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Mirrors > Home > MPE Home > Th. List > 2tp1odd | Structured version Visualization version GIF version |
Description: A number which is twice an integer increased by 1 is odd. (Contributed by AV, 16-Jul-2021.) |
Ref | Expression |
---|---|
2tp1odd | ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = ((2 · 𝐴) + 1)) → ¬ 2 ∥ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . 5 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℤ) | |
2 | oveq2 7153 | . . . . . . . 8 ⊢ (𝑘 = 𝐴 → (2 · 𝑘) = (2 · 𝐴)) | |
3 | 2 | oveq1d 7160 | . . . . . . 7 ⊢ (𝑘 = 𝐴 → ((2 · 𝑘) + 1) = ((2 · 𝐴) + 1)) |
4 | 3 | eqeq1d 2820 | . . . . . 6 ⊢ (𝑘 = 𝐴 → (((2 · 𝑘) + 1) = ((2 · 𝐴) + 1) ↔ ((2 · 𝐴) + 1) = ((2 · 𝐴) + 1))) |
5 | 4 | adantl 482 | . . . . 5 ⊢ ((𝐴 ∈ ℤ ∧ 𝑘 = 𝐴) → (((2 · 𝑘) + 1) = ((2 · 𝐴) + 1) ↔ ((2 · 𝐴) + 1) = ((2 · 𝐴) + 1))) |
6 | eqidd 2819 | . . . . 5 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) + 1) = ((2 · 𝐴) + 1)) | |
7 | 1, 5, 6 | rspcedvd 3623 | . . . 4 ⊢ (𝐴 ∈ ℤ → ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = ((2 · 𝐴) + 1)) |
8 | 2z 12002 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
9 | 8 | a1i 11 | . . . . . . 7 ⊢ (𝐴 ∈ ℤ → 2 ∈ ℤ) |
10 | 9, 1 | zmulcld 12081 | . . . . . 6 ⊢ (𝐴 ∈ ℤ → (2 · 𝐴) ∈ ℤ) |
11 | 10 | peano2zd 12078 | . . . . 5 ⊢ (𝐴 ∈ ℤ → ((2 · 𝐴) + 1) ∈ ℤ) |
12 | odd2np1 15678 | . . . . 5 ⊢ (((2 · 𝐴) + 1) ∈ ℤ → (¬ 2 ∥ ((2 · 𝐴) + 1) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = ((2 · 𝐴) + 1))) | |
13 | 11, 12 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℤ → (¬ 2 ∥ ((2 · 𝐴) + 1) ↔ ∃𝑘 ∈ ℤ ((2 · 𝑘) + 1) = ((2 · 𝐴) + 1))) |
14 | 7, 13 | mpbird 258 | . . 3 ⊢ (𝐴 ∈ ℤ → ¬ 2 ∥ ((2 · 𝐴) + 1)) |
15 | 14 | adantr 481 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = ((2 · 𝐴) + 1)) → ¬ 2 ∥ ((2 · 𝐴) + 1)) |
16 | breq2 5061 | . . 3 ⊢ (𝐵 = ((2 · 𝐴) + 1) → (2 ∥ 𝐵 ↔ 2 ∥ ((2 · 𝐴) + 1))) | |
17 | 16 | adantl 482 | . 2 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = ((2 · 𝐴) + 1)) → (2 ∥ 𝐵 ↔ 2 ∥ ((2 · 𝐴) + 1))) |
18 | 15, 17 | mtbird 326 | 1 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 = ((2 · 𝐴) + 1)) → ¬ 2 ∥ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∃wrex 3136 class class class wbr 5057 (class class class)co 7145 1c1 10526 + caddc 10528 · cmul 10530 2c2 11680 ℤcz 11969 ∥ cdvds 15595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-n0 11886 df-z 11970 df-dvds 15596 |
This theorem is referenced by: 2lgslem3b1 25904 2lgslem3c1 25905 limsup10exlem 41929 |
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