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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 19586 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | abvf.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
4 | eqid 2821 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
5 | eqid 2821 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
6 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
7 | 1, 3, 4, 5, 6 | isabv 19584 | . . . 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 269 | . 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 497 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝐹:𝐵⟶(0[,)+∞)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5059 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 0cc0 10531 + caddc 10534 · cmul 10536 +∞cpnf 10666 ≤ cle 10670 [,)cico 12734 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 0gc0g 16707 Ringcrg 19291 AbsValcabv 19581 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-abv 19582 |
This theorem is referenced by: abvf 19588 abvge0 19590 |
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