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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 18640 | . . . 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 5548 | . . 3 ⊢ dom AbsVal ⊆ Ring |
3 | elfvdm 6130 | . . 3 ⊢ (𝐹 ∈ (AbsVal‘𝑅) → 𝑅 ∈ dom AbsVal) | |
4 | 2, 3 | sseldi 3566 | . 2 ⊢ (𝐹 ∈ (AbsVal‘𝑅) → 𝑅 ∈ Ring) |
5 | abvf.a | . 2 ⊢ 𝐴 = (AbsVal‘𝑅) | |
6 | 4, 5 | eleq2s 2706 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 class class class wbr 4583 dom cdm 5038 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 0cc0 9815 + caddc 9818 · cmul 9820 +∞cpnf 9950 ≤ cle 9954 [,)cico 12048 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 0gc0g 15923 Ringcrg 18370 AbsValcabv 18639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fv 5812 df-abv 18640 |
This theorem is referenced by: abvfge0 18645 abveq0 18649 abvmul 18652 abvtri 18653 abv0 18654 abv1z 18655 abvneg 18657 abvsubtri 18658 abvpropd 18665 abvmet 22190 nrgring 22277 tngnrg 22288 abvcxp 25104 |
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