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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 11189 | . . . . . . . . 9 ⊢ 5 ∈ ℕ0 | |
2 | 1 | nn0zi 11279 | . . . . . . . 8 ⊢ 5 ∈ ℤ |
3 | 2z 11286 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
4 | dvdsmul2 14842 | . . . . . . . 8 ⊢ ((5 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (5 · 2)) | |
5 | 2, 3, 4 | mp2an 704 | . . . . . . 7 ⊢ 2 ∥ (5 · 2) |
6 | 5t2e10 11510 | . . . . . . 7 ⊢ (5 · 2) = ;10 | |
7 | 5, 6 | breqtri 4608 | . . . . . 6 ⊢ 2 ∥ ;10 |
8 | 10nn0 11392 | . . . . . . . 8 ⊢ ;10 ∈ ℕ0 | |
9 | 8 | nn0zi 11279 | . . . . . . 7 ⊢ ;10 ∈ ℤ |
10 | dec2dvds.1 | . . . . . . . 8 ⊢ 𝐴 ∈ ℕ0 | |
11 | 10 | nn0zi 11279 | . . . . . . 7 ⊢ 𝐴 ∈ ℤ |
12 | dvdsmultr1 14857 | . . . . . . 7 ⊢ ((2 ∈ ℤ ∧ ;10 ∈ ℤ ∧ 𝐴 ∈ ℤ) → (2 ∥ ;10 → 2 ∥ (;10 · 𝐴))) | |
13 | 3, 9, 11, 12 | mp3an 1416 | . . . . . 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 11279 | . . . . . . 7 ⊢ 𝐵 ∈ ℤ |
17 | dvdsmul2 14842 | . . . . . . 7 ⊢ ((𝐵 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (𝐵 · 2)) | |
18 | 16, 3, 17 | mp2an 704 | . . . . . 6 ⊢ 2 ∥ (𝐵 · 2) |
19 | dec2dvds.3 | . . . . . 6 ⊢ (𝐵 · 2) = 𝐶 | |
20 | 18, 19 | breqtri 4608 | . . . . 5 ⊢ 2 ∥ 𝐶 |
21 | 8, 10 | nn0mulcli 11208 | . . . . . . 7 ⊢ (;10 · 𝐴) ∈ ℕ0 |
22 | 21 | nn0zi 11279 | . . . . . 6 ⊢ (;10 · 𝐴) ∈ ℤ |
23 | 2nn0 11186 | . . . . . . . . 9 ⊢ 2 ∈ ℕ0 | |
24 | 15, 23 | nn0mulcli 11208 | . . . . . . . 8 ⊢ (𝐵 · 2) ∈ ℕ0 |
25 | 19, 24 | eqeltrri 2685 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 |
26 | 25 | nn0zi 11279 | . . . . . 6 ⊢ 𝐶 ∈ ℤ |
27 | dvds2add 14853 | . . . . . 6 ⊢ ((2 ∈ ℤ ∧ (;10 · 𝐴) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶))) | |
28 | 3, 22, 26, 27 | mp3an 1416 | . . . . 5 ⊢ ((2 ∥ (;10 · 𝐴) ∧ 2 ∥ 𝐶) → 2 ∥ ((;10 · 𝐴) + 𝐶)) |
29 | 14, 20, 28 | mp2an 704 | . . . 4 ⊢ 2 ∥ ((;10 · 𝐴) + 𝐶) |
30 | dfdec10 11373 | . . . 4 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
31 | 29, 30 | breqtrri 4610 | . . 3 ⊢ 2 ∥ ;𝐴𝐶 |
32 | 10, 25 | deccl 11388 | . . . . 5 ⊢ ;𝐴𝐶 ∈ ℕ0 |
33 | 32 | nn0zi 11279 | . . . 4 ⊢ ;𝐴𝐶 ∈ ℤ |
34 | 2nn 11062 | . . . 4 ⊢ 2 ∈ ℕ | |
35 | 1lt2 11071 | . . . 4 ⊢ 1 < 2 | |
36 | ndvdsp1 14973 | . . . 4 ⊢ ((;𝐴𝐶 ∈ ℤ ∧ 2 ∈ ℕ ∧ 1 < 2) → (2 ∥ ;𝐴𝐶 → ¬ 2 ∥ (;𝐴𝐶 + 1))) | |
37 | 33, 34, 35, 36 | mp3an 1416 | . . 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 2619 | . . . 4 ⊢ (𝐶 + 1) = 𝐷 |
41 | eqid 2610 | . . . 4 ⊢ ;𝐴𝐶 = ;𝐴𝐶 | |
42 | 10, 25, 40, 41 | decsuc 11411 | . . 3 ⊢ (;𝐴𝐶 + 1) = ;𝐴𝐷 |
43 | 42 | breq2i 4591 | . 2 ⊢ (2 ∥ (;𝐴𝐶 + 1) ↔ 2 ∥ ;𝐴𝐷) |
44 | 38, 43 | mtbi 311 | 1 ⊢ ¬ 2 ∥ ;𝐴𝐷 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ℕcn 10897 2c2 10947 5c5 10950 ℕ0cn0 11169 ℤcz 11254 ;cdc 11369 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 |
This theorem is referenced by: 11prm 15660 13prm 15661 17prm 15662 19prm 15663 23prm 15664 37prm 15666 43prm 15667 83prm 15668 139prm 15669 163prm 15670 317prm 15671 631prm 15672 257prm 40011 139prmALT 40049 31prm 40050 127prm 40053 |
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