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 2819 | . . . 4 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2819 | . . . 4 ⊢ ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) = ((dist‘𝑅) ↾ ((Base‘𝑅) × (Base‘𝑅))) | |
3 | eqid 2819 | . . . 4 ⊢ (ℤMod‘𝑅) = (ℤMod‘𝑅) | |
4 | 1, 2, 3 | isrrext 31234 | . . 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 1141 | . 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 1531 ∈ wcel 2108 × cxp 5546 ↾ cres 5550 ‘cfv 6348 0cc0 10529 Basecbs 16475 distcds 16566 DivRingcdr 19494 metUnifcmetu 20528 ℤModczlm 20640 chrcchr 20641 UnifStcuss 22854 CUnifSpccusp 22898 NrmRingcnrg 23181 NrmModcnlm 23182 ℝExt crrext 31228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-rex 3142 df-rab 3145 df-v 3495 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-res 5560 df-iota 6307 df-fv 6356 df-rrext 31233 |
This theorem is referenced by: rrhfe 31246 rrhcne 31247 rrhqima 31248 rrh0 31249 sitgclg 31593 |
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