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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 12890 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) | |
| 2 | 1 | recnd 11290 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℂ) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ‘cfv 6560 ℂcc 11154 ℤ≥cuz 12879 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-cnex 11212 ax-resscn 11213 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-ov 7435 df-neg 11496 df-z 12616 df-uz 12880 | 
| This theorem is referenced by: uzp1 12920 peano2uzr 12946 uzaddcl 12947 ge2halflem1 13151 eluzgtdifelfzo 13767 fzosplitpr 13816 fldiv4lem1div2uz2 13877 mulp1mod1 13953 seqm1 14061 bcval5 14358 swrdfv2 14700 relexpaddg 15093 shftuz 15109 seqshft 15125 climshftlem 15611 climshft 15613 isumshft 15876 dvdsexp 16366 pclem 16877 efgtlen 19745 dvradcnv 26465 logbgcd1irr 26838 clwwlkext2edg 30076 clwwlknonex2lem1 30127 clwwlknonex2lem2 30128 clwwlknonex2 30129 2clwwlk2clwwlk 30370 numclwwlk1lem2foalem 30371 numclwwlk1lem2fo 30378 numclwwlk2 30401 nn0prpwlem 36324 aks4d1p1p1 42065 fimgmcyc 42549 rmspecsqrtnq 42922 rmxm1 42951 rmym1 42952 rmxluc 42953 rmyluc 42954 rmyluc2 42955 jm2.17a 42977 relexpaddss 43736 trclfvdecomr 43746 binomcxplemnn0 44373 stoweidlem14 46034 fmtnorec3 47540 lighneallem4a 47600 lighneallem4b 47601 evengpop3 47790 evengpoap3 47791 nnsum4primeseven 47792 nnsum4primesevenALTV 47793 2tceilhalfelfzo1 48023 gpgedgvtx1 48025 expnegico01 48440 dignn0ldlem 48528 dignnld 48529 digexp 48533 dig1 48534 nn0sumshdiglemB 48546 | 
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