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Theorem lcmfunsnlem1 16602
Description: Lemma for lcmfdvds 16607 and lcmfunsnlem 16606 (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 1909 . . 3 𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2 nfra1 3272 . . . 4 𝑘𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)
3 nfv 1909 . . . 4 𝑘𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)
42, 3nfan 1894 . . 3 𝑘(∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))
51, 4nfan 1894 . 2 𝑘((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))
6 breq2 5148 . . . . . . . 8 (𝑘 = 𝑙 → (𝑚𝑘𝑚𝑙))
76ralbidv 3168 . . . . . . 7 (𝑘 = 𝑙 → (∀𝑚𝑦 𝑚𝑘 ↔ ∀𝑚𝑦 𝑚𝑙))
8 breq2 5148 . . . . . . 7 (𝑘 = 𝑙 → ((lcm𝑦) ∥ 𝑘 ↔ (lcm𝑦) ∥ 𝑙))
97, 8imbi12d 343 . . . . . 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 5148 . . . . . . . . . 10 (𝑙 = 𝑘 → (𝑚𝑙𝑚𝑘))
1211ralbidv 3168 . . . . . . . . 9 (𝑙 = 𝑘 → (∀𝑚𝑦 𝑚𝑙 ↔ ∀𝑚𝑦 𝑚𝑘))
13 breq2 5148 . . . . . . . . 9 (𝑙 = 𝑘 → ((lcm𝑦) ∥ 𝑙 ↔ (lcm𝑦) ∥ 𝑘))
1412, 13imbi12d 343 . . . . . . . 8 (𝑙 = 𝑘 → ((∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) ↔ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
1514rspcv 3599 . . . . . . 7 (𝑘 ∈ ℤ → (∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) → (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
1615adantl 480 . . . . . 6 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) → (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
17 sneq 4635 . . . . . . . . . . . . 13 (𝑛 = 𝑧 → {𝑛} = {𝑧})
1817uneq2d 4157 . . . . . . . . . . . 12 (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
1918fveq2d 6894 . . . . . . . . . . 11 (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
20 oveq2 7421 . . . . . . . . . . 11 (𝑛 = 𝑧 → ((lcm𝑦) lcm 𝑛) = ((lcm𝑦) lcm 𝑧))
2119, 20eqeq12d 2741 . . . . . . . . . 10 (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
2221rspcv 3599 . . . . . . . . 9 (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
23223ad2ant1 1130 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
2423adantr 479 . . . . . . 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 483 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
26 lcmfcl 16593 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
2726nn0zd 12609 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
28273adant1 1127 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
2928adantr 479 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (lcm𝑦) ∈ ℤ)
30 simpl1 1188 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑧 ∈ ℤ)
3125, 29, 303jca 1125 . . . . . . . . . . . . . 14 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
3231adantr 479 . . . . . . . . . . . . 13 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
3332adantr 479 . . . . . . . . . . . 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 4167 . . . . . . . . . . . . . . . 16 𝑦 ⊆ (𝑦 ∪ {𝑧})
35 ssralv 4042 . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . 13 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → (lcm𝑦) ∥ 𝑘)
39 snidg 4659 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℤ → 𝑧 ∈ {𝑧})
4039olcd 872 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℤ → (𝑧𝑦𝑧 ∈ {𝑧}))
41 elun 4142 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧𝑦𝑧 ∈ {𝑧}))
4240, 41sylibr 233 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℤ → 𝑧 ∈ (𝑦 ∪ {𝑧}))
43 breq1 5147 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑧 → (𝑚𝑘𝑧𝑘))
4443rspcv 3599 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4542, 44syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
46453ad2ant1 1130 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4746adantr 479 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4847adantr 479 . . . . . . . . . . . . . 14 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4948imp 405 . . . . . . . . . . . . 13 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → 𝑧𝑘)
5038, 49jca 510 . . . . . . . . . . . 12 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → ((lcm𝑦) ∥ 𝑘𝑧𝑘))
51 lcmdvds 16573 . . . . . . . . . . . 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 5147 . . . . . . . . . . 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 411 . . . . . . . . 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 411 . . . . . . 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 409 . . 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 450 . 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 3245 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 394  wo 845  w3a 1084   = wceq 1533  wcel 2098  wral 3051  cun 3939  wss 3941  {csn 4625   class class class wbr 5144  cfv 6543  (class class class)co 7413  Fincfn 8957  cz 12583  cdvds 16225   lcm clcm 16553  lcmclcmf 16554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-inf2 9659  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460  df-inf 9461  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-n0 12498  df-z 12584  df-uz 12848  df-rp 13002  df-fz 13512  df-fzo 13655  df-fl 13784  df-mod 13862  df-seq 13994  df-exp 14054  df-hash 14317  df-cj 15073  df-re 15074  df-im 15075  df-sqrt 15209  df-abs 15210  df-clim 15459  df-prod 15877  df-dvds 16226  df-gcd 16464  df-lcm 16555  df-lcmf 16556
This theorem is referenced by:  lcmfunsnlem  16606
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