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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 20810 | . . 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 7025 | . 2 ⊢ (𝐹 ∈ (AbsVal‘𝑅) → 𝑅 ∈ Ring) |
| 3 | abvf.a | . 2 ⊢ 𝐴 = (AbsVal‘𝑅) | |
| 4 | 2, 3 | eleq2s 2859 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {crab 3436 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 0cc0 11155 + caddc 11158 · cmul 11160 +∞cpnf 11292 ≤ cle 11296 [,)cico 13389 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 0gc0g 17484 Ringcrg 20230 AbsValcabv 20809 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-abv 20810 |
| This theorem is referenced by: abvfge0 20815 abveq0 20819 abvmul 20822 abvtri 20823 abv0 20824 abv1z 20825 abvneg 20827 abvsubtri 20828 abvpropd 20836 abvmet 24588 nrgring 24684 tngnrg 24695 abvcxp 27659 |
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