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Theorem lcmfunsnlem2lem1 15974
Description: Lemma 1 for lcmfunsnlem2 15976. (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 1908 . . 3 𝑘(0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)
2 nfv 1908 . . . 4 𝑘 𝑛 ∈ ℤ
3 nfv 1908 . . . . 5 𝑘(𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)
4 nfra1 3223 . . . . . 6 𝑘𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)
5 nfv 1908 . . . . . 6 𝑘𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)
64, 5nfan 1893 . . . . 5 𝑘(∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))
73, 6nfan 1893 . . . 4 𝑘((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))
82, 7nfan 1893 . . 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 1893 . 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 769 . . . . . . . . . . . 12 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑘 ∈ ℕ)
11 simp2 1131 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
12 snssi 4739 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
13123ad2ant1 1127 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
1411, 13unssd 4165 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
15 simp3 1132 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
16 snfi 8586 . . . . . . . . . . . . . . . . . 18 {𝑧} ∈ Fin
17 unfi 8777 . . . . . . . . . . . . . . . . . 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 15964 . . . . . . . . . . . . . . . 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 12077 . . . . . . . . . . . . . 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 767 . . . . . . . . . . . 12 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑛 ∈ ℤ)
2610, 24, 253jca 1122 . . . . . . . . . . 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 3128 . . . . . . . . . . . . . . . . . . . . . . 23 (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
3029biimpi 217 . . . . . . . . . . . . . . . . . . . . . 22 (0 ∉ 𝑦 → ¬ 0 ∈ 𝑦)
31 elsni 4580 . . . . . . . . . . . . . . . . . . . . . . . 24 (0 ∈ {𝑧} → 0 = 𝑧)
3231eqcomd 2830 . . . . . . . . . . . . . . . . . . . . . . 23 (0 ∈ {𝑧} → 𝑧 = 0)
3332necon3ai 3045 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ≠ 0 → ¬ 0 ∈ {𝑧})
3430, 33anim12i 612 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∉ 𝑦𝑧 ≠ 0) → (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
35343adant3 1126 . . . . . . . . . . . . . . . . . . . 20 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
36 df-nel 3128 . . . . . . . . . . . . . . . . . . . . 21 (0 ∉ (𝑦 ∪ {𝑧}) ↔ ¬ 0 ∈ (𝑦 ∪ {𝑧}))
37 ioran 979 . . . . . . . . . . . . . . . . . . . . . 22 (¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
38 elun 4128 . . . . . . . . . . . . . . . . . . . . . 22 (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
3937, 38xchnxbir 334 . . . . . . . . . . . . . . . . . . . . 21 (¬ 0 ∈ (𝑦 ∪ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
4036, 39bitri 276 . . . . . . . . . . . . . . . . . . . 20 (0 ∉ (𝑦 ∪ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
4135, 40sylibr 235 . . . . . . . . . . . . . . . . . . 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 1122 . . . . . . . . . . . . . . . . 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 15962 . . . . . . . . . . . . . . . 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 11679 . . . . . . . . . . . . . 14 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm‘(𝑦 ∪ {𝑧})) ≠ 0)
4847neneqd 3025 . . . . . . . . . . . . 13 ((((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ∧ (𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → ¬ (lcm‘(𝑦 ∪ {𝑧})) = 0)
49 neneq 3026 . . . . . . . . . . . . . . 15 (𝑛 ≠ 0 → ¬ 𝑛 = 0)
50493ad2ant3 1129 . . . . . . . . . . . . . 14 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 𝑛 = 0)
5150ad2antrr 722 . . . . . . . . . . . . 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 979 . . . . . . . . . . . 12 (¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0) ↔ (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
5452, 53sylibr 235 . . . . . . . . . . 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 4573 . . . . . . . . . . . . . . . . . . . . 21 (𝑛 = 𝑧 → {𝑛} = {𝑧})
6362uneq2d 4142 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
6463fveq2d 6670 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
65 oveq2 7159 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → ((lcm𝑦) lcm 𝑛) = ((lcm𝑦) lcm 𝑧))
6664, 65eqeq12d 2840 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
6766rspcv 3621 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
68673ad2ant1 1127 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
69 nnz 11996 . . . . . . . . . . . . . . . . . . . . . . . . 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 15964 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
7372nn0zd 12077 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
74733adant1 1124 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
7574ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (lcm𝑦) ∈ ℤ)
76 simpll1 1206 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → 𝑧 ∈ ℤ)
7771, 75, 763jca 1122 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)) ∧ (𝑛 ∈ ℤ ∧ 𝑘 ∈ ℕ)) → (𝑘 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
7877ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 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 4155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑚𝑦𝑚 ∈ (𝑦 ∪ {𝑧}))
8079orcd 871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑚𝑦 → (𝑚 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑚 ∈ {𝑛}))
81 elun 4128 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑚 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑚 ∈ {𝑛}))
8280, 81sylibr 235 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚𝑦𝑚 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
83 breq1 5065 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑖 = 𝑚 → (𝑖𝑘𝑚𝑘))
8483rspcv 3621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3185 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑘 = 𝑙 → (𝑚𝑘𝑚𝑙))
9190ralbidv 3201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑙 → (∀𝑚𝑦 𝑚𝑘 ↔ ∀𝑚𝑦 𝑚𝑙))
92 breq2 5066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑘 = 𝑙 → ((lcm𝑦) ∥ 𝑘 ↔ (lcm𝑦) ∥ 𝑙))
9391, 92imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑘 = 𝑙 → ((∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ↔ (∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙)))
9493cbvralvw 3454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑙 = 𝑘 → (𝑚𝑙𝑚𝑘))
9897ralbidv 3201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑙 = 𝑘 → (∀𝑚𝑦 𝑚𝑙 ↔ ∀𝑚𝑦 𝑚𝑘))
99 breq2 5066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑙 = 𝑘 → ((lcm𝑦) ∥ 𝑙 ↔ (lcm𝑦) ∥ 𝑘))
10098, 99imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑙 = 𝑘 → ((∀𝑚𝑦 𝑚𝑙 → (lcm𝑦) ∥ 𝑙) ↔ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
101100rspcv 3621 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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, 102syl5bi 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4598 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑧 ∈ {𝑧}
111110olci 862 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧𝑦𝑧 ∈ {𝑧})
112 elun 4128 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑧𝑦𝑧 ∈ {𝑧}))
113111, 112mpbir 232 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑧 ∈ (𝑦 ∪ {𝑧})
114113orci 861 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑧 ∈ {𝑛})
115 elun 4128 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑧 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑧 ∈ {𝑛}))
116114, 115mpbir 232 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑧 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})
117 breq1 5065 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 = 𝑧 → (𝑖𝑘𝑧𝑘))
118117rspcv 3621 . . . . . . . . . . . . . . . . . . . . . . . 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 15944 . . . . . . . . . . . . . . . . . . . . 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 5065 . . . . . . . . . . . . . . . . . . . 20 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘 ↔ ((lcm𝑦) lcm 𝑧) ∥ 𝑘))
125123, 124syl5ibr 247 . . . . . . . . . . . . . . . . . . 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 4598 . . . . . . . . 9 𝑛 ∈ {𝑛}
141140olci 862 . . . . . . . 8 (𝑛 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑛 ∈ {𝑛})
142 elun 4128 . . . . . . . 8 (𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑛 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑛 ∈ {𝑛}))
143141, 142mpbir 232 . . . . . . 7 𝑛 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})
144 breq1 5065 . . . . . . . 8 (𝑖 = 𝑛 → (𝑖𝑘𝑛𝑘))
145144rspcv 3621 . . . . . . 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 15935 . . . 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 3220 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 843  w3a 1081   = wceq 1530  wcel 2106  wne 3020  wnel 3127  wral 3142  cun 3937  wss 3939  {csn 4563   class class class wbr 5062  cfv 6351  (class class class)co 7151  Fincfn 8501  0cc0 10529  cle 10668  cn 11630  0cn0 11889  cz 11973  cdvds 15599   lcm clcm 15924  lcmclcmf 15925
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-inf2 9096  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8282  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-sup 8898  df-inf 8899  df-oi 8966  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-rp 12383  df-fz 12886  df-fzo 13027  df-fl 13155  df-mod 13231  df-seq 13363  df-exp 13423  df-hash 13684  df-cj 14451  df-re 14452  df-im 14453  df-sqrt 14587  df-abs 14588  df-clim 14838  df-prod 15252  df-dvds 15600  df-gcd 15836  df-lcm 15926  df-lcmf 15927
This theorem is referenced by:  lcmfunsnlem2lem2  15975
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