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Mirrors > Home > MPE Home > Th. List > abvfge0 | Structured version Visualization version GIF version |
Description: An absolute value is a function from the ring to the nonnegative real numbers. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
abvf.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
abvf.b | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
abvfge0 | ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶(0[,)+∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abvf.a | . . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅) | |
2 | 1 | abvrcl 19521 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | abvf.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
4 | eqid 2818 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
5 | eqid 2818 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | eqid 2818 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 1, 3, 4, 5, 6 | isabv 19519 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))) |
8 | 2, 7 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ 𝐴 ↔ (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))))))) |
9 | 8 | ibi 268 | . 2 ⊢ (𝐹 ∈ 𝐴 → (𝐹:𝐵⟶(0[,)+∞) ∧ ∀𝑥 ∈ 𝐵 (((𝐹‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑅)) ∧ ∀𝑦 ∈ 𝐵 ((𝐹‘(𝑥(.r‘𝑅)𝑦)) = ((𝐹‘𝑥) · (𝐹‘𝑦)) ∧ (𝐹‘(𝑥(+g‘𝑅)𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦)))))) |
10 | 9 | simpld 495 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶(0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0cc0 10525 + caddc 10528 · cmul 10530 +∞cpnf 10660 ≤ cle 10664 [,)cico 12728 Basecbs 16471 +gcplusg 16553 .rcmulr 16554 0gc0g 16701 Ringcrg 19226 AbsValcabv 19516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-abv 19517 |
This theorem is referenced by: abvf 19523 abvge0 19525 |
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