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Theorem isabvd 18643
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 5944 . . . . . 6 (𝜑 → (𝐹:𝐵⟶ℝ ↔ 𝐹:(Base‘𝑅)⟶ℝ))
41, 3mpbid 221 . . . . 5 (𝜑𝐹:(Base‘𝑅)⟶ℝ)
5 ffn 5958 . . . . 5 (𝐹:(Base‘𝑅)⟶ℝ → 𝐹 Fn (Base‘𝑅))
64, 5syl 17 . . . 4 (𝜑𝐹 Fn (Base‘𝑅))
74ffvelrnda 6267 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
8 0le0 10987 . . . . . . . . . 10 0 ≤ 0
9 isabvd.z . . . . . . . . . . . 12 (𝜑0 = (0g𝑅))
109fveq2d 6107 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = (𝐹‘(0g𝑅)))
11 isabvd.3 . . . . . . . . . . 11 (𝜑 → (𝐹0 ) = 0)
1210, 11eqtr3d 2646 . . . . . . . . . 10 (𝜑 → (𝐹‘(0g𝑅)) = 0)
138, 12syl5breqr 4621 . . . . . . . . 9 (𝜑 → 0 ≤ (𝐹‘(0g𝑅)))
1413adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹‘(0g𝑅)))
15 fveq2 6103 . . . . . . . . 9 (𝑥 = (0g𝑅) → (𝐹𝑥) = (𝐹‘(0g𝑅)))
1615breq2d 4595 . . . . . . . 8 (𝑥 = (0g𝑅) → (0 ≤ (𝐹𝑥) ↔ 0 ≤ (𝐹‘(0g𝑅))))
1714, 16syl5ibrcom 236 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g𝑅) → 0 ≤ (𝐹𝑥)))
18 simp1 1054 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝜑)
19 simp2 1055 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ∈ (Base‘𝑅))
2023ad2ant1 1075 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝐵 = (Base‘𝑅))
2119, 20eleqtrrd 2691 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥𝐵)
22 simp3 1056 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥 ≠ (0g𝑅))
2393ad2ant1 1075 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 = (0g𝑅))
2422, 23neeqtrrd 2856 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 𝑥0 )
25 isabvd.4 . . . . . . . . . 10 ((𝜑𝑥𝐵𝑥0 ) → 0 < (𝐹𝑥))
2618, 21, 24, 25syl3anc 1318 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 < (𝐹𝑥))
27 0re 9919 . . . . . . . . . 10 0 ∈ ℝ
2873adant3 1074 . . . . . . . . . 10 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ∈ ℝ)
29 ltle 10005 . . . . . . . . . 10 ((0 ∈ ℝ ∧ (𝐹𝑥) ∈ ℝ) → (0 < (𝐹𝑥) → 0 ≤ (𝐹𝑥)))
3027, 28, 29sylancr 694 . . . . . . . . 9 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (0 < (𝐹𝑥) → 0 ≤ (𝐹𝑥)))
3126, 30mpd 15 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → 0 ≤ (𝐹𝑥))
32313expia 1259 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g𝑅) → 0 ≤ (𝐹𝑥)))
3317, 32pm2.61dne 2868 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → 0 ≤ (𝐹𝑥))
34 elrege0 12149 . . . . . 6 ((𝐹𝑥) ∈ (0[,)+∞) ↔ ((𝐹𝑥) ∈ ℝ ∧ 0 ≤ (𝐹𝑥)))
357, 33, 34sylanbrc 695 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ (0[,)+∞))
3635ralrimiva 2949 . . . 4 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞))
37 ffnfv 6295 . . . 4 (𝐹:(Base‘𝑅)⟶(0[,)+∞) ↔ (𝐹 Fn (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)(𝐹𝑥) ∈ (0[,)+∞)))
386, 36, 37sylanbrc 695 . . 3 (𝜑𝐹:(Base‘𝑅)⟶(0[,)+∞))
3926gt0ne0d 10471 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑥 ≠ (0g𝑅)) → (𝐹𝑥) ≠ 0)
40393expia 1259 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 ≠ (0g𝑅) → (𝐹𝑥) ≠ 0))
4140necon4d 2806 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((𝐹𝑥) = 0 → 𝑥 = (0g𝑅)))
4212adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝐹‘(0g𝑅)) = 0)
4315eqeq1d 2612 . . . . . . 7 (𝑥 = (0g𝑅) → ((𝐹𝑥) = 0 ↔ (𝐹‘(0g𝑅)) = 0))
4442, 43syl5ibrcom 236 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑥 = (0g𝑅) → (𝐹𝑥) = 0))
4541, 44impbid 201 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → ((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)))
46123ad2ant1 1075 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(0g𝑅)) = 0)
4746adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(0g𝑅)) = 0)
48 oveq1 6556 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = ((0g𝑅)(.r𝑅)𝑦))
49 isabvd.1 . . . . . . . . . . . . . 14 (𝜑𝑅 ∈ Ring)
50493ad2ant1 1075 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Ring)
51 simp3 1056 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
52 eqid 2610 . . . . . . . . . . . . . 14 (Base‘𝑅) = (Base‘𝑅)
53 eqid 2610 . . . . . . . . . . . . . 14 (.r𝑅) = (.r𝑅)
54 eqid 2610 . . . . . . . . . . . . . 14 (0g𝑅) = (0g𝑅)
5552, 53, 54ringlz 18410 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)𝑦) = (0g𝑅))
5650, 51, 55syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)𝑦) = (0g𝑅))
5748, 56sylan9eqr 2666 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
5857fveq2d 6107 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
5915, 46sylan9eqr 2666 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑥) = 0)
6059oveq1d 6564 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = (0 · (𝐹𝑦)))
6143ad2ant1 1075 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝐹:(Base‘𝑅)⟶ℝ)
6261, 51ffvelrnd 6268 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℝ)
6362recnd 9947 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑦) ∈ ℂ)
6463adantr 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑦) ∈ ℂ)
6564mul02d 10113 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (0 · (𝐹𝑦)) = 0)
6660, 65eqtrd 2644 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
6747, 58, 663eqtr4d 2654 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
6846adantr 480 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(0g𝑅)) = 0)
69 oveq2 6557 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(.r𝑅)𝑦) = (𝑥(.r𝑅)(0g𝑅)))
70 simp2 1055 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
7152, 53, 54ringrz 18411 . . . . . . . . . . . . 13 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)(0g𝑅)) = (0g𝑅))
7250, 70, 71syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑅)(0g𝑅)) = (0g𝑅))
7369, 72sylan9eqr 2666 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(.r𝑅)𝑦) = (0g𝑅))
7473fveq2d 6107 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = (𝐹‘(0g𝑅)))
75 fveq2 6103 . . . . . . . . . . . . 13 (𝑦 = (0g𝑅) → (𝐹𝑦) = (𝐹‘(0g𝑅)))
7675, 46sylan9eqr 2666 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑦) = 0)
7776oveq2d 6565 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = ((𝐹𝑥) · 0))
7861, 70ffvelrnd 6268 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℝ)
7978recnd 9947 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹𝑥) ∈ ℂ)
8079adantr 480 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑥) ∈ ℂ)
8180mul01d 10114 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · 0) = 0)
8277, 81eqtrd 2644 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → ((𝐹𝑥) · (𝐹𝑦)) = 0)
8368, 74, 823eqtr4d 2654 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
84 simpl1 1057 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝜑)
85 isabvd.t . . . . . . . . . . . . 13 (𝜑· = (.r𝑅))
8684, 85syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → · = (.r𝑅))
8786oveqd 6566 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 · 𝑦) = (𝑥(.r𝑅)𝑦))
8887fveq2d 6107 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = (𝐹‘(𝑥(.r𝑅)𝑦)))
89 simpl2 1058 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥 ∈ (Base‘𝑅))
9084, 2syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝐵 = (Base‘𝑅))
9189, 90eleqtrrd 2691 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥𝐵)
92 simprl 790 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥 ≠ (0g𝑅))
9384, 9syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 0 = (0g𝑅))
9492, 93neeqtrrd 2856 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑥0 )
95 simpl3 1059 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦 ∈ (Base‘𝑅))
9695, 90eleqtrrd 2691 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦𝐵)
97 simprr 792 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦 ≠ (0g𝑅))
9897, 93neeqtrrd 2856 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → 𝑦0 )
99 isabvd.5 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10084, 91, 94, 96, 98, 99syl122anc 1327 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10188, 100eqtr3d 2646 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
10267, 83, 101pm2.61da2ne 2870 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)))
103 oveq1 6556 . . . . . . . . . . . 12 (𝑥 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = ((0g𝑅)(+g𝑅)𝑦))
104 ringgrp 18375 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑅 ∈ Grp)
10550, 104syl 17 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅 ∈ Grp)
106 eqid 2610 . . . . . . . . . . . . . 14 (+g𝑅) = (+g𝑅)
10752, 106, 54grplid 17275 . . . . . . . . . . . . 13 ((𝑅 ∈ Grp ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g𝑅)(+g𝑅)𝑦) = 𝑦)
108105, 51, 107syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((0g𝑅)(+g𝑅)𝑦) = 𝑦)
109103, 108sylan9eqr 2666 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑦)
110109fveq2d 6107 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑦))
1118, 59syl5breqr 4621 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → 0 ≤ (𝐹𝑥))
11262, 78addge02d 10495 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦))))
113112adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (0 ≤ (𝐹𝑥) ↔ (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦))))
114111, 113mpbid 221 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹𝑦) ≤ ((𝐹𝑥) + (𝐹𝑦)))
115110, 114eqbrtrd 4605 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑥 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
116 oveq2 6557 . . . . . . . . . . . 12 (𝑦 = (0g𝑅) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑅)(0g𝑅)))
11752, 106, 54grprid 17276 . . . . . . . . . . . . 13 ((𝑅 ∈ Grp ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)(0g𝑅)) = 𝑥)
118105, 70, 117syl2anc 691 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(+g𝑅)(0g𝑅)) = 𝑥)
119116, 118sylan9eqr 2666 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝑥(+g𝑅)𝑦) = 𝑥)
120119fveq2d 6107 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) = (𝐹𝑥))
1218, 76syl5breqr 4621 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → 0 ≤ (𝐹𝑦))
12278, 62addge01d 10494 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (0 ≤ (𝐹𝑦) ↔ (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦))))
123122adantr 480 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (0 ≤ (𝐹𝑦) ↔ (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦))))
124121, 123mpbid 221 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹𝑥) ≤ ((𝐹𝑥) + (𝐹𝑦)))
125120, 124eqbrtrd 4605 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ 𝑦 = (0g𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
126 isabvd.p . . . . . . . . . . . . 13 (𝜑+ = (+g𝑅))
12784, 126syl 17 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → + = (+g𝑅))
128127oveqd 6566 . . . . . . . . . . 11 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝑥 + 𝑦) = (𝑥(+g𝑅)𝑦))
129128fveq2d 6107 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝑥(+g𝑅)𝑦)))
130 isabvd.6 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
13184, 91, 94, 96, 98, 130syl122anc 1327 . . . . . . . . . 10 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥 + 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
132129, 131eqbrtrrd 4607 . . . . . . . . 9 (((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) ∧ (𝑥 ≠ (0g𝑅) ∧ 𝑦 ≠ (0g𝑅))) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
133115, 125, 132pm2.61da2ne 2870 . . . . . . . 8 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))
134102, 133jca 553 . . . . . . 7 ((𝜑𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))
1351343expia 1259 . . . . . 6 ((𝜑𝑥 ∈ (Base‘𝑅)) → (𝑦 ∈ (Base‘𝑅) → ((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
136135ralrimiv 2948 . . . . 5 ((𝜑𝑥 ∈ (Base‘𝑅)) → ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦))))
13745, 136jca 553 . . . 4 ((𝜑𝑥 ∈ (Base‘𝑅)) → (((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
138137ralrimiva 2949 . . 3 (𝜑 → ∀𝑥 ∈ (Base‘𝑅)(((𝐹𝑥) = 0 ↔ 𝑥 = (0g𝑅)) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝐹‘(𝑥(.r𝑅)𝑦)) = ((𝐹𝑥) · (𝐹𝑦)) ∧ (𝐹‘(𝑥(+g𝑅)𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))))
139 eqid 2610 . . . . 5 (AbsVal‘𝑅) = (AbsVal‘𝑅)
140139, 52, 106, 53, 54isabv 18642 . . . 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)))))))
14149, 140syl 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)))))))
14238, 138, 141mpbir2and 959 . 2 (𝜑𝐹 ∈ (AbsVal‘𝑅))
143 isabvd.a . 2 (𝜑𝐴 = (AbsVal‘𝑅))
144142, 143eleqtrrd 2691 1 (𝜑𝐹𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896   class class class wbr 4583   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  cc 9813  cr 9814  0cc0 9815   + caddc 9818   · cmul 9820  +∞cpnf 9950   < clt 9953  cle 9954  [,)cico 12048  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  0gc0g 15923  Grpcgrp 17245  Ringcrg 18370  AbsValcabv 18639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ico 12052  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mgp 18313  df-ring 18372  df-abv 18640
This theorem is referenced by:  abvres  18662  abvtrivd  18663  absabv  19622  abvcxp  25104  padicabv  25119
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