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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 11711 | . . . 4 ⊢ 5 ∈ ℕ | |
2 | dec5nprm.1 | . . . 4 ⊢ 𝐴 ∈ ℕ | |
3 | 1, 2 | nnmulcli 11650 | . . 3 ⊢ (5 · 𝐴) ∈ ℕ |
4 | dec2nprm.2 | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | nnnn0addcl 11915 | . . 3 ⊢ (((5 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℕ0) → ((5 · 𝐴) + 𝐵) ∈ ℕ) | |
6 | 3, 4, 5 | mp2an 688 | . 2 ⊢ ((5 · 𝐴) + 𝐵) ∈ ℕ |
7 | 2nn 11698 | . 2 ⊢ 2 ∈ ℕ | |
8 | 1nn0 11901 | . . 3 ⊢ 1 ∈ ℕ0 | |
9 | 1lt5 11805 | . . 3 ⊢ 1 < 5 | |
10 | 1, 2, 4, 8, 9 | numlti 12123 | . 2 ⊢ 1 < ((5 · 𝐴) + 𝐵) |
11 | 1lt2 11796 | . 2 ⊢ 1 < 2 | |
12 | 1 | nncni 11636 | . . . . . 6 ⊢ 5 ∈ ℂ |
13 | 2 | nncni 11636 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
14 | 2cn 11700 | . . . . . 6 ⊢ 2 ∈ ℂ | |
15 | 12, 13, 14 | mul32i 10824 | . . . . 5 ⊢ ((5 · 𝐴) · 2) = ((5 · 2) · 𝐴) |
16 | 5t2e10 12186 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
17 | 16 | oveq1i 7155 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (;10 · 𝐴) |
18 | 15, 17 | eqtri 2841 | . . . 4 ⊢ ((5 · 𝐴) · 2) = (;10 · 𝐴) |
19 | dec2nprm.3 | . . . 4 ⊢ (𝐵 · 2) = 𝐶 | |
20 | 18, 19 | oveq12i 7157 | . . 3 ⊢ (((5 · 𝐴) · 2) + (𝐵 · 2)) = ((;10 · 𝐴) + 𝐶) |
21 | 3 | nncni 11636 | . . . 4 ⊢ (5 · 𝐴) ∈ ℂ |
22 | 4 | nn0cni 11897 | . . . 4 ⊢ 𝐵 ∈ ℂ |
23 | 21, 22, 14 | adddiri 10642 | . . 3 ⊢ (((5 · 𝐴) + 𝐵) · 2) = (((5 · 𝐴) · 2) + (𝐵 · 2)) |
24 | dfdec10 12089 | . . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
25 | 20, 23, 24 | 3eqtr4i 2851 | . 2 ⊢ (((5 · 𝐴) + 𝐵) · 2) = ;𝐴𝐶 |
26 | 6, 7, 10, 11, 25 | nprmi 16021 | 1 ⊢ ¬ ;𝐴𝐶 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1528 ∈ wcel 2105 (class class class)co 7145 0cc0 10525 1c1 10526 + caddc 10528 · cmul 10530 ℕcn 11626 2c2 11680 5c5 11683 ℕ0cn0 11885 ;cdc 12086 ℙcprime 16003 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-prm 16004 |
This theorem is referenced by: (None) |
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