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Theorem lcmfunsnlem1 16518
Description: Lemma for lcmfdvds 16523 and lcmfunsnlem 16522 (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 1918 . . 3 𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
2 nfra1 3266 . . . 4 𝑘𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)
3 nfv 1918 . . . 4 𝑘𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)
42, 3nfan 1903 . . 3 𝑘(∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))
51, 4nfan 1903 . 2 𝑘((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))
6 breq2 5110 . . . . . . . 8 (𝑘 = 𝑙 → (𝑚𝑘𝑚𝑙))
76ralbidv 3171 . . . . . . 7 (𝑘 = 𝑙 → (∀𝑚𝑦 𝑚𝑘 ↔ ∀𝑚𝑦 𝑚𝑙))
8 breq2 5110 . . . . . . 7 (𝑘 = 𝑙 → ((lcm𝑦) ∥ 𝑘 ↔ (lcm𝑦) ∥ 𝑙))
97, 8imbi12d 345 . . . . . 6 (𝑘 = 𝑙 → ((∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ↔ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙)))
109cbvralvw 3224 . . . . 5 (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ↔ ∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙))
11 breq2 5110 . . . . . . . . . 10 (𝑙 = 𝑘 → (𝑚𝑙𝑚𝑘))
1211ralbidv 3171 . . . . . . . . 9 (𝑙 = 𝑘 → (∀𝑚𝑦 𝑚𝑙 ↔ ∀𝑚𝑦 𝑚𝑘))
13 breq2 5110 . . . . . . . . 9 (𝑙 = 𝑘 → ((lcm𝑦) ∥ 𝑙 ↔ (lcm𝑦) ∥ 𝑘))
1412, 13imbi12d 345 . . . . . . . 8 (𝑙 = 𝑘 → ((∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) ↔ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
1514rspcv 3576 . . . . . . 7 (𝑘 ∈ ℤ → (∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) → (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
1615adantl 483 . . . . . 6 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) → (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
17 sneq 4597 . . . . . . . . . . . . 13 (𝑛 = 𝑧 → {𝑛} = {𝑧})
1817uneq2d 4124 . . . . . . . . . . . 12 (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
1918fveq2d 6847 . . . . . . . . . . 11 (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
20 oveq2 7366 . . . . . . . . . . 11 (𝑛 = 𝑧 → ((lcm𝑦) lcm 𝑛) = ((lcm𝑦) lcm 𝑧))
2119, 20eqeq12d 2749 . . . . . . . . . 10 (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
2221rspcv 3576 . . . . . . . . 9 (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
23223ad2ant1 1134 . . . . . . . 8 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
2423adantr 482 . . . . . . 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 486 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
26 lcmfcl 16509 . . . . . . . . . . . . . . . . . 18 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
2726nn0zd 12530 . . . . . . . . . . . . . . . . 17 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
28273adant1 1131 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
2928adantr 482 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (lcm𝑦) ∈ ℤ)
30 simpl1 1192 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → 𝑧 ∈ ℤ)
3125, 29, 303jca 1129 . . . . . . . . . . . . . 14 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
3231adantr 482 . . . . . . . . . . . . 13 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
3332adantr 482 . . . . . . . . . . . 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 4133 . . . . . . . . . . . . . . . 16 𝑦 ⊆ (𝑦 ∪ {𝑧})
35 ssralv 4011 . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . 13 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → (lcm𝑦) ∥ 𝑘)
39 snidg 4621 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ ℤ → 𝑧 ∈ {𝑧})
4039olcd 873 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℤ → (𝑧𝑦𝑧 ∈ {𝑧}))
41 elun 4109 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧𝑦𝑧 ∈ {𝑧}))
4240, 41sylibr 233 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ ℤ → 𝑧 ∈ (𝑦 ∪ {𝑧}))
43 breq1 5109 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑧 → (𝑚𝑘𝑧𝑘))
4443rspcv 3576 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ (𝑦 ∪ {𝑧}) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4542, 44syl 17 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
46453ad2ant1 1134 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4746adantr 482 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4847adantr 482 . . . . . . . . . . . . . 14 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) → (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘𝑧𝑘))
4948imp 408 . . . . . . . . . . . . 13 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → 𝑧𝑘)
5038, 49jca 513 . . . . . . . . . . . 12 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑘 ∈ ℤ) ∧ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘) → ((lcm𝑦) ∥ 𝑘𝑧𝑘))
51 lcmdvds 16489 . . . . . . . . . . . 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 5109 . . . . . . . . . . 11 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ↔ ((lcm𝑦) lcm 𝑧) ∥ 𝑘))
5452, 53syl5ibrcom 247 . . . . . . . . . 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 414 . . . . . . . . 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 414 . . . . . . 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 412 . . 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 453 . 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 3239 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 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3061  cun 3909  wss 3911  {csn 4587   class class class wbr 5106  cfv 6497  (class class class)co 7358  Fincfn 8886  cz 12504  cdvds 16141   lcm clcm 16469  lcmclcmf 16470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9582  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-n0 12419  df-z 12505  df-uz 12769  df-rp 12921  df-fz 13431  df-fzo 13574  df-fl 13703  df-mod 13781  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-clim 15376  df-prod 15794  df-dvds 16142  df-gcd 16380  df-lcm 16471  df-lcmf 16472
This theorem is referenced by:  lcmfunsnlem  16522
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