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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 (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 6722 . . . . . 6 (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
41, 3mpbid 232 . . . . 5 (𝜑𝐹:(Base‘𝑅)⟶ℝ)
54ffnd 6737 . . . 4 (𝜑𝐹 Fn (Base‘𝑅))
64ffvelcdmda 7104 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
7 0le0 12367 . . . . . . . . . 10 0 ≤ 0
8 isabvd.z . . . . . . . . . . . 12 (𝜑0 = (0g𝑅))
98fveq2d 6910 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = (𝐹‘(0g𝑅)))
10 isabvd.3 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = 0)
119, 10eqtr3d 2779 . . . . . . . . . 10 (𝜑 → (𝐹‘(0g𝑅)) = 0)
127, 11breqtrrid 5181 . . . . . . . . 9 (𝜑 → 0 ≤ (𝐹‘(0g𝑅)))
1312adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g𝑅)))
14 fveq2 6906 . . . . . . . . 9 (𝑥 = (0g𝑅) → (𝐹𝑥) = (𝐹‘(0g𝑅)))
1514breq2d 5155 . . . . . . . 8 (𝑥 = (0g𝑅) → (0 ≤ (𝐹𝑥) ↔ 0 ≤ (𝐹‘(0g𝑅))))
1613, 15syl5ibrcom 247 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g𝑅) → 0 ≤ (𝐹𝑥)))
17 simp1 1137 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝜑)
18 simp2 1138 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ∈ (Base‘𝑅))
1923ad2ant1 1134 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝐵 = (Base‘𝑅))
2018, 19eleqtrrd 2844 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥𝐵)
21 simp3 1139 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ≠ (0g𝑅))
2283ad2ant1 1134 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 = (0g𝑅))
2321, 22neeqtrrd 3015 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥0 )
24 isabvd.4 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑥0 ) → 0 < (𝐹𝑥))
2517, 20, 23, 24syl3anc 1373 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 < (𝐹𝑥))
26 0re 11263 . . . . . . . . . 10 0 ∈ ℝ
2763adant3 1133 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ∈ ℝ)
28 ltle 11349 . . . . . . . . . 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 1122 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g𝑅) → 0 ≤ (𝐹𝑥)))
3216, 31pm2.61dne 3028 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹𝑥))
33 elrege0 13494 . . . . . 6 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
346, 32, 33sylanbrc 583 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ (0[,)+∞))
3534ralrimiva 3146 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞))
36 ffnfv 7139 . . . 4 (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞)))
375, 35, 36sylanbrc 583 . . 3 (𝜑𝐹:(Base‘𝑅)⟶(0[,)+∞))
3825gt0ne0d 11827 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ≠ 0)
39383expia 1122 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g𝑅) → (𝐹𝑥) ≠ 0))
4039necon4d 2964 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((𝐹𝑥) = 0 → 𝑥 = (0g𝑅)))
4111adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹‘(0g𝑅)) = 0)
42 fveqeq2 6915 . . . . . . 7 (𝑥 = (0g𝑅) → ((𝐹𝑥) = 0 ↔ (𝐹‘(0g𝑅)) = 0))
4341, 42syl5ibrcom 247 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g𝑅) → (𝐹𝑥) = 0))
4440, 43impbid 212 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)))
45113ad2ant1 1134 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(0g𝑅)) = 0)
4645adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(0g𝑅)) = 0)
47 oveq1 7438 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = ((0g𝑅)(.r𝑅)𝑦))
48 isabvd.1 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
49483ad2ant1 1134 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
50 simp3 1139 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
51 eqid 2737 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
52 eqid 2737 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
53 eqid 2737 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5451, 52, 53ringlz 20290 . . . . . . . . . . . . 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 2799 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
5756fveq2d 6910 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
5814, 45sylan9eqr 2799 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑥) = 0)
5958oveq1d 7446 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = (0 · (𝐹𝑦)))
6043ad2ant1 1134 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹:(Base‘𝑅)⟶ℝ)
6160, 50ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℝ)
6261recnd 11289 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℂ)
6362adantr 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑦) ∈ ℂ)
6463mul02d 11459 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (0 · (𝐹𝑦)) = 0)
6559, 64eqtrd 2777 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
6646, 57, 653eqtr4d 2787 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
6745adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(0g𝑅)) = 0)
68 oveq2 7439 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = (𝑥(.r𝑅)(0g𝑅)))
69 simp2 1138 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
7051, 52, 53ringrz 20291 . . . . . . . . . . . . 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 2799 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
7372fveq2d 6910 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
74 fveq2 6906 . . . . . . . . . . . . 13 (𝑦 = (0g𝑅) → (𝐹𝑦) = (𝐹‘(0g𝑅)))
7574, 45sylan9eqr 2799 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑦) = 0)
7675oveq2d 7447 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = ((𝐹𝑥) · 0))
7760, 69ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
7877recnd 11289 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℂ)
7978adantr 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑥) ∈ ℂ)
8079mul01d 11460 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · 0) = 0)
8176, 80eqtrd 2777 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
8267, 73, 813eqtr4d 2787 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
83 simpl1 1192 . . . . . . . . . . . . 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 7448 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 · 𝑦) = (𝑥(.r𝑅)𝑦))
8786fveq2d 6910 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑥(.r𝑅)𝑦)))
88 simpl2 1193 . . . . . . . . . . . 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 2844 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥𝐵)
91 simprl 771 . . . . . . . . . . . 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 3015 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥0 )
94 simpl3 1194 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦 ∈ (Base‘𝑅))
9594, 89eleqtrrd 2844 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦𝐵)
96 simprr 773 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦 ≠ (0g𝑅))
9796, 92neeqtrrd 3015 . . . . . . . . . . 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 1381 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10087, 99eqtr3d 2779 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10166, 82, 100pm2.61da2ne 3030 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
102 oveq1 7438 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = ((0g𝑅)(+g𝑅)𝑦))
103 ringgrp 20235 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
10449, 103syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
105 eqid 2737 . . . . . . . . . . . . . 14 (+g𝑅) = (+g𝑅)
10651, 105, 53grplid 18985 . . . . . . . . . . . . 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 2799 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑦)
109108fveq2d 6910 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑦))
1107, 58breqtrrid 5181 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → 0 ≤ (𝐹𝑥))
11161, 77addge02d 11852 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦))))
112111adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (0 ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦))))
113110, 112mpbid 232 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦)))
114109, 113eqbrtrd 5165 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
115 oveq2 7439 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑅)(0g𝑅)))
11651, 105, 53grprid 18986 . . . . . . . . . . . . 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 2799 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑥)
119118fveq2d 6910 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑥))
1207, 75breqtrrid 5181 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → 0 ≤ (𝐹𝑦))
12177, 61addge01d 11851 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹𝑦) ↔ (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦))))
122121adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (0 ≤ (𝐹𝑦) ↔ (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦))))
123120, 122mpbid 232 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦)))
124119, 123eqbrtrd 5165 . . . . . . . . 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 7448 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
128127fveq2d 6910 . . . . . . . . . 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 1381 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
131128, 130eqbrtrrd 5167 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
132114, 124, 131pm2.61da2ne 3030 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
133101, 132jca 511 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))
1341333expia 1122 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑦 ∈ (Base‘𝑅) → ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
135134ralrimiv 3145 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))
13644, 135jca 511 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑅)) → (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
137136ralrimiva 3146 . . 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 2737 . . . . 5 (AbsVal‘𝑅) = (AbsVal‘𝑅)
139138, 51, 105, 52, 53isabv 20812 . . . 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 713 . 2 (𝜑𝐹 ∈ (AbsVal‘𝑅))
142 isabvd.a . 2 (𝜑𝐴 = (AbsVal‘𝑅))
143141, 142eleqtrrd 2844 1 (𝜑𝐹𝐴)
Colors of variables: wff setvar class
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  cc 11153  cr 11154  0cc0 11155   + caddc 11158   · cmul 11160  +∞cpnf 11292   < clt 11295  cle 11296  [,)cico 13389  Basecbs 17247  +gcplusg 17297  .rcmulr 17298  0gc0g 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
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