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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 19027 | . . . 4 ⊢ 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 | dmmptss 5774 | . . 3 ⊢ dom AbsVal ⊆ Ring |
3 | elfvdm 6363 | . . 3 ⊢ (𝐹 ∈ (AbsVal‘𝑅) → 𝑅 ∈ dom AbsVal) | |
4 | 2, 3 | sseldi 3750 | . 2 ⊢ (𝐹 ∈ (AbsVal‘𝑅) → 𝑅 ∈ Ring) |
5 | abvf.a | . 2 ⊢ 𝐴 = (AbsVal‘𝑅) | |
6 | 4, 5 | eleq2s 2868 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 {crab 3065 class class class wbr 4787 dom cdm 5250 ‘cfv 6030 (class class class)co 6796 ↑𝑚 cmap 8013 0cc0 10142 + caddc 10145 · cmul 10147 +∞cpnf 10277 ≤ cle 10281 [,)cico 12382 Basecbs 16064 +gcplusg 16149 .rcmulr 16150 0gc0g 16308 Ringcrg 18755 AbsValcabv 19026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-xp 5256 df-rel 5257 df-cnv 5258 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fv 6038 df-abv 19027 |
This theorem is referenced by: abvfge0 19032 abveq0 19036 abvmul 19039 abvtri 19040 abv0 19041 abv1z 19042 abvneg 19044 abvsubtri 19045 abvpropd 19052 abvmet 22600 nrgring 22687 tngnrg 22698 abvcxp 25525 |
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