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Mirrors > Home > MPE Home > Th. List > nndivtr | Structured version Visualization version GIF version |
Description: Transitive property of divisibility: if 𝐴 divides 𝐵 and 𝐵 divides 𝐶, then 𝐴 divides 𝐶. Typically, 𝐶 would be an integer, although the theorem holds for complex 𝐶. (Contributed by NM, 3-May-2005.) |
Ref | Expression |
---|---|
nndivtr | ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 / 𝐴) ∈ ℕ ∧ (𝐶 / 𝐵) ∈ ℕ)) → (𝐶 / 𝐴) ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnmulcl 11655 | . . 3 ⊢ (((𝐵 / 𝐴) ∈ ℕ ∧ (𝐶 / 𝐵) ∈ ℕ) → ((𝐵 / 𝐴) · (𝐶 / 𝐵)) ∈ ℕ) | |
2 | nncn 11640 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → 𝐵 ∈ ℂ) | |
3 | 2 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → 𝐵 ∈ ℂ) |
4 | simp3 1134 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → 𝐶 ∈ ℂ) | |
5 | nncn 11640 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ∈ ℂ) | |
6 | nnne0 11665 | . . . . . . . 8 ⊢ (𝐴 ∈ ℕ → 𝐴 ≠ 0) | |
7 | 5, 6 | jca 514 | . . . . . . 7 ⊢ (𝐴 ∈ ℕ → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
8 | 7 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) |
9 | nnne0 11665 | . . . . . . . 8 ⊢ (𝐵 ∈ ℕ → 𝐵 ≠ 0) | |
10 | 2, 9 | jca 514 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
11 | 10 | 3ad2ant2 1130 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) |
12 | divmul24 11338 | . . . . . 6 ⊢ (((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) ∧ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))) → ((𝐵 / 𝐴) · (𝐶 / 𝐵)) = ((𝐵 / 𝐵) · (𝐶 / 𝐴))) | |
13 | 3, 4, 8, 11, 12 | syl22anc 836 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → ((𝐵 / 𝐴) · (𝐶 / 𝐵)) = ((𝐵 / 𝐵) · (𝐶 / 𝐴))) |
14 | 2, 9 | dividd 11408 | . . . . . . 7 ⊢ (𝐵 ∈ ℕ → (𝐵 / 𝐵) = 1) |
15 | 14 | oveq1d 7165 | . . . . . 6 ⊢ (𝐵 ∈ ℕ → ((𝐵 / 𝐵) · (𝐶 / 𝐴)) = (1 · (𝐶 / 𝐴))) |
16 | 15 | 3ad2ant2 1130 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → ((𝐵 / 𝐵) · (𝐶 / 𝐴)) = (1 · (𝐶 / 𝐴))) |
17 | divcl 11298 | . . . . . . . . . 10 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐶 / 𝐴) ∈ ℂ) | |
18 | 17 | 3expb 1116 | . . . . . . . . 9 ⊢ ((𝐶 ∈ ℂ ∧ (𝐴 ∈ ℂ ∧ 𝐴 ≠ 0)) → (𝐶 / 𝐴) ∈ ℂ) |
19 | 7, 18 | sylan2 594 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℕ) → (𝐶 / 𝐴) ∈ ℂ) |
20 | 19 | ancoms 461 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℂ) → (𝐶 / 𝐴) ∈ ℂ) |
21 | 20 | mulid2d 10653 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℂ) → (1 · (𝐶 / 𝐴)) = (𝐶 / 𝐴)) |
22 | 21 | 3adant2 1127 | . . . . 5 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → (1 · (𝐶 / 𝐴)) = (𝐶 / 𝐴)) |
23 | 13, 16, 22 | 3eqtrd 2860 | . . . 4 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → ((𝐵 / 𝐴) · (𝐶 / 𝐵)) = (𝐶 / 𝐴)) |
24 | 23 | eleq1d 2897 | . . 3 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → (((𝐵 / 𝐴) · (𝐶 / 𝐵)) ∈ ℕ ↔ (𝐶 / 𝐴) ∈ ℕ)) |
25 | 1, 24 | syl5ib 246 | . 2 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) → (((𝐵 / 𝐴) ∈ ℕ ∧ (𝐶 / 𝐵) ∈ ℕ) → (𝐶 / 𝐴) ∈ ℕ)) |
26 | 25 | imp 409 | 1 ⊢ (((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℂ) ∧ ((𝐵 / 𝐴) ∈ ℕ ∧ (𝐶 / 𝐵) ∈ ℕ)) → (𝐶 / 𝐴) ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 (class class class)co 7150 ℂcc 10529 0cc0 10531 1c1 10532 · cmul 10536 / cdiv 11291 ℕcn 11632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 |
This theorem is referenced by: permnn 13680 infpnlem1 16240 |
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