MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isabvd Structured version   Visualization version   GIF version

Theorem isabvd 20080
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 6586 . . . . . 6 (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
41, 3mpbid 231 . . . . 5 (𝜑𝐹:(Base‘𝑅)⟶ℝ)
54ffnd 6601 . . . 4 (𝜑𝐹 Fn (Base‘𝑅))
64ffvelrnda 6961 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
7 0le0 12074 . . . . . . . . . 10 0 ≤ 0
8 isabvd.z . . . . . . . . . . . 12 (𝜑0 = (0g𝑅))
98fveq2d 6778 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = (𝐹‘(0g𝑅)))
10 isabvd.3 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = 0)
119, 10eqtr3d 2780 . . . . . . . . . 10 (𝜑 → (𝐹‘(0g𝑅)) = 0)
127, 11breqtrrid 5112 . . . . . . . . 9 (𝜑 → 0 ≤ (𝐹‘(0g𝑅)))
1312adantr 481 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g𝑅)))
14 fveq2 6774 . . . . . . . . 9 (𝑥 = (0g𝑅) → (𝐹𝑥) = (𝐹‘(0g𝑅)))
1514breq2d 5086 . . . . . . . 8 (𝑥 = (0g𝑅) → (0 ≤ (𝐹𝑥) ↔ 0 ≤ (𝐹‘(0g𝑅))))
1613, 15syl5ibrcom 246 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g𝑅) → 0 ≤ (𝐹𝑥)))
17 simp1 1135 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝜑)
18 simp2 1136 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ∈ (Base‘𝑅))
1923ad2ant1 1132 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝐵 = (Base‘𝑅))
2018, 19eleqtrrd 2842 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥𝐵)
21 simp3 1137 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ≠ (0g𝑅))
2283ad2ant1 1132 . . . . . . . . . . 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 1370 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 < (𝐹𝑥))
26 0re 10977 . . . . . . . . . 10 0 ∈ ℝ
2763adant3 1131 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ∈ ℝ)
28 ltle 11063 . . . . . . . . . 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 1120 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g𝑅) → 0 ≤ (𝐹𝑥)))
3216, 31pm2.61dne 3031 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹𝑥))
33 elrege0 13186 . . . . . 6 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
346, 32, 33sylanbrc 583 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ (0[,)+∞))
3534ralrimiva 3103 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞))
36 ffnfv 6992 . . . 4 (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞)))
375, 35, 36sylanbrc 583 . . 3 (𝜑𝐹:(Base‘𝑅)⟶(0[,)+∞))
3825gt0ne0d 11539 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ≠ 0)
39383expia 1120 . . . . . . 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 6783 . . . . . . 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 1132 . . . . . . . . . . 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 7282 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = ((0g𝑅)(.r𝑅)𝑦))
48 isabvd.1 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
49483ad2ant1 1132 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
50 simp3 1137 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
51 eqid 2738 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
52 eqid 2738 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
53 eqid 2738 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5451, 52, 53ringlz 19826 . . . . . . . . . . . . 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 2800 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
5756fveq2d 6778 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
5814, 45sylan9eqr 2800 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑥) = 0)
5958oveq1d 7290 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = (0 · (𝐹𝑦)))
6043ad2ant1 1132 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹:(Base‘𝑅)⟶ℝ)
6160, 50ffvelrnd 6962 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℝ)
6261recnd 11003 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℂ)
6362adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑦) ∈ ℂ)
6463mul02d 11173 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (0 · (𝐹𝑦)) = 0)
6559, 64eqtrd 2778 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
6646, 57, 653eqtr4d 2788 . . . . . . . . 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 7283 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = (𝑥(.r𝑅)(0g𝑅)))
69 simp2 1136 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
7051, 52, 53ringrz 19827 . . . . . . . . . . . . 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 2800 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
7372fveq2d 6778 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
74 fveq2 6774 . . . . . . . . . . . . 13 (𝑦 = (0g𝑅) → (𝐹𝑦) = (𝐹‘(0g𝑅)))
7574, 45sylan9eqr 2800 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑦) = 0)
7675oveq2d 7291 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = ((𝐹𝑥) · 0))
7760, 69ffvelrnd 6962 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
7877recnd 11003 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℂ)
7978adantr 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑥) ∈ ℂ)
8079mul01d 11174 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · 0) = 0)
8176, 80eqtrd 2778 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
8267, 73, 813eqtr4d 2788 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
83 simpl1 1190 . . . . . . . . . . . . 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 7292 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 · 𝑦) = (𝑥(.r𝑅)𝑦))
8786fveq2d 6778 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑥(.r𝑅)𝑦)))
88 simpl2 1191 . . . . . . . . . . . 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 2842 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥𝐵)
91 simprl 768 . . . . . . . . . . . 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 1192 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦 ∈ (Base‘𝑅))
9594, 89eleqtrrd 2842 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦𝐵)
96 simprr 770 . . . . . . . . . . . 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 1378 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10087, 99eqtr3d 2780 . . . . . . . . 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 7282 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = ((0g𝑅)(+g𝑅)𝑦))
103 ringgrp 19788 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
10449, 103syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
105 eqid 2738 . . . . . . . . . . . . . 14 (+g𝑅) = (+g𝑅)
10651, 105, 53grplid 18609 . . . . . . . . . . . . 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 2800 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑦)
109108fveq2d 6778 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑦))
1107, 58breqtrrid 5112 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → 0 ≤ (𝐹𝑥))
11161, 77addge02d 11564 . . . . . . . . . . . 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 5096 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
115 oveq2 7283 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑅)(0g𝑅)))
11651, 105, 53grprid 18610 . . . . . . . . . . . . 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 2800 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑥)
119118fveq2d 6778 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑥))
1207, 75breqtrrid 5112 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → 0 ≤ (𝐹𝑦))
12177, 61addge01d 11563 . . . . . . . . . . . 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 5096 . . . . . . . . 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 7292 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
128127fveq2d 6778 . . . . . . . . . 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 1378 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
131128, 130eqbrtrrd 5098 . . . . . . . . 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 1120 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑦 ∈ (Base‘𝑅) → ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
135134ralrimiv 3102 . . . . 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 3103 . . 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 2738 . . . . 5 (AbsVal‘𝑅) = (AbsVal‘𝑅)
139138, 51, 105, 52, 53isabv 20079 . . . 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 710 . 2 (𝜑𝐹 ∈ (AbsVal‘𝑅))
142 isabvd.a . 2 (𝜑𝐴 = (AbsVal‘𝑅))
143141, 142eleqtrrd 2842 1 (𝜑𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  wral 3064   class class class wbr 5074   Fn wfn 6428  wf 6429  cfv 6433  (class class class)co 7275  cc 10869  cr 10870  0cc0 10871   + caddc 10874   · cmul 10876  +∞cpnf 11006   < clt 11009  cle 11010  [,)cico 13081  Basecbs 16912  +gcplusg 16962  .rcmulr 16963  0gc0g 17150  Grpcgrp 18577  Ringcrg 19783  AbsValcabv 20076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-ico 13085  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-plusg 16975  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-mgp 19721  df-ring 19785  df-abv 20077
This theorem is referenced by:  abvres  20099  abvtrivd  20100  absabv  20655  abvcxp  26763  padicabv  26778
  Copyright terms: Public domain W3C validator