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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 12338 | . . . 4 ⊢ 5 ∈ ℕ | |
2 | dec5nprm.1 | . . . 4 ⊢ 𝐴 ∈ ℕ | |
3 | 1, 2 | nnmulcli 12277 | . . 3 ⊢ (5 · 𝐴) ∈ ℕ |
4 | dec2nprm.2 | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | nnnn0addcl 12542 | . . 3 ⊢ (((5 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℕ0) → ((5 · 𝐴) + 𝐵) ∈ ℕ) | |
6 | 3, 4, 5 | mp2an 690 | . 2 ⊢ ((5 · 𝐴) + 𝐵) ∈ ℕ |
7 | 2nn 12325 | . 2 ⊢ 2 ∈ ℕ | |
8 | 1nn0 12528 | . . 3 ⊢ 1 ∈ ℕ0 | |
9 | 1lt5 12432 | . . 3 ⊢ 1 < 5 | |
10 | 1, 2, 4, 8, 9 | numlti 12754 | . 2 ⊢ 1 < ((5 · 𝐴) + 𝐵) |
11 | 1lt2 12423 | . 2 ⊢ 1 < 2 | |
12 | 1 | nncni 12262 | . . . . . 6 ⊢ 5 ∈ ℂ |
13 | 2 | nncni 12262 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
14 | 2cn 12327 | . . . . . 6 ⊢ 2 ∈ ℂ | |
15 | 12, 13, 14 | mul32i 11450 | . . . . 5 ⊢ ((5 · 𝐴) · 2) = ((5 · 2) · 𝐴) |
16 | 5t2e10 12817 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
17 | 16 | oveq1i 7436 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (;10 · 𝐴) |
18 | 15, 17 | eqtri 2756 | . . . 4 ⊢ ((5 · 𝐴) · 2) = (;10 · 𝐴) |
19 | dec2nprm.3 | . . . 4 ⊢ (𝐵 · 2) = 𝐶 | |
20 | 18, 19 | oveq12i 7438 | . . 3 ⊢ (((5 · 𝐴) · 2) + (𝐵 · 2)) = ((;10 · 𝐴) + 𝐶) |
21 | 3 | nncni 12262 | . . . 4 ⊢ (5 · 𝐴) ∈ ℂ |
22 | 4 | nn0cni 12524 | . . . 4 ⊢ 𝐵 ∈ ℂ |
23 | 21, 22, 14 | adddiri 11267 | . . 3 ⊢ (((5 · 𝐴) + 𝐵) · 2) = (((5 · 𝐴) · 2) + (𝐵 · 2)) |
24 | dfdec10 12720 | . . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
25 | 20, 23, 24 | 3eqtr4i 2766 | . 2 ⊢ (((5 · 𝐴) + 𝐵) · 2) = ;𝐴𝐶 |
26 | 6, 7, 10, 11, 25 | nprmi 16669 | 1 ⊢ ¬ ;𝐴𝐶 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 (class class class)co 7426 0cc0 11148 1c1 11149 + caddc 11151 · cmul 11153 ℕcn 12252 2c2 12307 5c5 12310 ℕ0cn0 12512 ;cdc 12717 ℙcprime 16651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7879 df-2nd 8002 df-frecs 8295 df-wrecs 8326 df-recs 8400 df-rdg 8439 df-1o 8495 df-2o 8496 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-fin 8976 df-sup 9475 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12253 df-2 12315 df-3 12316 df-4 12317 df-5 12318 df-6 12319 df-7 12320 df-8 12321 df-9 12322 df-n0 12513 df-z 12599 df-dec 12718 df-uz 12863 df-rp 13017 df-seq 14009 df-exp 14069 df-cj 15088 df-re 15089 df-im 15090 df-sqrt 15224 df-abs 15225 df-dvds 16241 df-prm 16652 |
This theorem is referenced by: (None) |
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