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Mirrors > Home > MPE Home > Th. List > Mathboxes > evenz | Structured version Visualization version GIF version |
Description: An even number is an integer. (Contributed by AV, 14-Jun-2020.) |
Ref | Expression |
---|---|
evenz | ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iseven 43800 | . 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ)) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 (class class class)co 7158 / cdiv 11299 2c2 11695 ℤcz 11984 Even ceven 43796 |
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 2795 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-ov 7161 df-even 43798 |
This theorem is referenced by: evenm1odd 43811 evenp1odd 43812 bits0eALTV 43852 opeoALTV 43856 omeoALTV 43858 epoo 43875 emoo 43876 epee 43877 emee 43878 evensumeven 43879 evenltle 43889 even3prm2 43891 mogoldbblem 43892 sbgoldbalt 43953 sgoldbeven3prm 43955 mogoldbb 43957 bgoldbachlt 43985 tgblthelfgott 43987 |
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