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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfeven4 | Structured version Visualization version GIF version |
Description: Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.) |
Ref | Expression |
---|---|
dfeven4 | ⊢ Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-even 42041 | . 2 ⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ} | |
2 | simpr 479 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 / 2) ∈ ℤ) → (𝑧 / 2) ∈ ℤ) | |
3 | oveq2 6813 | . . . . . . . 8 ⊢ (𝑖 = (𝑧 / 2) → (2 · 𝑖) = (2 · (𝑧 / 2))) | |
4 | 3 | eqeq2d 2762 | . . . . . . 7 ⊢ (𝑖 = (𝑧 / 2) → (𝑧 = (2 · 𝑖) ↔ 𝑧 = (2 · (𝑧 / 2)))) |
5 | 4 | adantl 473 | . . . . . 6 ⊢ (((𝑧 ∈ ℤ ∧ (𝑧 / 2) ∈ ℤ) ∧ 𝑖 = (𝑧 / 2)) → (𝑧 = (2 · 𝑖) ↔ 𝑧 = (2 · (𝑧 / 2)))) |
6 | zcn 11566 | . . . . . . . . 9 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ) | |
7 | 6 | adantr 472 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 / 2) ∈ ℤ) → 𝑧 ∈ ℂ) |
8 | 2cnd 11277 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 / 2) ∈ ℤ) → 2 ∈ ℂ) | |
9 | 2ne0 11297 | . . . . . . . . 9 ⊢ 2 ≠ 0 | |
10 | 9 | a1i 11 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 / 2) ∈ ℤ) → 2 ≠ 0) |
11 | 7, 8, 10 | divcan2d 10987 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 / 2) ∈ ℤ) → (2 · (𝑧 / 2)) = 𝑧) |
12 | 11 | eqcomd 2758 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 / 2) ∈ ℤ) → 𝑧 = (2 · (𝑧 / 2))) |
13 | 2, 5, 12 | rspcedvd 3448 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ (𝑧 / 2) ∈ ℤ) → ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)) |
14 | 13 | ex 449 | . . . 4 ⊢ (𝑧 ∈ ℤ → ((𝑧 / 2) ∈ ℤ → ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖))) |
15 | oveq1 6812 | . . . . . . . 8 ⊢ (𝑧 = (2 · 𝑖) → (𝑧 / 2) = ((2 · 𝑖) / 2)) | |
16 | zcn 11566 | . . . . . . . . . 10 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℂ) | |
17 | 16 | adantl 473 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 ∈ ℂ) |
18 | 2cnd 11277 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 2 ∈ ℂ) | |
19 | 9 | a1i 11 | . . . . . . . . 9 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 2 ≠ 0) |
20 | 17, 18, 19 | divcan3d 10990 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → ((2 · 𝑖) / 2) = 𝑖) |
21 | 15, 20 | sylan9eqr 2808 | . . . . . . 7 ⊢ (((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝑧 = (2 · 𝑖)) → (𝑧 / 2) = 𝑖) |
22 | simpr 479 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → 𝑖 ∈ ℤ) | |
23 | 22 | adantr 472 | . . . . . . 7 ⊢ (((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝑧 = (2 · 𝑖)) → 𝑖 ∈ ℤ) |
24 | 21, 23 | eqeltrd 2831 | . . . . . 6 ⊢ (((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝑧 = (2 · 𝑖)) → (𝑧 / 2) ∈ ℤ) |
25 | 24 | ex 449 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (𝑧 = (2 · 𝑖) → (𝑧 / 2) ∈ ℤ)) |
26 | 25 | rexlimdva 3161 | . . . 4 ⊢ (𝑧 ∈ ℤ → (∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖) → (𝑧 / 2) ∈ ℤ)) |
27 | 14, 26 | impbid 202 | . . 3 ⊢ (𝑧 ∈ ℤ → ((𝑧 / 2) ∈ ℤ ↔ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖))) |
28 | 27 | rabbiia 3316 | . 2 ⊢ {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ} = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)} |
29 | 1, 28 | eqtri 2774 | 1 ⊢ Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 383 = wceq 1624 ∈ wcel 2131 ≠ wne 2924 ∃wrex 3043 {crab 3046 (class class class)co 6805 ℂcc 10118 0cc0 10120 · cmul 10125 / cdiv 10868 2c2 11254 ℤcz 11561 Even ceven 42039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-br 4797 df-opab 4857 df-mpt 4874 df-id 5166 df-po 5179 df-so 5180 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-er 7903 df-en 8114 df-dom 8115 df-sdom 8116 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-2 11263 df-z 11562 df-even 42041 |
This theorem is referenced by: m1expevenALTV 42062 dfeven2 42064 opoeALTV 42096 opeoALTV 42097 |
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