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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 11265 | . . 3 ⊢ 0 ∈ ℤ | |
2 | 2cn 10968 | . . . 4 ⊢ 2 ∈ ℂ | |
3 | 0zd 11266 | . . . . 5 ⊢ (2 ∈ ℂ → 0 ∈ ℤ) | |
4 | oveq2 6557 | . . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0)) | |
5 | 4 | eqeq2d 2620 | . . . . . 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 10094 | . . . . . 6 ⊢ (2 ∈ ℂ → (2 · 0) = 0) | |
8 | 7 | eqcomd 2616 | . . . . 5 ⊢ (2 ∈ ℂ → 0 = (2 · 0)) |
9 | 3, 6, 8 | rspcedvd 3289 | . . . 4 ⊢ (2 ∈ ℂ → ∃𝑥 ∈ ℤ 0 = (2 · 𝑥)) |
10 | 2, 9 | ax-mp 5 | . . 3 ⊢ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥) |
11 | eqeq1 2614 | . . . . 5 ⊢ (𝑧 = 0 → (𝑧 = (2 · 𝑥) ↔ 0 = (2 · 𝑥))) | |
12 | 11 | rexbidv 3034 | . . . 4 ⊢ (𝑧 = 0 → (∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥) ↔ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥))) |
13 | 12 | elrab 3331 | . . 3 ⊢ (0 ∈ {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)} ↔ (0 ∈ ℤ ∧ ∃𝑥 ∈ ℤ 0 = (2 · 𝑥))) |
14 | 1, 10, 13 | mpbir2an 957 | . 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 2687 | 1 ⊢ 0 ∈ 𝐸 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 {crab 2900 (class class class)co 6549 ℂcc 9813 0cc0 9815 · cmul 9820 2c2 10947 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-neg 10148 df-2 10956 df-z 11255 |
This theorem is referenced by: 2zlidl 41724 2zrng0 41728 2zrngamnd 41731 2zrngacmnd 41732 2zrngmmgm 41736 |
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