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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0even | Structured version Visualization version GIF version |
Description: 0 is an even integer. (Contributed by AV, 11-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
Ref | Expression |
---|---|
0even | ⊢ 0 ∈ 𝐸 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 11980 | . . 3 ⊢ 0 ∈ ℤ | |
2 | 2cn 11700 | . . . 4 ⊢ 2 ∈ ℂ | |
3 | 0zd 11981 | . . . . 5 ⊢ (2 ∈ ℂ → 0 ∈ ℤ) | |
4 | oveq2 7143 | . . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0)) | |
5 | 4 | eqeq2d 2809 | . . . . . 6 ⊢ (𝑥 = 0 → (0 = (2 · 𝑥) ↔ 0 = (2 · 0))) |
6 | 5 | adantl 485 | . . . . 5 ⊢ ((2 ∈ ℂ ∧ 𝑥 = 0) → (0 = (2 · 𝑥) ↔ 0 = (2 · 0))) |
7 | mul01 10808 | . . . . . 6 ⊢ (2 ∈ ℂ → (2 · 0) = 0) | |
8 | 7 | eqcomd 2804 | . . . . 5 ⊢ (2 ∈ ℂ → 0 = (2 · 0)) |
9 | 3, 6, 8 | rspcedvd 3574 | . . . 4 ⊢ (2 ∈ ℂ → ∃𝑥 ∈ ℤ 0 = (2 · 𝑥)) |
10 | 2, 9 | ax-mp 5 | . . 3 ⊢ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥) |
11 | eqeq1 2802 | . . . . 5 ⊢ (𝑧 = 0 → (𝑧 = (2 · 𝑥) ↔ 0 = (2 · 𝑥))) | |
12 | 11 | rexbidv 3256 | . . . 4 ⊢ (𝑧 = 0 → (∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥) ↔ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥))) |
13 | 12 | elrab 3628 | . . 3 ⊢ (0 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ↔ (0 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥))) |
14 | 1, 10, 13 | mpbir2an 710 | . 2 ⊢ 0 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
15 | 2zrng.e | . 2 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
16 | 14, 15 | eleqtrri 2889 | 1 ⊢ 0 ∈ 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 (class class class)co 7135 ℂcc 10524 0cc0 10526 · cmul 10531 2c2 11680 ℤcz 11969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-neg 10862 df-2 11688 df-z 11970 |
This theorem is referenced by: 2zlidl 44558 2zrng0 44562 2zrngamnd 44565 2zrngacmnd 44566 2zrngmmgm 44570 |
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