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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 11711 | . . . 4 ⊢ 2 ∈ ℕ | |
2 | dec5nprm.1 | . . . 4 ⊢ 𝐴 ∈ ℕ | |
3 | 1, 2 | nnmulcli 11663 | . . 3 ⊢ (2 · 𝐴) ∈ ℕ |
4 | peano2nn 11650 | . . 3 ⊢ ((2 · 𝐴) ∈ ℕ → ((2 · 𝐴) + 1) ∈ ℕ) | |
5 | 3, 4 | ax-mp 5 | . 2 ⊢ ((2 · 𝐴) + 1) ∈ ℕ |
6 | 5nn 11724 | . 2 ⊢ 5 ∈ ℕ | |
7 | 1nn0 11914 | . . 3 ⊢ 1 ∈ ℕ0 | |
8 | 1lt2 11809 | . . 3 ⊢ 1 < 2 | |
9 | 1, 2, 7, 7, 8 | numlti 12136 | . 2 ⊢ 1 < ((2 · 𝐴) + 1) |
10 | 1lt5 11818 | . 2 ⊢ 1 < 5 | |
11 | 1 | nncni 11648 | . . . . . 6 ⊢ 2 ∈ ℂ |
12 | 2 | nncni 11648 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
13 | 5cn 11726 | . . . . . 6 ⊢ 5 ∈ ℂ | |
14 | 11, 12, 13 | mul32i 10836 | . . . . 5 ⊢ ((2 · 𝐴) · 5) = ((2 · 5) · 𝐴) |
15 | 5t2e10 12199 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
16 | 13, 11, 15 | mulcomli 10650 | . . . . . 6 ⊢ (2 · 5) = ;10 |
17 | 16 | oveq1i 7166 | . . . . 5 ⊢ ((2 · 5) · 𝐴) = (;10 · 𝐴) |
18 | 14, 17 | eqtri 2844 | . . . 4 ⊢ ((2 · 𝐴) · 5) = (;10 · 𝐴) |
19 | 13 | mulid2i 10646 | . . . 4 ⊢ (1 · 5) = 5 |
20 | 18, 19 | oveq12i 7168 | . . 3 ⊢ (((2 · 𝐴) · 5) + (1 · 5)) = ((;10 · 𝐴) + 5) |
21 | 3 | nncni 11648 | . . . 4 ⊢ (2 · 𝐴) ∈ ℂ |
22 | ax-1cn 10595 | . . . 4 ⊢ 1 ∈ ℂ | |
23 | 21, 22, 13 | adddiri 10654 | . . 3 ⊢ (((2 · 𝐴) + 1) · 5) = (((2 · 𝐴) · 5) + (1 · 5)) |
24 | dfdec10 12102 | . . 3 ⊢ ;𝐴5 = ((;10 · 𝐴) + 5) | |
25 | 20, 23, 24 | 3eqtr4i 2854 | . 2 ⊢ (((2 · 𝐴) + 1) · 5) = ;𝐴5 |
26 | 5, 6, 9, 10, 25 | nprmi 16033 | 1 ⊢ ¬ ;𝐴5 ∈ ℙ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∈ wcel 2114 (class class class)co 7156 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℕcn 11638 2c2 11693 5c5 11696 ;cdc 12099 ℙcprime 16015 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-dvds 15608 df-prm 16016 |
This theorem is referenced by: (None) |
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