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Mirrors > Home > MPE Home > Th. List > Mathboxes > nn0onn0ex | Structured version Visualization version GIF version |
Description: For each odd nonnegative integer there is a nonnegative integer which, multiplied by 2 and increased by 1, results in the odd nonnegative integer. (Contributed by AV, 30-May-2020.) |
Ref | Expression |
---|---|
nn0onn0ex | ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ∃𝑚 ∈ ℕ0 𝑁 = ((2 · 𝑚) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0o 16367 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0) | |
2 | simpr 483 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 − 1) / 2) ∈ ℕ0) → ((𝑁 − 1) / 2) ∈ ℕ0) | |
3 | oveq2 7434 | . . . . . 6 ⊢ (𝑚 = ((𝑁 − 1) / 2) → (2 · 𝑚) = (2 · ((𝑁 − 1) / 2))) | |
4 | 3 | oveq1d 7441 | . . . . 5 ⊢ (𝑚 = ((𝑁 − 1) / 2) → ((2 · 𝑚) + 1) = ((2 · ((𝑁 − 1) / 2)) + 1)) |
5 | 4 | eqeq2d 2739 | . . . 4 ⊢ (𝑚 = ((𝑁 − 1) / 2) → (𝑁 = ((2 · 𝑚) + 1) ↔ 𝑁 = ((2 · ((𝑁 − 1) / 2)) + 1))) |
6 | 5 | adantl 480 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ ((𝑁 − 1) / 2) ∈ ℕ0) ∧ 𝑚 = ((𝑁 − 1) / 2)) → (𝑁 = ((2 · 𝑚) + 1) ↔ 𝑁 = ((2 · ((𝑁 − 1) / 2)) + 1))) |
7 | nn0cn 12520 | . . . . . . . 8 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
8 | peano2cnm 11564 | . . . . . . . 8 ⊢ (𝑁 ∈ ℂ → (𝑁 − 1) ∈ ℂ) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → (𝑁 − 1) ∈ ℂ) |
10 | 2cnd 12328 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) | |
11 | 2ne0 12354 | . . . . . . . 8 ⊢ 2 ≠ 0 | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 2 ≠ 0) |
13 | 9, 10, 12 | divcan2d 12030 | . . . . . 6 ⊢ (𝑁 ∈ ℕ0 → (2 · ((𝑁 − 1) / 2)) = (𝑁 − 1)) |
14 | 13 | oveq1d 7441 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((2 · ((𝑁 − 1) / 2)) + 1) = ((𝑁 − 1) + 1)) |
15 | npcan1 11677 | . . . . . 6 ⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) | |
16 | 7, 15 | syl 17 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((𝑁 − 1) + 1) = 𝑁) |
17 | 14, 16 | eqtr2d 2769 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 𝑁 = ((2 · ((𝑁 − 1) / 2)) + 1)) |
18 | 17 | adantr 479 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 − 1) / 2) ∈ ℕ0) → 𝑁 = ((2 · ((𝑁 − 1) / 2)) + 1)) |
19 | 2, 6, 18 | rspcedvd 3613 | . 2 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 − 1) / 2) ∈ ℕ0) → ∃𝑚 ∈ ℕ0 𝑁 = ((2 · 𝑚) + 1)) |
20 | 1, 19 | syldan 589 | 1 ⊢ ((𝑁 ∈ ℕ0 ∧ ((𝑁 + 1) / 2) ∈ ℕ0) → ∃𝑚 ∈ ℕ0 𝑁 = ((2 · 𝑚) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 ∃wrex 3067 (class class class)co 7426 ℂcc 11144 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 − cmin 11482 / cdiv 11909 2c2 12305 ℕ0cn0 12510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 |
This theorem is referenced by: (None) |
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