![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dec2nprm | Structured version Visualization version GIF version |
Description: Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
dec5nprm.1 | ⊢ 𝐴 ∈ ℕ |
dec2nprm.2 | ⊢ 𝐵 ∈ ℕ0 |
dec2nprm.3 | ⊢ (𝐵 · 2) = 𝐶 |
Ref | Expression |
---|---|
dec2nprm | ⊢ ¬ ;𝐴𝐶 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn 12246 | . . . 4 ⊢ 5 ∈ ℕ | |
2 | dec5nprm.1 | . . . 4 ⊢ 𝐴 ∈ ℕ | |
3 | 1, 2 | nnmulcli 12185 | . . 3 ⊢ (5 · 𝐴) ∈ ℕ |
4 | dec2nprm.2 | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | nnnn0addcl 12450 | . . 3 ⊢ (((5 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℕ0) → ((5 · 𝐴) + 𝐵) ∈ ℕ) | |
6 | 3, 4, 5 | mp2an 691 | . 2 ⊢ ((5 · 𝐴) + 𝐵) ∈ ℕ |
7 | 2nn 12233 | . 2 ⊢ 2 ∈ ℕ | |
8 | 1nn0 12436 | . . 3 ⊢ 1 ∈ ℕ0 | |
9 | 1lt5 12340 | . . 3 ⊢ 1 < 5 | |
10 | 1, 2, 4, 8, 9 | numlti 12662 | . 2 ⊢ 1 < ((5 · 𝐴) + 𝐵) |
11 | 1lt2 12331 | . 2 ⊢ 1 < 2 | |
12 | 1 | nncni 12170 | . . . . . 6 ⊢ 5 ∈ ℂ |
13 | 2 | nncni 12170 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
14 | 2cn 12235 | . . . . . 6 ⊢ 2 ∈ ℂ | |
15 | 12, 13, 14 | mul32i 11358 | . . . . 5 ⊢ ((5 · 𝐴) · 2) = ((5 · 2) · 𝐴) |
16 | 5t2e10 12725 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
17 | 16 | oveq1i 7372 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (;10 · 𝐴) |
18 | 15, 17 | eqtri 2765 | . . . 4 ⊢ ((5 · 𝐴) · 2) = (;10 · 𝐴) |
19 | dec2nprm.3 | . . . 4 ⊢ (𝐵 · 2) = 𝐶 | |
20 | 18, 19 | oveq12i 7374 | . . 3 ⊢ (((5 · 𝐴) · 2) + (𝐵 · 2)) = ((;10 · 𝐴) + 𝐶) |
21 | 3 | nncni 12170 | . . . 4 ⊢ (5 · 𝐴) ∈ ℂ |
22 | 4 | nn0cni 12432 | . . . 4 ⊢ 𝐵 ∈ ℂ |
23 | 21, 22, 14 | adddiri 11175 | . . 3 ⊢ (((5 · 𝐴) + 𝐵) · 2) = (((5 · 𝐴) · 2) + (𝐵 · 2)) |
24 | dfdec10 12628 | . . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
25 | 20, 23, 24 | 3eqtr4i 2775 | . 2 ⊢ (((5 · 𝐴) + 𝐵) · 2) = ;𝐴𝐶 |
26 | 6, 7, 10, 11, 25 | nprmi 16572 | 1 ⊢ ¬ ;𝐴𝐶 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 (class class class)co 7362 0cc0 11058 1c1 11059 + caddc 11061 · cmul 11063 ℕcn 12160 2c2 12215 5c5 12218 ℕ0cn0 12420 ;cdc 12625 ℙcprime 16554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-rp 12923 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-dvds 16144 df-prm 16555 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |