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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrextnlm | Structured version Visualization version GIF version |
Description: The norm of an extension of ℝ is absolutely homogeneous. (Contributed by Thierry Arnoux, 2-May-2018.) |
Ref | Expression |
---|---|
rrextnlm.z | ⊢ 𝑍 = (ℤMod‘𝑅) |
Ref | Expression |
---|---|
rrextnlm | ⊢ (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2740 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2740 | . . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
3 | rrextnlm.z | . . . 4 ⊢ 𝑍 = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 31946 | . . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))) |
5 | 4 | simp2bi 1145 | . 2 ⊢ (𝑅 ∈ ℝExt → (𝑍 ∈ NrmMod ∧ (chr‘𝑅) = 0)) |
6 | 5 | simpld 495 | 1 ⊢ (𝑅 ∈ ℝExt → 𝑍 ∈ NrmMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 × cxp 5588 ↾ cres 5592 ‘cfv 6432 0cc0 10872 Basecbs 16910 distcds 16969 DivRingcdr 19989 metUnifcmetu 20586 ℤModczlm 20700 chrcchr 20701 UnifStcuss 23403 CUnifSpccusp 23447 NrmRingcnrg 23733 NrmModcnlm 23734 ℝExt crrext 31940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-ext 2711 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2072 df-clab 2718 df-cleq 2732 df-clel 2818 df-rab 3075 df-v 3433 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-xp 5596 df-res 5602 df-iota 6390 df-fv 6440 df-rrext 31945 |
This theorem is referenced by: rrhfe 31958 rrhcne 31959 rrhqima 31960 sitgclg 32305 |
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