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Mirrors > Home > ILE Home > Th. List > n2dvdsm1 | GIF version |
Description: 2 does not divide -1. That means -1 is odd. (Contributed by AV, 15-Aug-2021.) |
Ref | Expression |
---|---|
n2dvdsm1 | ⊢ ¬ 2 ∥ -1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | z0even 10311 | . . 3 ⊢ 2 ∥ 0 | |
2 | ax-1cn 7069 | . . . 4 ⊢ 1 ∈ ℂ | |
3 | neg1cn 8144 | . . . 4 ⊢ -1 ∈ ℂ | |
4 | 1pneg1e0 8150 | . . . 4 ⊢ (1 + -1) = 0 | |
5 | 2, 3, 4 | addcomli 7253 | . . 3 ⊢ (-1 + 1) = 0 |
6 | 1, 5 | breqtrri 3810 | . 2 ⊢ 2 ∥ (-1 + 1) |
7 | neg1z 8383 | . . 3 ⊢ -1 ∈ ℤ | |
8 | oddp1even 10275 | . . 3 ⊢ (-1 ∈ ℤ → (¬ 2 ∥ -1 ↔ 2 ∥ (-1 + 1))) | |
9 | 7, 8 | ax-mp 7 | . 2 ⊢ (¬ 2 ∥ -1 ↔ 2 ∥ (-1 + 1)) |
10 | 6, 9 | mpbir 144 | 1 ⊢ ¬ 2 ∥ -1 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 0cc0 6981 1c1 6982 + caddc 6984 -cneg 7280 2c2 8089 ℤcz 8351 ∥ cdvds 10195 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-xor 1307 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-n0 8289 df-z 8352 df-dvds 10196 |
This theorem is referenced by: (None) |
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