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Theorem lcmfunsnlem2lem1 16514
Description: Lemma 1 for lcmfunsnlem2 16516. (Contributed by AV, 26-Aug-2020.)
Assertion
Ref Expression
lcmfunsnlem2lem1 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) → ∀𝑘 ∈ ℕ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))
Distinct variable groups:   𝑦,𝑚,𝑧   𝑘,𝑛,𝑦,𝑧,𝑚,𝑖

Proof of Theorem lcmfunsnlem2lem1
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 nfv 1917 . . 3 𝑘(0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)
2 nfv 1917 . . . 4 𝑘 𝑛 ∈ ℤ
3 nfv 1917 . . . . 5 𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
4 nfra1 3267 . . . . . 6 𝑘𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)
5 nfv 1917 . . . . . 6 𝑘𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)
64, 5nfan 1902 . . . . 5 𝑘(∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))
73, 6nfan 1902 . . . 4 𝑘((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))
82, 7nfan 1902 . . 3 𝑘(𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))
91, 8nfan 1902 . 2 𝑘((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))))
10 simprr 771 . . . . . . . . . . . 12 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ)
11 simp2 1137 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
12 snssi 4768 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
13123ad2ant1 1133 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
1411, 13unssd 4146 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
15 simp3 1138 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
16 snfi 8988 . . . . . . . . . . . . . . . . . 18 {𝑧} ∈ Fin
17 unfi 9116 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
1815, 16, 17sylancl 586 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
1914, 18jca 512 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin))
20 lcmfcl 16504 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ0)
2119, 20syl 17 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ0)
2221nn0zd 12525 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
2322adantl 482 . . . . . . . . . . . . 13 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
2423adantr 481 . . . . . . . . . . . 12 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
25 simprl 769 . . . . . . . . . . . 12 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑛 ∈ ℤ)
2610, 24, 253jca 1128 . . . . . . . . . . 11 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
2714adantl 482 . . . . . . . . . . . . . . . . . 18 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
2818adantl 482 . . . . . . . . . . . . . . . . . 18 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → (𝑦 ∪ {𝑧}) ∈ Fin)
29 df-nel 3050 . . . . . . . . . . . . . . . . . . . . . . 23 (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
3029biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 (0 ∉ 𝑦 → ¬ 0 ∈ 𝑦)
31 elsni 4603 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ∈ {𝑧} → 0 = 𝑧)
3231eqcomd 2742 . . . . . . . . . . . . . . . . . . . . . . 23 (0 ∈ {𝑧} → 𝑧 = 0)
3332necon3ai 2968 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ≠ 0 → ¬ 0 ∈ {𝑧})
3430, 33anim12i 613 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∉ 𝑦𝑧 ≠ 0) → (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
35343adant3 1132 . . . . . . . . . . . . . . . . . . . 20 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
36 df-nel 3050 . . . . . . . . . . . . . . . . . . . . 21 (0 ∉ (𝑦 ∪ {𝑧}) ↔ ¬ 0 ∈ (𝑦 ∪ {𝑧}))
37 ioran 982 . . . . . . . . . . . . . . . . . . . . . 22 (¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
38 elun 4108 . . . . . . . . . . . . . . . . . . . . . 22 (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
3937, 38xchnxbir 332 . . . . . . . . . . . . . . . . . . . . 21 (¬ 0 ∈ (𝑦 ∪ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
4036, 39bitri 274 . . . . . . . . . . . . . . . . . . . 20 (0 ∉ (𝑦 ∪ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
4135, 40sylibr 233 . . . . . . . . . . . . . . . . . . 19 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ (𝑦 ∪ {𝑧}))
4241adantr 481 . . . . . . . . . . . . . . . . . 18 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → 0 ∉ (𝑦 ∪ {𝑧}))
4327, 28, 423jca 1128 . . . . . . . . . . . . . . . . 17 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) → ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})))
4443adantr 481 . . . . . . . . . . . . . . . 16 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})))
45 lcmfn0cl 16502 . . . . . . . . . . . . . . . 16 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
4644, 45syl 17 . . . . . . . . . . . . . . 15 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
4746nnne0d 12203 . . . . . . . . . . . . . 14 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘(𝑦 ∪ {𝑧})) ≠ 0)
4847neneqd 2948 . . . . . . . . . . . . 13 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ¬ (lcm‘(𝑦 ∪ {𝑧})) = 0)
49 neneq 2949 . . . . . . . . . . . . . . 15 (𝑛 ≠ 0 → ¬ 𝑛 = 0)
50493ad2ant3 1135 . . . . . . . . . . . . . 14 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 𝑛 = 0)
5150ad2antrr 724 . . . . . . . . . . . . 13 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ¬ 𝑛 = 0)
5248, 51jca 512 . . . . . . . . . . . 12 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
53 ioran 982 . . . . . . . . . . . 12 (¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0) ↔ (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
5452, 53sylibr 233 . . . . . . . . . . 11 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))
5526, 54jca 512 . . . . . . . . . 10 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)))
5655exp43 437 . . . . . . . . 9 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 ∈ ℤ → (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))))))
5756adantrd 492 . . . . . . . 8 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → (𝑛 ∈ ℤ → (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))))))
5857com23 86 . . . . . . 7 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (𝑛 ∈ ℤ → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))))))
5958imp32 419 . . . . . 6 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) → (𝑘 ∈ ℕ → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))))
6059imp 407 . . . . 5 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) ∧ 𝑘 ∈ ℕ) → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)))
6160adantr 481 . . . 4 (((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) ∧ 𝑘 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → ((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)))
62 sneq 4596 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑧 → {𝑛} = {𝑧})
6362uneq2d 4123 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
6463fveq2d 6846 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
65 oveq2 7365 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → ((lcm𝑦) lcm 𝑛) = ((lcm𝑦) lcm 𝑧))
6664, 65eqeq12d 2752 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
6766rspcv 3577 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
68673ad2ant1 1133 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
69 nnz 12520 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
7069adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℤ)
7170adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℤ)
72 lcmfcl 16504 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
7372nn0zd 12525 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
74733adant1 1130 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
7574ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm𝑦) ∈ ℤ)
76 simpll1 1212 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑧 ∈ ℤ)
7771, 75, 763jca 1128 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
7877ad2antrr 724 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
79 elun1 4136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑚𝑦𝑚 ∈ (𝑦 ∪ {𝑧}))
8079orcd 871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑚𝑦 → (𝑚 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑚 ∈ {𝑛}))
81 elun 4108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑚 ∈ {𝑛}))
8280, 81sylibr 233 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚𝑦𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
83 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑖 = 𝑚 → (𝑖𝑘𝑚𝑘))
8483rspcv 3577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘𝑚𝑘))
8582, 84syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑚𝑦 → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘𝑚𝑘))
8685com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (𝑚𝑦𝑚𝑘))
8786adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → (𝑚𝑦𝑚𝑘))
8887ralrimiv 3142 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → ∀𝑚𝑦 𝑚𝑘)
8988adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → ∀𝑚𝑦 𝑚𝑘)
90 breq2 5109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑙 → (𝑚𝑘𝑚𝑙))
9190ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑙 → (∀𝑚𝑦 𝑚𝑘 ↔ ∀𝑚𝑦 𝑚𝑙))
92 breq2 5109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑙 → ((lcm𝑦) ∥ 𝑘 ↔ (lcm𝑦) ∥ 𝑙))
9391, 92imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 = 𝑙 → ((∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ↔ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙)))
9493cbvralvw 3225 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ↔ ∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙))
9570adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → 𝑘 ∈ ℤ)
9695adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → 𝑘 ∈ ℤ)
97 breq2 5109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑙 = 𝑘 → (𝑚𝑙𝑚𝑘))
9897ralbidv 3174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑙 = 𝑘 → (∀𝑚𝑦 𝑚𝑙 ↔ ∀𝑚𝑦 𝑚𝑘))
99 breq2 5109 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑙 = 𝑘 → ((lcm𝑦) ∥ 𝑙 ↔ (lcm𝑦) ∥ 𝑘))
10098, 99imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑙 = 𝑘 → ((∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) ↔ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
101100rspcv 3577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 ∈ ℤ → (∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) → (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
10296, 101syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → (∀𝑙 ∈ ℤ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) → (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
10394, 102biimtrid 241 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
10489, 103mpid 44 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) ∧ ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0))) → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → (lcm𝑦) ∥ 𝑘))
105104exp31 420 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → (lcm𝑦) ∥ 𝑘))))
106105com24 95 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → (((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm𝑦) ∥ 𝑘))))
107106imp 407 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) → (((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm𝑦) ∥ 𝑘)))
108107impl 456 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm𝑦) ∥ 𝑘))
109108imp 407 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → (lcm𝑦) ∥ 𝑘)
110 vsnid 4623 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑧 ∈ {𝑧}
111110olci 864 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧𝑦𝑧 ∈ {𝑧})
112 elun 4108 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧𝑦𝑧 ∈ {𝑧}))
113111, 112mpbir 230 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑧 ∈ (𝑦 ∪ {𝑧})
114113orci 863 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑧 ∈ {𝑛})
115 elun 4108 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑧 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑧 ∈ {𝑛}))
116114, 115mpbir 230 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})
117 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑧 → (𝑖𝑘𝑧𝑘))
118117rspcv 3577 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘𝑧𝑘))
119116, 118mp1i 13 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘𝑧𝑘))
120119imp 407 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → 𝑧𝑘)
121109, 120jca 512 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → ((lcm𝑦) ∥ 𝑘𝑧𝑘))
122 lcmdvds 16484 . . . . . . . . . . . . . . . . . . . . 21 ((𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → (((lcm𝑦) ∥ 𝑘𝑧𝑘) → ((lcm𝑦) lcm 𝑧) ∥ 𝑘))
12378, 121, 122sylc 65 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → ((lcm𝑦) lcm 𝑧) ∥ 𝑘)
124 breq1 5108 . . . . . . . . . . . . . . . . . . . 20 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ↔ ((lcm𝑦) lcm 𝑧) ∥ 𝑘))
125123, 124syl5ibr 245 . . . . . . . . . . . . . . . . . . 19 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → ((((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
126125expd 416 . . . . . . . . . . . . . . . . . 18 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → (((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
127126exp5j 446 . . . . . . . . . . . . . . . . 17 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))))
128127com12 32 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))))
12968, 128syld 47 . . . . . . . . . . . . . . 15 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))))
130129com23 86 . . . . . . . . . . . . . 14 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))))
131130imp32 419 . . . . . . . . . . . . 13 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → ((𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ) → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
132131expd 416 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → (𝑛 ∈ ℤ → (𝑘 ∈ ℕ → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))))
133132com34 91 . . . . . . . . . . 11 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → (𝑛 ∈ ℤ → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (𝑘 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))))
134133com12 32 . . . . . . . . . 10 (𝑛 ∈ ℤ → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (𝑘 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))))
135134imp 407 . . . . . . . . 9 ((𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))) → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (𝑘 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
136135com12 32 . . . . . . . 8 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ((𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))) → (𝑘 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))))
137136imp 407 . . . . . . 7 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) → (𝑘 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
138137imp 407 . . . . . 6 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) ∧ 𝑘 ∈ ℕ) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
139138imp 407 . . . . 5 (((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) ∧ 𝑘 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)
140 vsnid 4623 . . . . . . . . 9 𝑛 ∈ {𝑛}
141140olci 864 . . . . . . . 8 (𝑛 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑛 ∈ {𝑛})
142 elun 4108 . . . . . . . 8 (𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑛 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑛 ∈ {𝑛}))
143141, 142mpbir 230 . . . . . . 7 𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})
144 breq1 5108 . . . . . . . 8 (𝑖 = 𝑛 → (𝑖𝑘𝑛𝑘))
145144rspcv 3577 . . . . . . 7 (𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘𝑛𝑘))
146143, 145mp1i 13 . . . . . 6 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) ∧ 𝑘 ∈ ℕ) → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘𝑛𝑘))
147146imp 407 . . . . 5 (((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) ∧ 𝑘 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → 𝑛𝑘)
148139, 147jca 512 . . . 4 (((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) ∧ 𝑘 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘𝑛𝑘))
149 lcmledvds 16475 . . . 4 (((𝑘 ∈ ℕ ∧ (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)) → (((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘𝑛𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))
15061, 148, 149sylc 65 . . 3 (((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) ∧ 𝑘 ∈ ℕ) ∧ ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘)
151150exp31 420 . 2 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) → (𝑘 ∈ ℕ → (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘)))
1529, 151ralrimi 3240 1 (((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))) → ∀𝑘 ∈ ℕ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wnel 3049  wral 3064  cun 3908  wss 3910  {csn 4586   class class class wbr 5105  cfv 6496  (class class class)co 7357  Fincfn 8883  0cc0 11051  cle 11190  cn 12153  0cn0 12413  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:  lcmfunsnlem2lem2  16515
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