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Mirrors > Home > MPE Home > Th. List > abvrcl | Structured version Visualization version GIF version |
Description: Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.) |
Ref | Expression |
---|---|
abvf.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
Ref | Expression |
---|---|
abvrcl | ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-abv 20709 | . . 3 ⊢ AbsVal = (𝑟 ∈ Ring ↦ {𝑓 ∈ ((0[,)+∞) ↑m (Base‘𝑟)) ∣ ∀𝑥 ∈ (Base‘𝑟)(((𝑓‘𝑥) = 0 ↔ 𝑥 = (0g‘𝑟)) ∧ ∀𝑦 ∈ (Base‘𝑟)((𝑓‘(𝑥(.r‘𝑟)𝑦)) = ((𝑓‘𝑥) · (𝑓‘𝑦)) ∧ (𝑓‘(𝑥(+g‘𝑟)𝑦)) ≤ ((𝑓‘𝑥) + (𝑓‘𝑦))))}) | |
2 | 1 | mptrcl 7013 | . 2 ⊢ (𝐹 ∈ (AbsVal‘𝑅) → 𝑅 ∈ Ring) |
3 | abvf.a | . 2 ⊢ 𝐴 = (AbsVal‘𝑅) | |
4 | 2, 3 | eleq2s 2843 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 {crab 3418 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 0cc0 11140 + caddc 11143 · cmul 11145 +∞cpnf 11277 ≤ cle 11281 [,)cico 13361 Basecbs 17183 +gcplusg 17236 .rcmulr 17237 0gc0g 17424 Ringcrg 20185 AbsValcabv 20708 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-xp 5684 df-rel 5685 df-cnv 5686 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fv 6557 df-abv 20709 |
This theorem is referenced by: abvfge0 20714 abveq0 20718 abvmul 20721 abvtri 20722 abv0 20723 abv1z 20724 abvneg 20726 abvsubtri 20727 abvpropd 20734 abvmet 24528 nrgring 24624 tngnrg 24635 abvcxp 27593 |
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