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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 43813 | . 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 7156 / cdiv 11297 2c2 11693 ℤcz 11982 Even ceven 43809 |
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 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-even 43811 |
This theorem is referenced by: evenm1odd 43824 evenp1odd 43825 bits0eALTV 43865 opeoALTV 43869 omeoALTV 43871 epoo 43888 emoo 43889 epee 43890 emee 43891 evensumeven 43892 evenltle 43902 even3prm2 43904 mogoldbblem 43905 sbgoldbalt 43966 sgoldbeven3prm 43968 mogoldbb 43970 bgoldbachlt 43998 tgblthelfgott 44000 |
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