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Mirrors > Home > ILE Home > Th. List > dvdsmul1 | GIF version |
Description: An integer divides a multiple of itself. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
dvdsmul1 | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zcn 9083 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ) | |
2 | zcn 9083 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
3 | mulcom 7773 | . . 3 ⊢ ((𝑁 ∈ ℂ ∧ 𝑀 ∈ ℂ) → (𝑁 · 𝑀) = (𝑀 · 𝑁)) | |
4 | 1, 2, 3 | syl2anr 288 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 · 𝑀) = (𝑀 · 𝑁)) |
5 | zmulcl 9131 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
6 | dvds0lem 11539 | . . . . 5 ⊢ (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) ∧ (𝑁 · 𝑀) = (𝑀 · 𝑁)) → 𝑀 ∥ (𝑀 · 𝑁)) | |
7 | 6 | ex 114 | . . . 4 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → ((𝑁 · 𝑀) = (𝑀 · 𝑁) → 𝑀 ∥ (𝑀 · 𝑁))) |
8 | 7 | 3com12 1186 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → ((𝑁 · 𝑀) = (𝑀 · 𝑁) → 𝑀 ∥ (𝑀 · 𝑁))) |
9 | 5, 8 | mpd3an3 1317 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 · 𝑀) = (𝑀 · 𝑁) → 𝑀 ∥ (𝑀 · 𝑁))) |
10 | 4, 9 | mpd 13 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 · 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 = wceq 1332 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 ℂcc 7642 · cmul 7649 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-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-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-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
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-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-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 df-dvds 11530 |
This theorem is referenced by: dvdsmultr1 11567 3dvdsdec 11598 3dvds2dec 11599 2teven 11620 opoe 11628 omoe 11629 z4even 11649 ndvdsi 11666 mulgcd 11740 dvdsmulgcd 11749 lcmval 11780 lcmcllem 11784 lcmgcdlem 11794 qredeq 11813 cncongr2 11821 nprm 11840 exprmfct 11854 evenennn 11942 |
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