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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 41541 | . 2 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ)) | |
2 | 1 | simplbi 476 | 1 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 (class class class)co 6650 / cdiv 10684 2c2 11070 ℤcz 11377 Even ceven 41537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-even 41539 |
This theorem is referenced by: evenm1odd 41552 evenp1odd 41553 bits0eALTV 41591 opeoALTV 41595 omeoALTV 41597 epoo 41612 emoo 41613 epee 41614 emee 41615 evensumeven 41616 evenltle 41626 even3prm2 41628 mogoldbblem 41629 sbgoldbalt 41669 sgoldbeven3prm 41671 mogoldbb 41673 bgoldbachlt 41701 tgblthelfgott 41703 bgoldbachltOLD 41707 tgblthelfgottOLD 41709 |
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