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Mirrors > Home > MPE Home > Th. List > oddp1even | Structured version Visualization version GIF version |
Description: An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) |
Ref | Expression |
---|---|
oddp1even | ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 + 1))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oddm1even 15684 | . 2 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 − 1))) | |
2 | 2z 12006 | . . 3 ⊢ 2 ∈ ℤ | |
3 | peano2zm 12017 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈ ℤ) | |
4 | dvdsadd 15644 | . . 3 ⊢ ((2 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ) → (2 ∥ (𝑁 − 1) ↔ 2 ∥ (2 + (𝑁 − 1)))) | |
5 | 2, 3, 4 | sylancr 589 | . 2 ⊢ (𝑁 ∈ ℤ → (2 ∥ (𝑁 − 1) ↔ 2 ∥ (2 + (𝑁 − 1)))) |
6 | 2cnd 11707 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 2 ∈ ℂ) | |
7 | zcn 11978 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
8 | 1cnd 10628 | . . . . 5 ⊢ (𝑁 ∈ ℤ → 1 ∈ ℂ) | |
9 | 6, 7, 8 | addsub12d 11012 | . . . 4 ⊢ (𝑁 ∈ ℤ → (2 + (𝑁 − 1)) = (𝑁 + (2 − 1))) |
10 | 2m1e1 11755 | . . . . 5 ⊢ (2 − 1) = 1 | |
11 | 10 | oveq2i 7159 | . . . 4 ⊢ (𝑁 + (2 − 1)) = (𝑁 + 1) |
12 | 9, 11 | syl6eq 2870 | . . 3 ⊢ (𝑁 ∈ ℤ → (2 + (𝑁 − 1)) = (𝑁 + 1)) |
13 | 12 | breq2d 5069 | . 2 ⊢ (𝑁 ∈ ℤ → (2 ∥ (2 + (𝑁 − 1)) ↔ 2 ∥ (𝑁 + 1))) |
14 | 1, 5, 13 | 3bitrd 307 | 1 ⊢ (𝑁 ∈ ℤ → (¬ 2 ∥ 𝑁 ↔ 2 ∥ (𝑁 + 1))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∈ wcel 2108 class class class wbr 5057 (class class class)co 7148 1c1 10530 + caddc 10532 − cmin 10862 2c2 11684 ℤcz 11973 ∥ cdvds 15599 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 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 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-n0 11890 df-z 11974 df-dvds 15600 |
This theorem is referenced by: zeo5 15697 oddp1d2 15699 n2dvdsm1 15711 sumodd 15731 knoppndvlem10 33853 stirlinglem5 42354 fouriersw 42507 2dvdsoddp1 43812 0dig2nn0o 44664 |
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