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Mirrors > Home > MPE Home > Th. List > dec5nprm | Structured version Visualization version GIF version |
Description: Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
dec5nprm.1 | ⊢ 𝐴 ∈ ℕ |
Ref | Expression |
---|---|
dec5nprm | ⊢ ¬ ;𝐴5 ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2nn 12044 | . . . 4 ⊢ 2 ∈ ℕ | |
2 | dec5nprm.1 | . . . 4 ⊢ 𝐴 ∈ ℕ | |
3 | 1, 2 | nnmulcli 11996 | . . 3 ⊢ (2 · 𝐴) ∈ ℕ |
4 | peano2nn 11983 | . . 3 ⊢ ((2 · 𝐴) ∈ ℕ → ((2 · 𝐴) + 1) ∈ ℕ) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ((2 · 𝐴) + 1) ∈ ℕ |
6 | 5nn 12057 | . 2 ⊢ 5 ∈ ℕ | |
7 | 1nn0 12247 | . . 3 ⊢ 1 ∈ ℕ0 | |
8 | 1lt2 12142 | . . 3 ⊢ 1 < 2 | |
9 | 1, 2, 7, 7, 8 | numlti 12471 | . 2 ⊢ 1 < ((2 · 𝐴) + 1) |
10 | 1lt5 12151 | . 2 ⊢ 1 < 5 | |
11 | 1 | nncni 11981 | . . . . . 6 ⊢ 2 ∈ ℂ |
12 | 2 | nncni 11981 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
13 | 5cn 12059 | . . . . . 6 ⊢ 5 ∈ ℂ | |
14 | 11, 12, 13 | mul32i 11169 | . . . . 5 ⊢ ((2 · 𝐴) · 5) = ((2 · 5) · 𝐴) |
15 | 5t2e10 12534 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
16 | 13, 11, 15 | mulcomli 10983 | . . . . . 6 ⊢ (2 · 5) = ;10 |
17 | 16 | oveq1i 7279 | . . . . 5 ⊢ ((2 · 5) · 𝐴) = (;10 · 𝐴) |
18 | 14, 17 | eqtri 2768 | . . . 4 ⊢ ((2 · 𝐴) · 5) = (;10 · 𝐴) |
19 | 13 | mulid2i 10979 | . . . 4 ⊢ (1 · 5) = 5 |
20 | 18, 19 | oveq12i 7281 | . . 3 ⊢ (((2 · 𝐴) · 5) + (1 · 5)) = ((;10 · 𝐴) + 5) |
21 | 3 | nncni 11981 | . . . 4 ⊢ (2 · 𝐴) ∈ ℂ |
22 | ax-1cn 10928 | . . . 4 ⊢ 1 ∈ ℂ | |
23 | 21, 22, 13 | adddiri 10987 | . . 3 ⊢ (((2 · 𝐴) + 1) · 5) = (((2 · 𝐴) · 5) + (1 · 5)) |
24 | dfdec10 12437 | . . 3 ⊢ ;𝐴5 = ((;10 · 𝐴) + 5) | |
25 | 20, 23, 24 | 3eqtr4i 2778 | . 2 ⊢ (((2 · 𝐴) + 1) · 5) = ;𝐴5 |
26 | 5, 6, 9, 10, 25 | nprmi 16390 | 1 ⊢ ¬ ;𝐴5 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2110 (class class class)co 7269 0cc0 10870 1c1 10871 + caddc 10873 · cmul 10875 ℕcn 11971 2c2 12026 5c5 12029 ;cdc 12434 ℙcprime 16372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-er 8479 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-sup 9177 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-rp 12728 df-seq 13718 df-exp 13779 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-dvds 15960 df-prm 16373 |
This theorem is referenced by: (None) |
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