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Mirrors > Home > MPE Home > Th. List > eluzelcn | Structured version Visualization version GIF version |
Description: A member of an upper set of integers is a complex number. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
Ref | Expression |
---|---|
eluzelcn | ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluzelre 11910 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) | |
2 | 1 | recnd 10280 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2139 ‘cfv 6049 ℂcc 10146 ℤ≥cuz 11899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-cnex 10204 ax-resscn 10205 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-fv 6057 df-ov 6817 df-neg 10481 df-z 11590 df-uz 11900 |
This theorem is referenced by: uzp1 11934 peano2uzr 11956 uzaddcl 11957 eluzgtdifelfzo 12744 fzosplitpr 12791 fldiv4lem1div2uz2 12851 mulp1mod1 12925 seqm1 13032 bcval5 13319 swrdfv2 13666 relexpaddg 14012 shftuz 14028 seqshft 14044 climshftlem 14524 climshft 14526 isumshft 14790 dvdsexp 15271 pclem 15765 efgtlen 18359 dvradcnv 24394 clwwlkext2edg 27207 clwwlknonex2lem1 27277 clwwlknonex2lem2 27278 clwwlknonex2 27279 extwwlkfablem1OLD 27518 2clwwlk2clwwlk 27528 numclwlk1lem2foalem 27531 numclwlk1lem2fo 27538 numclwwlk2 27563 numclwwlk2OLD 27570 nn0prpwlem 32644 rmspecsqrtnq 37990 rmspecsqrtnqOLD 37991 rmxm1 38019 rmym1 38020 rmxluc 38021 rmyluc 38022 rmyluc2 38023 jm2.17a 38047 relexpaddss 38530 trclfvdecomr 38540 binomcxplemnn0 39068 stoweidlem14 40752 fmtnorec3 41988 lighneallem4a 42053 lighneallem4b 42054 evengpop3 42214 evengpoap3 42215 nnsum4primeseven 42216 nnsum4primesevenALTV 42217 expnegico01 42836 dignn0ldlem 42924 dignnld 42925 digexp 42929 dig1 42930 nn0sumshdiglemB 42942 |
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