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Mirrors > Home > MPE Home > Th. List > ex-dvds | Structured version Visualization version GIF version |
Description: Example for df-dvds 15610: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.) |
Ref | Expression |
---|---|
ex-dvds | ⊢ 3 ∥ 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 12017 | . . 3 ⊢ 2 ∈ ℤ | |
2 | 3z 12018 | . . 3 ⊢ 3 ∈ ℤ | |
3 | 6nn 11729 | . . . 4 ⊢ 6 ∈ ℕ | |
4 | 3 | nnzi 12009 | . . 3 ⊢ 6 ∈ ℤ |
5 | 1, 2, 4 | 3pm3.2i 1335 | . 2 ⊢ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) |
6 | 3cn 11721 | . . . 4 ⊢ 3 ∈ ℂ | |
7 | 6 | 2timesi 11778 | . . 3 ⊢ (2 · 3) = (3 + 3) |
8 | 3p3e6 11792 | . . 3 ⊢ (3 + 3) = 6 | |
9 | 7, 8 | eqtri 2846 | . 2 ⊢ (2 · 3) = 6 |
10 | dvds0lem 15622 | . 2 ⊢ (((2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) ∧ (2 · 3) = 6) → 3 ∥ 6) | |
11 | 5, 9, 10 | mp2an 690 | 1 ⊢ 3 ∥ 6 |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 (class class class)co 7158 + caddc 10542 · cmul 10544 2c2 11695 3c3 11696 6c6 11699 ℤcz 11984 ∥ cdvds 15609 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rrecex 10611 ax-cnre 10612 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-z 11985 df-dvds 15610 |
This theorem is referenced by: (None) |
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