![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ex-dvds | GIF version |
Description: Example for df-dvds 11530: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.) |
Ref | Expression |
---|---|
ex-dvds | ⊢ 3 ∥ 6 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2z 9106 | . . 3 ⊢ 2 ∈ ℤ | |
2 | 3z 9107 | . . 3 ⊢ 3 ∈ ℤ | |
3 | 6nn 8909 | . . . 4 ⊢ 6 ∈ ℕ | |
4 | 3 | nnzi 9099 | . . 3 ⊢ 6 ∈ ℤ |
5 | 1, 2, 4 | 3pm3.2i 1160 | . 2 ⊢ (2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) |
6 | 3cn 8819 | . . . 4 ⊢ 3 ∈ ℂ | |
7 | 6 | 2timesi 8874 | . . 3 ⊢ (2 · 3) = (3 + 3) |
8 | 3p3e6 8886 | . . 3 ⊢ (3 + 3) = 6 | |
9 | 7, 8 | eqtri 2161 | . 2 ⊢ (2 · 3) = 6 |
10 | dvds0lem 11539 | . 2 ⊢ (((2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 6 ∈ ℤ) ∧ (2 · 3) = 6) → 3 ∥ 6) | |
11 | 5, 9, 10 | mp2an 423 | 1 ⊢ 3 ∥ 6 |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 963 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 + caddc 7647 · cmul 7649 2c2 8795 3c3 8796 6c6 8799 ℤcz 9078 ∥ cdvds 11529 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-z 9079 df-dvds 11530 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |