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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2even | Structured version Visualization version GIF version |
Description: 2 is an even integer. (Contributed by AV, 12-Feb-2020.) |
Ref | Expression |
---|---|
2zrng.e | ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
Ref | Expression |
---|---|
2even | ⊢ 2 ∈ 𝐸 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12002 | . . 3 ⊢ 2 ∈ ℤ | |
2 | 2cn 11700 | . . . 4 ⊢ 2 ∈ ℂ | |
3 | 1zzd 12001 | . . . . 5 ⊢ (2 ∈ ℂ → 1 ∈ ℤ) | |
4 | oveq2 7153 | . . . . . . 7 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1)) | |
5 | 4 | eqeq2d 2829 | . . . . . 6 ⊢ (𝑥 = 1 → (2 = (2 · 𝑥) ↔ 2 = (2 · 1))) |
6 | 5 | adantl 482 | . . . . 5 ⊢ ((2 ∈ ℂ ∧ 𝑥 = 1) → (2 = (2 · 𝑥) ↔ 2 = (2 · 1))) |
7 | mulid1 10627 | . . . . . 6 ⊢ (2 ∈ ℂ → (2 · 1) = 2) | |
8 | 7 | eqcomd 2824 | . . . . 5 ⊢ (2 ∈ ℂ → 2 = (2 · 1)) |
9 | 3, 6, 8 | rspcedvd 3623 | . . . 4 ⊢ (2 ∈ ℂ → ∃𝑥 ∈ ℤ 2 = (2 · 𝑥)) |
10 | 2, 9 | ax-mp 5 | . . 3 ⊢ ∃𝑥 ∈ ℤ 2 = (2 · 𝑥) |
11 | eqeq1 2822 | . . . . 5 ⊢ (𝑧 = 2 → (𝑧 = (2 · 𝑥) ↔ 2 = (2 · 𝑥))) | |
12 | 11 | rexbidv 3294 | . . . 4 ⊢ (𝑧 = 2 → (∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥) ↔ ∃𝑥 ∈ ℤ 2 = (2 · 𝑥))) |
13 | 12 | elrab 3677 | . . 3 ⊢ (2 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ↔ (2 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 2 = (2 · 𝑥))) |
14 | 1, 10, 13 | mpbir2an 707 | . 2 ⊢ 2 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} |
15 | 2zrng.e | . 2 ⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} | |
16 | 14, 15 | eleqtrri 2909 | 1 ⊢ 2 ∈ 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 {crab 3139 (class class class)co 7145 ℂcc 10523 1c1 10526 · cmul 10530 2c2 11680 ℤcz 11969 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rrecex 10597 ax-cnre 10598 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-neg 10861 df-nn 11627 df-2 11688 df-z 11970 |
This theorem is referenced by: 2zrngnmlid 44148 |
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