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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 12566 | . . 3 ⊢ 0 ∈ ℤ | |
2 | 2cn 12284 | . . . 4 ⊢ 2 ∈ ℂ | |
3 | 0zd 12567 | . . . . 5 ⊢ (2 ∈ ℂ → 0 ∈ ℤ) | |
4 | oveq2 7414 | . . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0)) | |
5 | 4 | eqeq2d 2744 | . . . . . 6 ⊢ (𝑥 = 0 → (0 = (2 · 𝑥) ↔ 0 = (2 · 0))) |
6 | 5 | adantl 483 | . . . . 5 ⊢ ((2 ∈ ℂ ∧ 𝑥 = 0) → (0 = (2 · 𝑥) ↔ 0 = (2 · 0))) |
7 | mul01 11390 | . . . . . 6 ⊢ (2 ∈ ℂ → (2 · 0) = 0) | |
8 | 7 | eqcomd 2739 | . . . . 5 ⊢ (2 ∈ ℂ → 0 = (2 · 0)) |
9 | 3, 6, 8 | rspcedvd 3615 | . . . 4 ⊢ (2 ∈ ℂ → ∃𝑥 ∈ ℤ 0 = (2 · 𝑥)) |
10 | 2, 9 | ax-mp 5 | . . 3 ⊢ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥) |
11 | eqeq1 2737 | . . . . 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 3683 | . . 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 2833 | 1 ⊢ 0 ∈ 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 {crab 3433 (class class class)co 7406 ℂcc 11105 0cc0 11107 · cmul 11112 2c2 12264 ℤcz 12555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-ltxr 11250 df-neg 11444 df-2 12272 df-z 12556 |
This theorem is referenced by: 2zlidl 46786 2zrng0 46790 2zrngamnd 46793 2zrngacmnd 46794 2zrngmmgm 46798 |
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