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Mirrors > Home > MPE Home > Th. List > dec2dvds | Structured version Visualization version GIF version |
Description: Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
dec2dvds.1 | ⊢ 𝐴 ∈ ℕ0 |
dec2dvds.2 | ⊢ 𝐵 ∈ ℕ0 |
dec2dvds.3 | ⊢ (𝐵 · 2) = 𝐶 |
dec2dvds.4 | ⊢ 𝐷 = (𝐶 + 1) |
Ref | Expression |
---|---|
dec2dvds | ⊢ ¬ 2 ∥ ;𝐴𝐷 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 5nn0 12543 | . . . . . . . . 9 ⊢ 5 ∈ ℕ0 | |
2 | 1 | nn0zi 12639 | . . . . . . . 8 ⊢ 5 ∈ ℤ |
3 | 2z 12646 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
4 | dvdsmul2 16312 | . . . . . . . 8 ⊢ ((5 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (5 · 2)) | |
5 | 2, 3, 4 | mp2an 692 | . . . . . . 7 ⊢ 2 ∥ (5 · 2) |
6 | 5t2e10 12830 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
7 | 5, 6 | breqtri 5172 | . . . . . 6 ⊢ 2 ∥ ;10 |
8 | 10nn0 12748 | . . . . . . . 8 ⊢ ;10 ∈ ℕ0 | |
9 | 8 | nn0zi 12639 | . . . . . . 7 ⊢ ;10 ∈ ℤ |
10 | dec2dvds.1 | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
11 | 10 | nn0zi 12639 | . . . . . . 7 ⊢ 𝐴 ∈ ℤ |
12 | dvdsmultr1 16329 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ ;10 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (2 ∥ ;10 → 2 ∥ (;10 · 𝐴))) | |
13 | 3, 9, 11, 12 | mp3an 1460 | . . . . . 6 ⊢ (2 ∥ ;10 → 2 ∥ (;10 · 𝐴)) |
14 | 7, 13 | ax-mp 5 | . . . . 5 ⊢ 2 ∥ (;10 · 𝐴) |
15 | dec2dvds.2 | . . . . . . . 8 ⊢ 𝐵 ∈ ℕ0 | |
16 | 15 | nn0zi 12639 | . . . . . . 7 ⊢ 𝐵 ∈ ℤ |
17 | dvdsmul2 16312 | . . . . . . 7 ⊢ ((𝐵 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (𝐵 · 2)) | |
18 | 16, 3, 17 | mp2an 692 | . . . . . 6 ⊢ 2 ∥ (𝐵 · 2) |
19 | dec2dvds.3 | . . . . . 6 ⊢ (𝐵 · 2) = 𝐶 | |
20 | 18, 19 | breqtri 5172 | . . . . 5 ⊢ 2 ∥ 𝐶 |
21 | 8, 10 | nn0mulcli 12561 | . . . . . . 7 ⊢ (;10 · 𝐴) ∈ ℕ0 |
22 | 21 | nn0zi 12639 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℤ |
23 | 2nn0 12540 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
24 | 15, 23 | nn0mulcli 12561 | . . . . . . . 8 ⊢ (𝐵 · 2) ∈ ℕ0 |
25 | 19, 24 | eqeltrri 2835 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 |
26 | 25 | nn0zi 12639 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
27 | dvds2add 16323 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ (;10 · 𝐴) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶))) | |
28 | 3, 22, 26, 27 | mp3an 1460 | . . . . 5 ⊢ ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶)) |
29 | 14, 20, 28 | mp2an 692 | . . . 4 ⊢ 2 ∥ ((;10 · 𝐴) + 𝐶) |
30 | dfdec10 12733 | . . . 4 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
31 | 29, 30 | breqtrri 5174 | . . 3 ⊢ 2 ∥ ;𝐴𝐶 |
32 | 10, 25 | deccl 12745 | . . . . 5 ⊢ ;𝐴𝐶 ∈ ℕ0 |
33 | 32 | nn0zi 12639 | . . . 4 ⊢ ;𝐴𝐶 ∈ ℤ |
34 | 2nn 12336 | . . . 4 ⊢ 2 ∈ ℕ | |
35 | 1lt2 12434 | . . . 4 ⊢ 1 < 2 | |
36 | ndvdsp1 16444 | . . . 4 ⊢ ((;𝐴𝐶 ∈ ℤ ∧ 2 ∈ ℕ ∧ 1 < 2) → (2 ∥ ;𝐴𝐶 → ¬ 2 ∥ (;𝐴𝐶 + 1))) | |
37 | 33, 34, 35, 36 | mp3an 1460 | . . 3 ⊢ (2 ∥ ;𝐴𝐶 → ¬ 2 ∥ (;𝐴𝐶 + 1)) |
38 | 31, 37 | ax-mp 5 | . 2 ⊢ ¬ 2 ∥ (;𝐴𝐶 + 1) |
39 | dec2dvds.4 | . . . . 5 ⊢ 𝐷 = (𝐶 + 1) | |
40 | 39 | eqcomi 2743 | . . . 4 ⊢ (𝐶 + 1) = 𝐷 |
41 | eqid 2734 | . . . 4 ⊢ ;𝐴𝐶 = ;𝐴𝐶 | |
42 | 10, 25, 40, 41 | decsuc 12761 | . . 3 ⊢ (;𝐴𝐶 + 1) = ;𝐴𝐷 |
43 | 42 | breq2i 5155 | . 2 ⊢ (2 ∥ (;𝐴𝐶 + 1) ↔ 2 ∥ ;𝐴𝐷) |
44 | 38, 43 | mtbi 322 | 1 ⊢ ¬ 2 ∥ ;𝐴𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 (class class class)co 7430 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 ℕcn 12263 2c2 12318 5c5 12321 ℕ0cn0 12523 ℤcz 12610 ;cdc 12730 ∥ cdvds 16286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 |
This theorem is referenced by: 11prm 17148 13prm 17149 17prm 17150 19prm 17151 23prm 17152 37prm 17154 43prm 17155 83prm 17156 139prm 17157 163prm 17158 317prm 17159 631prm 17160 257prm 47485 139prmALT 47520 31prm 47521 127prm 47523 |
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