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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfeven2 | Structured version Visualization version GIF version |
Description: Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.) |
Ref | Expression |
---|---|
dfeven2 | ⊢ Even = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfeven4 44706 | . 2 ⊢ Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)} | |
2 | eqcom 2743 | . . . . . 6 ⊢ (𝑧 = (2 · 𝑖) ↔ (2 · 𝑖) = 𝑧) | |
3 | 2cnd 11873 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 2 ∈ ℂ) | |
4 | zcn 12146 | . . . . . . . . 9 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ) | |
5 | 4 | adantl 485 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 ∈ ℂ) |
6 | 3, 5 | mulcomd 10819 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (2 · 𝑖) = (𝑖 · 2)) |
7 | 6 | eqeq1d 2738 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((2 · 𝑖) = 𝑧 ↔ (𝑖 · 2) = 𝑧)) |
8 | 2, 7 | syl5bb 286 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑧 = (2 · 𝑖) ↔ (𝑖 · 2) = 𝑧)) |
9 | 8 | rexbidva 3205 | . . . 4 ⊢ (𝑧 ∈ ℤ → (∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖) ↔ ∃𝑖 ∈ ℤ (𝑖 · 2) = 𝑧)) |
10 | 2z 12174 | . . . . 5 ⊢ 2 ∈ ℤ | |
11 | divides 15780 | . . . . 5 ⊢ ((2 ∈ ℤ ∧ 𝑧 ∈ ℤ) → (2 ∥ 𝑧 ↔ ∃𝑖 ∈ ℤ (𝑖 · 2) = 𝑧)) | |
12 | 10, 11 | mpan 690 | . . . 4 ⊢ (𝑧 ∈ ℤ → (2 ∥ 𝑧 ↔ ∃𝑖 ∈ ℤ (𝑖 · 2) = 𝑧)) |
13 | 9, 12 | bitr4d 285 | . . 3 ⊢ (𝑧 ∈ ℤ → (∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖) ↔ 2 ∥ 𝑧)) |
14 | 13 | rabbiia 3372 | . 2 ⊢ {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)} = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧} |
15 | 1, 14 | eqtri 2759 | 1 ⊢ Even = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 {crab 3055 class class class wbr 5039 (class class class)co 7191 ℂcc 10692 · cmul 10699 2c2 11850 ℤcz 12141 ∥ cdvds 15778 Even ceven 44692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-z 12142 df-dvds 15779 df-even 44694 |
This theorem is referenced by: iseven2 44719 |
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