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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 20923 | . . 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 6999 | . 2 ⊢ (𝐹 ∈ (AbsVal‘𝑅) → 𝑅 ∈ Ring) |
| 3 | abvf.a | . 2 ⊢ 𝐴 = (AbsVal‘𝑅) | |
| 4 | 2, 3 | eleq2s 2880 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1569 ∈ wcel 2142 ∀wral 3078 {crab 3415 class class class wbr 5108 ‘cfv 6536 (class class class)co 7412 ↑m cmap 8822 0cc0 11106 + caddc 11109 · cmul 11111 +∞cpnf 11246 ≤ cle 11250 [,)cico 13380 Basecbs 17275 +gcplusg 17316 .rcmulr 17317 0gc0g 17498 Ringcrg 20321 AbsValcabv 20922 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pr 5403 |
| This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-rab 3416 df-v 3456 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-xp 5666 df-rel 5667 df-cnv 5668 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fv 6544 df-abv 20923 |
| This theorem is used by: abvfge0 20928 abveq0 20932 abvmul 20935 abvtri 20936 abv0 20937 abv1z 20938 abvneg 20940 abvsubtri 20941 abvpropd 20949 abvmet 24743 nrgring 24831 tngnrg 24842 abvcxp 27790 |
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