Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrextchr | Structured version Visualization version GIF version |
Description: The ring characteristic of an extension of ℝ is zero. (Contributed by Thierry Arnoux, 2-May-2018.) |
Ref | Expression |
---|---|
rrextchr | ⊢ (𝑅 ∈ ℝExt → (chr‘𝑅) = 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2821 | . . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
3 | eqid 2821 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 31241 | . . 3 ⊢ (𝑅 ∈ ℝExt ↔ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) ∧ ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0) ∧ (𝑅 ∈ CUnifSp ∧ (UnifSt‘𝑅) = (metUnif‘((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))))))) |
5 | 4 | simp2bi 1142 | . 2 ⊢ (𝑅 ∈ ℝExt → ((ℤMod‘𝑅) ∈ NrmMod ∧ (chr‘𝑅) = 0)) |
6 | 5 | simprd 498 | 1 ⊢ (𝑅 ∈ ℝExt → (chr‘𝑅) = 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 × cxp 5553 ↾ cres 5557 ‘cfv 6355 0cc0 10537 Basecbs 16483 distcds 16574 DivRingcdr 19502 metUnifcmetu 20536 ℤModczlm 20648 chrcchr 20649 UnifStcuss 22862 CUnifSpccusp 22906 NrmRingcnrg 23189 NrmModcnlm 23190 ℝExt crrext 31235 |
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 2793 |
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-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-res 5567 df-iota 6314 df-fv 6363 df-rrext 31240 |
This theorem is referenced by: rrhfe 31253 rrhcne 31254 rrhqima 31255 rrh0 31256 sitgclg 31600 |
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