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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 19590 | . . 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 6779 | . 2 ⊢ (𝐹 ∈ (AbsVal‘𝑅) → 𝑅 ∈ Ring) |
3 | abvf.a | . 2 ⊢ 𝐴 = (AbsVal‘𝑅) | |
4 | 2, 3 | eleq2s 2933 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 ↑m cmap 8408 0cc0 10539 + caddc 10542 · cmul 10544 +∞cpnf 10674 ≤ cle 10678 [,)cico 12743 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 0gc0g 16715 Ringcrg 19299 AbsValcabv 19589 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fv 6365 df-abv 19590 |
This theorem is referenced by: abvfge0 19595 abveq0 19599 abvmul 19602 abvtri 19603 abv0 19604 abv1z 19605 abvneg 19607 abvsubtri 19608 abvpropd 19615 abvmet 23187 nrgring 23274 tngnrg 23285 abvcxp 26193 |
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