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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 12294 | . . . 4 ⊢ 5 ∈ ℕ | |
2 | dec5nprm.1 | . . . 4 ⊢ 𝐴 ∈ ℕ | |
3 | 1, 2 | nnmulcli 12233 | . . 3 ⊢ (5 · 𝐴) ∈ ℕ |
4 | dec2nprm.2 | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | nnnn0addcl 12498 | . . 3 ⊢ (((5 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℕ0) → ((5 · 𝐴) + 𝐵) ∈ ℕ) | |
6 | 3, 4, 5 | mp2an 689 | . 2 ⊢ ((5 · 𝐴) + 𝐵) ∈ ℕ |
7 | 2nn 12281 | . 2 ⊢ 2 ∈ ℕ | |
8 | 1nn0 12484 | . . 3 ⊢ 1 ∈ ℕ0 | |
9 | 1lt5 12388 | . . 3 ⊢ 1 < 5 | |
10 | 1, 2, 4, 8, 9 | numlti 12710 | . 2 ⊢ 1 < ((5 · 𝐴) + 𝐵) |
11 | 1lt2 12379 | . 2 ⊢ 1 < 2 | |
12 | 1 | nncni 12218 | . . . . . 6 ⊢ 5 ∈ ℂ |
13 | 2 | nncni 12218 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
14 | 2cn 12283 | . . . . . 6 ⊢ 2 ∈ ℂ | |
15 | 12, 13, 14 | mul32i 11406 | . . . . 5 ⊢ ((5 · 𝐴) · 2) = ((5 · 2) · 𝐴) |
16 | 5t2e10 12773 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
17 | 16 | oveq1i 7411 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (;10 · 𝐴) |
18 | 15, 17 | eqtri 2752 | . . . 4 ⊢ ((5 · 𝐴) · 2) = (;10 · 𝐴) |
19 | dec2nprm.3 | . . . 4 ⊢ (𝐵 · 2) = 𝐶 | |
20 | 18, 19 | oveq12i 7413 | . . 3 ⊢ (((5 · 𝐴) · 2) + (𝐵 · 2)) = ((;10 · 𝐴) + 𝐶) |
21 | 3 | nncni 12218 | . . . 4 ⊢ (5 · 𝐴) ∈ ℂ |
22 | 4 | nn0cni 12480 | . . . 4 ⊢ 𝐵 ∈ ℂ |
23 | 21, 22, 14 | adddiri 11223 | . . 3 ⊢ (((5 · 𝐴) + 𝐵) · 2) = (((5 · 𝐴) · 2) + (𝐵 · 2)) |
24 | dfdec10 12676 | . . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
25 | 20, 23, 24 | 3eqtr4i 2762 | . 2 ⊢ (((5 · 𝐴) + 𝐵) · 2) = ;𝐴𝐶 |
26 | 6, 7, 10, 11, 25 | nprmi 16622 | 1 ⊢ ¬ ;𝐴𝐶 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2098 (class class class)co 7401 0cc0 11105 1c1 11106 + caddc 11108 · cmul 11110 ℕcn 12208 2c2 12263 5c5 12266 ℕ0cn0 12468 ;cdc 12673 ℙcprime 16604 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-pre-sup 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-2o 8462 df-er 8698 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-sup 9432 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-rp 12971 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-prm 16605 |
This theorem is referenced by: (None) |
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