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 · 𝑦) = (𝑥(.r𝑅)𝑦))
8786fveq2d 6670 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑥(.r𝑅)𝑦)))
88 simpl2 1186 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥 ∈ (Base‘𝑅))
8983, 2syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝐵 = (Base‘𝑅))
9088, 89eleqtrrd 2920 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥𝐵)
91 simprl 767 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥 ≠ (0g𝑅))
9283, 8syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 0 = (0g𝑅))
9391, 92neeqtrrd 3094 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥0 )
94 simpl3 1187 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦 ∈ (Base‘𝑅))
9594, 89eleqtrrd 2920 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦𝐵)
96 simprr 769 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦 ≠ (0g𝑅))
9796, 92neeqtrrd 3094 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦0 )
98 isabvd.5 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
9983, 90, 93, 95, 97, 98syl122anc 1373 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10087, 99eqtr3d 2862 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10166, 82, 100pm2.61da2ne 3109 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
102 oveq1 7158 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = ((0g𝑅)(+g𝑅)𝑦))
103 ringgrp 19225 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
10449, 103syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
105 eqid 2825 . . . . . . . . . . . . . 14 (+g𝑅) = (+g𝑅)
10651, 105, 53grplid 18066 . . . . . . . . . . . . 13 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g𝑅)(+g𝑅)𝑦) = 𝑦)
107104, 50, 106syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g𝑅)(+g𝑅)𝑦) = 𝑦)
108102, 107sylan9eqr 2882 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑦)
109108fveq2d 6670 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑦))
1107, 58breqtrrid 5100 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → 0 ≤ (𝐹𝑥))
11161, 77addge02d 11221 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦))))
112111adantr 481 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (0 ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦))))
113110, 112mpbid 233 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦)))
114109, 113eqbrtrd 5084 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
115 oveq2 7159 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑅)(0g𝑅)))
11651, 105, 53grprid 18067 . . . . . . . . . . . . 13 ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)(0g𝑅)) = 𝑥)
117104, 69, 116syl2anc 584 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)(0g𝑅)) = 𝑥)
118115, 117sylan9eqr 2882 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑥)
119118fveq2d 6670 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑥))
1207, 75breqtrrid 5100 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → 0 ≤ (𝐹𝑦))
12177, 61addge01d 11220 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹𝑦) ↔ (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦))))
122121adantr 481 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (0 ≤ (𝐹𝑦) ↔ (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦))))
123120, 122mpbid 233 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦)))
124119, 123eqbrtrd 5084 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
125 isabvd.p . . . . . . . . . . . . 13 (𝜑+ = (+g𝑅))
12683, 125syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → + = (+g𝑅))
127126oveqd 7168 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
128127fveq2d 6670 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
129 isabvd.6 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
13083, 90, 93, 95, 97, 129syl122anc 1373 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
131128, 130eqbrtrrd 5086 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
132114, 124, 131pm2.61da2ne 3109 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
133101, 132jca 512 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))
1341333expia 1115 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑦 ∈ (Base‘𝑅) → ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
135134ralrimiv 3185 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))
13644, 135jca 512 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑅)) → (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
137136ralrimiva 3186 . . 3 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
138 eqid 2825 . . . . 5 (AbsVal‘𝑅) = (AbsVal‘𝑅)
139138, 51, 105, 52, 53isabv 19513 . . . 4 (𝑅 ∈ Ring → (𝐹 ∈ (AbsVal‘𝑅) ↔ (𝐹:(Base‘𝑅)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝑅)(((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
14048, 139syl 17 . . 3 (𝜑 → (𝐹 ∈ (AbsVal‘𝑅) ↔ (𝐹:(Base‘𝑅)⟶(0[,)+∞) ∧ ∀𝑥 ∈ (Base‘𝑅)(((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))))
14137, 137, 140mpbir2and 709 . 2 (𝜑𝐹 ∈ (AbsVal‘𝑅))
142 isabvd.a . 2 (𝜑𝐴 = (AbsVal‘𝑅))
143141, 142eleqtrrd 2920 1 (𝜑𝐹𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ≠ wne 3020  ∀wral 3142   class class class wbr 5062   Fn wfn 6346  ⟶wf 6347  ‘cfv 6351  (class class class)co 7151  ℂcc 10527  ℝcr 10528  0cc0 10529   + caddc 10532   · cmul 10534  +∞cpnf 10664   < clt 10667   ≤ cle 10668  [,)cico 12733  Basecbs 16476  +gcplusg 16558  .rcmulr 16559  0gc0g 16706  Grpcgrp 18036  Ringcrg 19220  AbsValcabv 19510 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-ico 12737  df-ndx 16479  df-slot 16480  df-base 16482  df-sets 16483  df-plusg 16571  df-0g 16708  df-mgm 17845  df-sgrp 17893  df-mnd 17904  df-grp 18039  df-minusg 18040  df-mgp 19163  df-ring 19222  df-abv 19511 This theorem is referenced by:  abvres  19533  abvtrivd  19534  absabv  20521  abvcxp  26108  padicabv  26123
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