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Mirrors > Home > MPE Home > Th. List > even2n | Structured version Visualization version GIF version |
Description: An integer is even iff it is twice another integer. (Contributed by AV, 25-Jun-2020.) |
Ref | Expression |
---|---|
even2n | ⊢ (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evenelz 15685 | . 2 ⊢ (2 ∥ 𝑁 → 𝑁 ∈ ℤ) | |
2 | 2z 12015 | . . . . . . 7 ⊢ 2 ∈ ℤ | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → 2 ∈ ℤ) |
4 | id 22 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℤ) | |
5 | 3, 4 | zmulcld 12094 | . . . . 5 ⊢ (𝑛 ∈ ℤ → (2 · 𝑛) ∈ ℤ) |
6 | 5 | adantr 483 | . . . 4 ⊢ ((𝑛 ∈ ℤ ∧ (2 · 𝑛) = 𝑁) → (2 · 𝑛) ∈ ℤ) |
7 | eleq1 2900 | . . . . 5 ⊢ ((2 · 𝑛) = 𝑁 → ((2 · 𝑛) ∈ ℤ ↔ 𝑁 ∈ ℤ)) | |
8 | 7 | adantl 484 | . . . 4 ⊢ ((𝑛 ∈ ℤ ∧ (2 · 𝑛) = 𝑁) → ((2 · 𝑛) ∈ ℤ ↔ 𝑁 ∈ ℤ)) |
9 | 6, 8 | mpbid 234 | . . 3 ⊢ ((𝑛 ∈ ℤ ∧ (2 · 𝑛) = 𝑁) → 𝑁 ∈ ℤ) |
10 | 9 | rexlimiva 3281 | . 2 ⊢ (∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁 → 𝑁 ∈ ℤ) |
11 | divides 15609 | . . . 4 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (𝑛 · 2) = 𝑁)) | |
12 | zcn 11987 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
13 | 2cnd 11716 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → 2 ∈ ℂ) | |
14 | 12, 13 | mulcomd 10662 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → (𝑛 · 2) = (2 · 𝑛)) |
15 | 14 | eqeq1d 2823 | . . . . 5 ⊢ (𝑛 ∈ ℤ → ((𝑛 · 2) = 𝑁 ↔ (2 · 𝑛) = 𝑁)) |
16 | 15 | rexbiia 3246 | . . . 4 ⊢ (∃𝑛 ∈ ℤ (𝑛 · 2) = 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁) |
17 | 11, 16 | syl6bb 289 | . . 3 ⊢ ((2 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)) |
18 | 2, 17 | mpan 688 | . 2 ⊢ (𝑁 ∈ ℤ → (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁)) |
19 | 1, 10, 18 | pm5.21nii 382 | 1 ⊢ (2 ∥ 𝑁 ↔ ∃𝑛 ∈ ℤ (2 · 𝑛) = 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 class class class wbr 5066 (class class class)co 7156 · cmul 10542 2c2 11693 ℤcz 11982 ∥ cdvds 15607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-ltxr 10680 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-dvds 15608 |
This theorem is referenced by: evennn02n 15699 evennn2n 15700 m1expe 15725 |
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