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Mirrors > Home > MPE Home > Th. List > dec5dvds2 | Structured version Visualization version GIF version |
Description: Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
dec5dvds.1 | ⊢ 𝐴 ∈ ℕ0 |
dec5dvds.2 | ⊢ 𝐵 ∈ ℕ |
dec5dvds.3 | ⊢ 𝐵 < 5 |
dec5dvds2.4 | ⊢ (5 + 𝐵) = 𝐶 |
Ref | Expression |
---|---|
dec5dvds2 | ⊢ ¬ 5 ∥ ;𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dec5dvds.1 | . . 3 ⊢ 𝐴 ∈ ℕ0 | |
2 | dec5dvds.2 | . . 3 ⊢ 𝐵 ∈ ℕ | |
3 | dec5dvds.3 | . . 3 ⊢ 𝐵 < 5 | |
4 | 1, 2, 3 | dec5dvds 16403 | . 2 ⊢ ¬ 5 ∥ ;𝐴𝐵 |
5 | 5nn0 11920 | . . . . 5 ⊢ 5 ∈ ℕ0 | |
6 | 5 | nn0zi 12010 | . . . 4 ⊢ 5 ∈ ℤ |
7 | 2 | nnnn0i 11908 | . . . . . 6 ⊢ 𝐵 ∈ ℕ0 |
8 | 1, 7 | deccl 12116 | . . . . 5 ⊢ ;𝐴𝐵 ∈ ℕ0 |
9 | 8 | nn0zi 12010 | . . . 4 ⊢ ;𝐴𝐵 ∈ ℤ |
10 | dvdsadd 15655 | . . . 4 ⊢ ((5 ∈ ℤ ∧ ;𝐴𝐵 ∈ ℤ) → (5 ∥ ;𝐴𝐵 ↔ 5 ∥ (5 + ;𝐴𝐵))) | |
11 | 6, 9, 10 | mp2an 690 | . . 3 ⊢ (5 ∥ ;𝐴𝐵 ↔ 5 ∥ (5 + ;𝐴𝐵)) |
12 | 0nn0 11915 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
13 | 5 | dec0h 12123 | . . . . 5 ⊢ 5 = ;05 |
14 | eqid 2824 | . . . . 5 ⊢ ;𝐴𝐵 = ;𝐴𝐵 | |
15 | 1 | nn0cni 11912 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
16 | 15 | addid2i 10831 | . . . . 5 ⊢ (0 + 𝐴) = 𝐴 |
17 | dec5dvds2.4 | . . . . 5 ⊢ (5 + 𝐵) = 𝐶 | |
18 | 12, 5, 1, 7, 13, 14, 16, 17 | decadd 12155 | . . . 4 ⊢ (5 + ;𝐴𝐵) = ;𝐴𝐶 |
19 | 18 | breq2i 5077 | . . 3 ⊢ (5 ∥ (5 + ;𝐴𝐵) ↔ 5 ∥ ;𝐴𝐶) |
20 | 11, 19 | bitri 277 | . 2 ⊢ (5 ∥ ;𝐴𝐵 ↔ 5 ∥ ;𝐴𝐶) |
21 | 4, 20 | mtbi 324 | 1 ⊢ ¬ 5 ∥ ;𝐴𝐶 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 (class class class)co 7159 0cc0 10540 + caddc 10543 < clt 10678 ℕcn 11641 5c5 11698 ℕ0cn0 11900 ℤcz 11984 ;cdc 12101 ∥ cdvds 15610 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-inf 8910 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 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-fz 12896 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-dvds 15611 |
This theorem is referenced by: 37prm 16457 139prm 16460 317prm 16462 257prm 43730 139prmALT 43766 127prm 43770 |
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