| Mathbox for Alexander van der Vekens |
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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 12624 | . . 3 ⊢ 0 ∈ ℤ | |
| 2 | 2cn 12341 | . . . 4 ⊢ 2 ∈ ℂ | |
| 3 | 0zd 12625 | . . . . 5 ⊢ (2 ∈ ℂ → 0 ∈ ℤ) | |
| 4 | oveq2 7439 | . . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0)) | |
| 5 | 4 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑥 = 0 → (0 = (2 · 𝑥) ↔ 0 = (2 · 0))) |
| 6 | 5 | adantl 481 | . . . . 5 ⊢ ((2 ∈ ℂ ∧ 𝑥 = 0) → (0 = (2 · 𝑥) ↔ 0 = (2 · 0))) |
| 7 | mul01 11440 | . . . . . 6 ⊢ (2 ∈ ℂ → (2 · 0) = 0) | |
| 8 | 7 | eqcomd 2743 | . . . . 5 ⊢ (2 ∈ ℂ → 0 = (2 · 0)) |
| 9 | 3, 6, 8 | rspcedvd 3624 | . . . 4 ⊢ (2 ∈ ℂ → ∃𝑥 ∈ ℤ 0 = (2 · 𝑥)) |
| 10 | 2, 9 | ax-mp 5 | . . 3 ⊢ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥) |
| 11 | eqeq1 2741 | . . . . 5 ⊢ (𝑧 = 0 → (𝑧 = (2 · 𝑥) ↔ 0 = (2 · 𝑥))) | |
| 12 | 11 | rexbidv 3179 | . . . 4 ⊢ (𝑧 = 0 → (∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥) ↔ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥))) |
| 13 | 12 | elrab 3692 | . . 3 ⊢ (0 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ↔ (0 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥))) |
| 14 | 1, 10, 13 | mpbir2an 711 | . 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 2840 | 1 ⊢ 0 ∈ 𝐸 |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 {crab 3436 (class class class)co 7431 ℂcc 11153 0cc0 11155 · cmul 11160 2c2 12321 ℤcz 12613 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-neg 11495 df-2 12329 df-z 12614 |
| This theorem is referenced by: 2zlidl 48156 2zrng0 48160 2zrngamnd 48163 2zrngacmnd 48164 2zrngmmgm 48168 |
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