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Mirrors > Home > MPE Home > Th. List > Mathboxes > nnennex | Structured version Visualization version GIF version |
Description: For each even positive integer there is a positive integer which, multiplied by 2, results in the even positive integer. (Contributed by AV, 5-Jun-2023.) |
Ref | Expression |
---|---|
nnennex | ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) → ∃𝑚 ∈ ℕ 𝑁 = (2 · 𝑚)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 488 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) → (𝑁 / 2) ∈ ℕ) | |
2 | oveq2 7181 | . . . 4 ⊢ (𝑚 = (𝑁 / 2) → (2 · 𝑚) = (2 · (𝑁 / 2))) | |
3 | 2 | eqeq2d 2750 | . . 3 ⊢ (𝑚 = (𝑁 / 2) → (𝑁 = (2 · 𝑚) ↔ 𝑁 = (2 · (𝑁 / 2)))) |
4 | 3 | adantl 485 | . 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) ∧ 𝑚 = (𝑁 / 2)) → (𝑁 = (2 · 𝑚) ↔ 𝑁 = (2 · (𝑁 / 2)))) |
5 | nncn 11727 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
6 | 2cnd 11797 | . . . 4 ⊢ (𝑁 ∈ ℕ → 2 ∈ ℂ) | |
7 | 2ne0 11823 | . . . . 5 ⊢ 2 ≠ 0 | |
8 | 7 | a1i 11 | . . . 4 ⊢ (𝑁 ∈ ℕ → 2 ≠ 0) |
9 | divcan2 11387 | . . . . 5 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → (2 · (𝑁 / 2)) = 𝑁) | |
10 | 9 | eqcomd 2745 | . . . 4 ⊢ ((𝑁 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0) → 𝑁 = (2 · (𝑁 / 2))) |
11 | 5, 6, 8, 10 | syl3anc 1372 | . . 3 ⊢ (𝑁 ∈ ℕ → 𝑁 = (2 · (𝑁 / 2))) |
12 | 11 | adantr 484 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) → 𝑁 = (2 · (𝑁 / 2))) |
13 | 1, 4, 12 | rspcedvd 3530 | 1 ⊢ ((𝑁 ∈ ℕ ∧ (𝑁 / 2) ∈ ℕ) → ∃𝑚 ∈ ℕ 𝑁 = (2 · 𝑚)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 ∃wrex 3055 (class class class)co 7173 ℂcc 10616 0cc0 10618 · cmul 10623 / cdiv 11378 ℕcn 11719 2c2 11774 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 |
This theorem is referenced by: (None) |
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