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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > emoo | Structured version Visualization version GIF version |
Description: The difference of an even and an odd is odd. (Contributed by AV, 24-Jul-2020.) |
Ref | Expression |
---|---|
emoo | ⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 − 𝐵) ∈ Odd ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | evenz 41868 | . . . 4 ⊢ (𝐴 ∈ Even → 𝐴 ∈ ℤ) | |
2 | 1 | zcnd 11521 | . . 3 ⊢ (𝐴 ∈ Even → 𝐴 ∈ ℂ) |
3 | oddz 41869 | . . . 4 ⊢ (𝐵 ∈ Odd → 𝐵 ∈ ℤ) | |
4 | 3 | zcnd 11521 | . . 3 ⊢ (𝐵 ∈ Odd → 𝐵 ∈ ℂ) |
5 | negsub 10367 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
6 | 2, 4, 5 | syl2an 493 | . 2 ⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) |
7 | onego 41884 | . . 3 ⊢ (𝐵 ∈ Odd → -𝐵 ∈ Odd ) | |
8 | epoo 41937 | . . 3 ⊢ ((𝐴 ∈ Even ∧ -𝐵 ∈ Odd ) → (𝐴 + -𝐵) ∈ Odd ) | |
9 | 7, 8 | sylan2 490 | . 2 ⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 + -𝐵) ∈ Odd ) |
10 | 6, 9 | eqeltrrd 2731 | 1 ⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → (𝐴 − 𝐵) ∈ Odd ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 (class class class)co 6690 ℂcc 9972 + caddc 9977 − cmin 10304 -cneg 10305 Even ceven 41862 Odd codd 41863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-n0 11331 df-z 11416 df-even 41864 df-odd 41865 |
This theorem is referenced by: even3prm2 41953 evengpop3 42011 evengpoap3 42012 nnsum4primeseven 42013 nnsum4primesevenALTV 42014 |
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