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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 12255 | . 2 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℝ) | |
2 | 1 | recnd 10669 | 1 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ‘cfv 6355 ℂcc 10535 ℤ≥cuz 12244 |
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 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-neg 10873 df-z 11983 df-uz 12245 |
This theorem is referenced by: uzp1 12280 peano2uzr 12304 uzaddcl 12305 eluzgtdifelfzo 13100 fzosplitpr 13147 fldiv4lem1div2uz2 13207 mulp1mod1 13281 seqm1 13388 bcval5 13679 swrdfv2 14023 relexpaddg 14412 shftuz 14428 seqshft 14444 climshftlem 14931 climshft 14933 isumshft 15194 dvdsexp 15677 pclem 16175 efgtlen 18852 dvradcnv 25009 logbgcd1irr 25372 clwwlkext2edg 27835 clwwlknonex2lem1 27886 clwwlknonex2lem2 27887 clwwlknonex2 27888 2clwwlk2clwwlk 28129 numclwwlk1lem2foalem 28130 numclwwlk1lem2fo 28137 numclwwlk2 28160 nn0prpwlem 33670 rmspecsqrtnq 39523 rmxm1 39551 rmym1 39552 rmxluc 39553 rmyluc 39554 rmyluc2 39555 jm2.17a 39577 relexpaddss 40083 trclfvdecomr 40093 binomcxplemnn0 40701 stoweidlem14 42319 fmtnorec3 43730 lighneallem4a 43793 lighneallem4b 43794 evengpop3 43983 evengpoap3 43984 nnsum4primeseven 43985 nnsum4primesevenALTV 43986 expnegico01 44593 dignn0ldlem 44682 dignnld 44683 digexp 44687 dig1 44688 nn0sumshdiglemB 44700 |
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