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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 11726 | . . . 4 ⊢ 5 ∈ ℕ | |
2 | dec5nprm.1 | . . . 4 ⊢ 𝐴 ∈ ℕ | |
3 | 1, 2 | nnmulcli 11665 | . . 3 ⊢ (5 · 𝐴) ∈ ℕ |
4 | dec2nprm.2 | . . 3 ⊢ 𝐵 ∈ ℕ0 | |
5 | nnnn0addcl 11930 | . . 3 ⊢ (((5 · 𝐴) ∈ ℕ ∧ 𝐵 ∈ ℕ0) → ((5 · 𝐴) + 𝐵) ∈ ℕ) | |
6 | 3, 4, 5 | mp2an 690 | . 2 ⊢ ((5 · 𝐴) + 𝐵) ∈ ℕ |
7 | 2nn 11713 | . 2 ⊢ 2 ∈ ℕ | |
8 | 1nn0 11916 | . . 3 ⊢ 1 ∈ ℕ0 | |
9 | 1lt5 11820 | . . 3 ⊢ 1 < 5 | |
10 | 1, 2, 4, 8, 9 | numlti 12138 | . 2 ⊢ 1 < ((5 · 𝐴) + 𝐵) |
11 | 1lt2 11811 | . 2 ⊢ 1 < 2 | |
12 | 1 | nncni 11650 | . . . . . 6 ⊢ 5 ∈ ℂ |
13 | 2 | nncni 11650 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
14 | 2cn 11715 | . . . . . 6 ⊢ 2 ∈ ℂ | |
15 | 12, 13, 14 | mul32i 10838 | . . . . 5 ⊢ ((5 · 𝐴) · 2) = ((5 · 2) · 𝐴) |
16 | 5t2e10 12201 | . . . . . 6 ⊢ (5 · 2) = ;10 | |
17 | 16 | oveq1i 7168 | . . . . 5 ⊢ ((5 · 2) · 𝐴) = (;10 · 𝐴) |
18 | 15, 17 | eqtri 2846 | . . . 4 ⊢ ((5 · 𝐴) · 2) = (;10 · 𝐴) |
19 | dec2nprm.3 | . . . 4 ⊢ (𝐵 · 2) = 𝐶 | |
20 | 18, 19 | oveq12i 7170 | . . 3 ⊢ (((5 · 𝐴) · 2) + (𝐵 · 2)) = ((;10 · 𝐴) + 𝐶) |
21 | 3 | nncni 11650 | . . . 4 ⊢ (5 · 𝐴) ∈ ℂ |
22 | 4 | nn0cni 11912 | . . . 4 ⊢ 𝐵 ∈ ℂ |
23 | 21, 22, 14 | adddiri 10656 | . . 3 ⊢ (((5 · 𝐴) + 𝐵) · 2) = (((5 · 𝐴) · 2) + (𝐵 · 2)) |
24 | dfdec10 12104 | . . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
25 | 20, 23, 24 | 3eqtr4i 2856 | . 2 ⊢ (((5 · 𝐴) + 𝐵) · 2) = ;𝐴𝐶 |
26 | 6, 7, 10, 11, 25 | nprmi 16035 | 1 ⊢ ¬ ;𝐴𝐶 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 ℕcn 11640 2c2 11695 5c5 11698 ℕ0cn0 11900 ;cdc 12101 ℙcprime 16017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-dvds 15610 df-prm 16018 |
This theorem is referenced by: (None) |
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