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Theorem lcmfunsnlem1 16513
Description: Lemma for lcmfdvds 16518 and lcmfunsnlem 16517 (Induction step part 1). (Contributed by AV, 25-Aug-2020.)
Assertion
Ref Expression
lcmfunsnlem1 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → ∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
Distinct variable groups:   𝑦,𝑚,𝑧   𝑘,𝑛,𝑦,𝑧   𝑘,𝑚

Proof of Theorem lcmfunsnlem1
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 nfv 1917 . . 3 𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2 nfra1 3267 . . . 4 𝑘𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)
3 nfv 1917 . . . 4 𝑘𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)
42, 3nfan 1902 . . 3 𝑘(∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))
51, 4nfan 1902 . 2 𝑘((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))
6 breq2 5109 . . . . . . . 8 (𝑘 = 𝑙 → (𝑚𝑘𝑚𝑙))
76ralbidv 3174 . . . . . . 7 (𝑘 = 𝑙 → (∀𝑚𝑦 𝑚𝑘 ↔ ∀𝑚𝑦 𝑚𝑙))
8 breq2 5109 . . . . . . 7 (𝑘 = 𝑙 → ((lcm𝑦) ∥ 𝑘 ↔ (lcm𝑦) ∥ 𝑙))
97, 8imbi12d 344 . . . . . 6 (𝑘 = 𝑙 → ((∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ↔ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙)))
109cbvralvw 3225 . . . . 5 (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ↔ ∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙))
11 breq2 5109 . . . . . . . . . 10 (𝑙 = 𝑘 → (𝑚𝑙𝑚𝑘))
1211ralbidv 3174 . . . . . . . . 9 (𝑙 = 𝑘 → (∀𝑚𝑦 𝑚𝑙 ↔ ∀𝑚𝑦 𝑚𝑘))
13 breq2 5109 . . . . . . . . 9 (𝑙 = 𝑘 → ((lcm𝑦) ∥ 𝑙 ↔ (lcm𝑦) ∥ 𝑘))
1412, 13imbi12d 344 . . . . . . . 8 (𝑙 = 𝑘 → ((∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) ↔ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
1514rspcv 3577 . . . . . . 7 (𝑘 ∈ ℤ → (∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) → (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
1615adantl 482 . . . . . 6 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) → (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
17 sneq 4596 . . . . . . . . . . . . 13 (𝑛 = 𝑧 → {𝑛} = {𝑧})
1817uneq2d 4123 . . . . . . . . . . . 12 (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
1918fveq2d 6846 . . . . . . . . . . 11 (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
20 oveq2 7365 . . . . . . . . . . 11 (𝑛 = 𝑧 → ((lcm𝑦) lcm 𝑛) = ((lcm𝑦) lcm 𝑧))
2119, 20eqeq12d 2752 . . . . . . . . . 10 (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
2221rspcv 3577 . . . . . . . . 9 (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
23223ad2ant1 1133 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
2423adantr 481 . . . . . . 7 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
25 simpr 485 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
26 lcmfcl 16504 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
2726nn0zd 12525 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
28273adant1 1130 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
2928adantr 481 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (lcm𝑦) ∈ ℤ)
30 simpl1 1191 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑧 ∈ ℤ)
3125, 29, 303jca 1128 . . . . . . . . . . . . . 14 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
3231adantr 481 . . . . . . . . . . . . 13 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
3332adantr 481 . . . . . . . . . . . 12 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
34 ssun1 4132 . . . . . . . . . . . . . . . 16 𝑦 ⊆ (𝑦 ∪ {𝑧})
35 ssralv 4010 . . . . . . . . . . . . . . . 16 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → ∀𝑚𝑦 𝑚𝑘))
3634, 35mp1i 13 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → ∀𝑚𝑦 𝑚𝑘))
3736imim1d 82 . . . . . . . . . . . . . 14 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → ((∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
3837imp31 418 . . . . . . . . . . . . 13 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → (lcm𝑦) ∥ 𝑘)
39 snidg 4620 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℤ → 𝑧 ∈ {𝑧})
4039olcd 872 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℤ → (𝑧𝑦𝑧 ∈ {𝑧}))
41 elun 4108 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧𝑦𝑧 ∈ {𝑧}))
4240, 41sylibr 233 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℤ → 𝑧 ∈ (𝑦 ∪ {𝑧}))
43 breq1 5108 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑧 → (𝑚𝑘𝑧𝑘))
4443rspcv 3577 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4542, 44syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
46453ad2ant1 1133 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4746adantr 481 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4847adantr 481 . . . . . . . . . . . . . 14 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4948imp 407 . . . . . . . . . . . . 13 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → 𝑧𝑘)
5038, 49jca 512 . . . . . . . . . . . 12 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → ((lcm𝑦) ∥ 𝑘𝑧𝑘))
51 lcmdvds 16484 . . . . . . . . . . . 12 ((𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm𝑦) ∥ 𝑘𝑧𝑘) → ((lcm𝑦) lcm 𝑧) ∥ 𝑘))
5233, 50, 51sylc 65 . . . . . . . . . . 11 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → ((lcm𝑦) lcm 𝑧) ∥ 𝑘)
53 breq1 5108 . . . . . . . . . . 11 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ↔ ((lcm𝑦) lcm 𝑧) ∥ 𝑘))
5452, 53syl5ibrcom 246 . . . . . . . . . 10 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
5554ex 413 . . . . . . . . 9 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
5655com23 86 . . . . . . . 8 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
5756ex 413 . . . . . . 7 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → ((∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
5824, 57syl5d 73 . . . . . 6 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → ((∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
5916, 58syld 47 . . . . 5 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
6010, 59biimtrid 241 . . . 4 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
6160impd 411 . . 3 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → ((∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
6261impancom 452 . 2 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → (𝑘 ∈ ℤ → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
635, 62ralrimi 3240 1 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → ∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wral 3064  cun 3908  wss 3910  {csn 4586   class class class wbr 5105  cfv 6496  (class class class)co 7357  Fincfn 8883  cz 12499  cdvds 16136   lcm clcm 16464  lcmclcmf 16465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fzo 13568  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-prod 15789  df-dvds 16137  df-gcd 16375  df-lcm 16466  df-lcmf 16467
This theorem is referenced by:  lcmfunsnlem  16517
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