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Mirrors > Home > MPE Home > Th. List > Mathboxes > evenp1odd | Structured version Visualization version GIF version |
Description: The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.) |
Ref | Expression |
---|---|
evenp1odd | ⊢ (𝑍 ∈ Even → (𝑍 + 1) ∈ Odd ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evenz 47236 | . . 3 ⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ) | |
2 | 1 | peano2zd 12713 | . 2 ⊢ (𝑍 ∈ Even → (𝑍 + 1) ∈ ℤ) |
3 | iseven 47234 | . . 3 ⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ)) | |
4 | zcn 12607 | . . . . . . . 8 ⊢ (𝑍 ∈ ℤ → 𝑍 ∈ ℂ) | |
5 | pncan1 11677 | . . . . . . . 8 ⊢ (𝑍 ∈ ℂ → ((𝑍 + 1) − 1) = 𝑍) | |
6 | 4, 5 | syl 17 | . . . . . . 7 ⊢ (𝑍 ∈ ℤ → ((𝑍 + 1) − 1) = 𝑍) |
7 | 6 | eqcomd 2732 | . . . . . 6 ⊢ (𝑍 ∈ ℤ → 𝑍 = ((𝑍 + 1) − 1)) |
8 | 7 | oveq1d 7429 | . . . . 5 ⊢ (𝑍 ∈ ℤ → (𝑍 / 2) = (((𝑍 + 1) − 1) / 2)) |
9 | 8 | eleq1d 2811 | . . . 4 ⊢ (𝑍 ∈ ℤ → ((𝑍 / 2) ∈ ℤ ↔ (((𝑍 + 1) − 1) / 2) ∈ ℤ)) |
10 | 9 | biimpa 475 | . . 3 ⊢ ((𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ) → (((𝑍 + 1) − 1) / 2) ∈ ℤ) |
11 | 3, 10 | sylbi 216 | . 2 ⊢ (𝑍 ∈ Even → (((𝑍 + 1) − 1) / 2) ∈ ℤ) |
12 | isodd2 47241 | . 2 ⊢ ((𝑍 + 1) ∈ Odd ↔ ((𝑍 + 1) ∈ ℤ ∧ (((𝑍 + 1) − 1) / 2) ∈ ℤ)) | |
13 | 2, 11, 12 | sylanbrc 581 | 1 ⊢ (𝑍 ∈ Even → (𝑍 + 1) ∈ Odd ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 (class class class)co 7414 ℂcc 11145 1c1 11148 + caddc 11150 − cmin 11483 / cdiv 11910 2c2 12311 ℤcz 12602 Even ceven 47230 Odd codd 47231 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-n0 12517 df-z 12603 df-even 47232 df-odd 47233 |
This theorem is referenced by: epee 47311 3odd 47314 5odd 47316 7odd 47318 evenltle 47323 9gbo 47380 |
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