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Theorem isabvd 20279
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 6654 . . . . . 6 (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
41, 3mpbid 231 . . . . 5 (𝜑𝐹:(Base‘𝑅)⟶ℝ)
54ffnd 6669 . . . 4 (𝜑𝐹 Fn (Base‘𝑅))
64ffvelcdmda 7035 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
7 0le0 12254 . . . . . . . . . 10 0 ≤ 0
8 isabvd.z . . . . . . . . . . . 12 (𝜑0 = (0g𝑅))
98fveq2d 6846 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = (𝐹‘(0g𝑅)))
10 isabvd.3 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = 0)
119, 10eqtr3d 2778 . . . . . . . . . 10 (𝜑 → (𝐹‘(0g𝑅)) = 0)
127, 11breqtrrid 5143 . . . . . . . . 9 (𝜑 → 0 ≤ (𝐹‘(0g𝑅)))
1312adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g𝑅)))
14 fveq2 6842 . . . . . . . . 9 (𝑥 = (0g𝑅) → (𝐹𝑥) = (𝐹‘(0g𝑅)))
1514breq2d 5117 . . . . . . . 8 (𝑥 = (0g𝑅) → (0 ≤ (𝐹𝑥) ↔ 0 ≤ (𝐹‘(0g𝑅))))
1613, 15syl5ibrcom 246 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g𝑅) → 0 ≤ (𝐹𝑥)))
17 simp1 1136 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝜑)
18 simp2 1137 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ∈ (Base‘𝑅))
1923ad2ant1 1133 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝐵 = (Base‘𝑅))
2018, 19eleqtrrd 2841 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥𝐵)
21 simp3 1138 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ≠ (0g𝑅))
2283ad2ant1 1133 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 = (0g𝑅))
2321, 22neeqtrrd 3018 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥0 )
24 isabvd.4 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑥0 ) → 0 < (𝐹𝑥))
2517, 20, 23, 24syl3anc 1371 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 < (𝐹𝑥))
26 0re 11157 . . . . . . . . . 10 0 ∈ ℝ
2763adant3 1132 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ∈ ℝ)
28 ltle 11243 . . . . . . . . . 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 1121 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g𝑅) → 0 ≤ (𝐹𝑥)))
3216, 31pm2.61dne 3031 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹𝑥))
33 elrege0 13371 . . . . . 6 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
346, 32, 33sylanbrc 583 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ (0[,)+∞))
3534ralrimiva 3143 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞))
36 ffnfv 7066 . . . 4 (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞)))
375, 35, 36sylanbrc 583 . . 3 (𝜑𝐹:(Base‘𝑅)⟶(0[,)+∞))
3825gt0ne0d 11719 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ≠ 0)
39383expia 1121 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g𝑅) → (𝐹𝑥) ≠ 0))
4039necon4d 2967 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((𝐹𝑥) = 0 → 𝑥 = (0g𝑅)))
4111adantr 481 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹‘(0g𝑅)) = 0)
42 fveqeq2 6851 . . . . . . 7 (𝑥 = (0g𝑅) → ((𝐹𝑥) = 0 ↔ (𝐹‘(0g𝑅)) = 0))
4341, 42syl5ibrcom 246 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g𝑅) → (𝐹𝑥) = 0))
4440, 43impbid 211 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)))
45113ad2ant1 1133 . . . . . . . . . . 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 7364 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = ((0g𝑅)(.r𝑅)𝑦))
48 isabvd.1 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
49483ad2ant1 1133 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
50 simp3 1138 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
51 eqid 2736 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
52 eqid 2736 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
53 eqid 2736 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5451, 52, 53ringlz 20011 . . . . . . . . . . . . 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 2798 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
5756fveq2d 6846 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
5814, 45sylan9eqr 2798 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑥) = 0)
5958oveq1d 7372 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = (0 · (𝐹𝑦)))
6043ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹:(Base‘𝑅)⟶ℝ)
6160, 50ffvelcdmd 7036 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℝ)
6261recnd 11183 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℂ)
6362adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑦) ∈ ℂ)
6463mul02d 11353 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (0 · (𝐹𝑦)) = 0)
6559, 64eqtrd 2776 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
6646, 57, 653eqtr4d 2786 . . . . . . . . 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 7365 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = (𝑥(.r𝑅)(0g𝑅)))
69 simp2 1137 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
7051, 52, 53ringrz 20012 . . . . . . . . . . . . 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 2798 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
7372fveq2d 6846 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
74 fveq2 6842 . . . . . . . . . . . . 13 (𝑦 = (0g𝑅) → (𝐹𝑦) = (𝐹‘(0g𝑅)))
7574, 45sylan9eqr 2798 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑦) = 0)
7675oveq2d 7373 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = ((𝐹𝑥) · 0))
7760, 69ffvelcdmd 7036 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
7877recnd 11183 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℂ)
7978adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑥) ∈ ℂ)
8079mul01d 11354 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · 0) = 0)
8176, 80eqtrd 2776 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
8267, 73, 813eqtr4d 2786 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
83 simpl1 1191 . . . . . . . . . . . . 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 7374 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 · 𝑦) = (𝑥(.r𝑅)𝑦))
8786fveq2d 6846 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑥(.r𝑅)𝑦)))
88 simpl2 1192 . . . . . . . . . . . 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 2841 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥𝐵)
91 simprl 769 . . . . . . . . . . . 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 3018 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥0 )
94 simpl3 1193 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦 ∈ (Base‘𝑅))
9594, 89eleqtrrd 2841 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦𝐵)
96 simprr 771 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦 ≠ (0g𝑅))
9796, 92neeqtrrd 3018 . . . . . . . . . . 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 1379 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10087, 99eqtr3d 2778 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10166, 82, 100pm2.61da2ne 3033 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
102 oveq1 7364 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = ((0g𝑅)(+g𝑅)𝑦))
103 ringgrp 19969 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
10449, 103syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
105 eqid 2736 . . . . . . . . . . . . . 14 (+g𝑅) = (+g𝑅)
10651, 105, 53grplid 18780 . . . . . . . . . . . . 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 2798 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑦)
109108fveq2d 6846 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑦))
1107, 58breqtrrid 5143 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → 0 ≤ (𝐹𝑥))
11161, 77addge02d 11744 . . . . . . . . . . . 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 231 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦)))
114109, 113eqbrtrd 5127 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
115 oveq2 7365 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑅)(0g𝑅)))
11651, 105, 53grprid 18781 . . . . . . . . . . . . 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 2798 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑥)
119118fveq2d 6846 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑥))
1207, 75breqtrrid 5143 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → 0 ≤ (𝐹𝑦))
12177, 61addge01d 11743 . . . . . . . . . . . 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 231 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦)))
124119, 123eqbrtrd 5127 . . . . . . . . 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 7374 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
128127fveq2d 6846 . . . . . . . . . 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 1379 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
131128, 130eqbrtrrd 5129 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
132114, 124, 131pm2.61da2ne 3033 . . . . . . . 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 1121 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑦 ∈ (Base‘𝑅) → ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
135134ralrimiv 3142 . . . . 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 3143 . . 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 2736 . . . . 5 (AbsVal‘𝑅) = (AbsVal‘𝑅)
139138, 51, 105, 52, 53isabv 20278 . . . 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 711 . 2 (𝜑𝐹 ∈ (AbsVal‘𝑅))
142 isabvd.a . 2 (𝜑𝐴 = (AbsVal‘𝑅))
143141, 142eleqtrrd 2841 1 (𝜑𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064   class class class wbr 5105   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cc 11049  cr 11050  0cc0 11051   + caddc 11054   · cmul 11056  +∞cpnf 11186   < clt 11189  cle 11190  [,)cico 13266  Basecbs 17083  +gcplusg 17133  .rcmulr 17134  0gc0g 17321  Grpcgrp 18748  Ringcrg 19964  AbsValcabv 20275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-ico 13270  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-mgp 19897  df-ring 19966  df-abv 20276
This theorem is referenced by:  abvres  20298  abvtrivd  20299  absabv  20854  abvcxp  26963  padicabv  26978
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