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Theorem lcmfunsnlem2lem2 16515
Description: Lemma 2 for lcmfunsnlem2 16516. (Contributed by AV, 26-Aug-2020.)
Assertion
Ref Expression
lcmfunsnlem2lem2 (((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))))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
Distinct variable groups:   𝑦,𝑚,𝑧   𝑘,𝑛,𝑦,𝑧,𝑚

Proof of Theorem lcmfunsnlem2lem2
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 elun 4108 . . . . . . . . . . 11 (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑖 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑖 ∈ {𝑛}))
2 elun 4108 . . . . . . . . . . . . 13 (𝑖 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑖𝑦𝑖 ∈ {𝑧}))
3 simp1 1136 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑧 ∈ ℤ)
43adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∈ ℤ)
54adantl 482 . . . . . . . . . . . . . . . 16 (((𝑖𝑦𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑧 ∈ ℤ)
6 sneq 4596 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑧 → {𝑛} = {𝑧})
76uneq2d 4123 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
87fveq2d 6846 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
9 oveq2 7365 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 → ((lcm𝑦) lcm 𝑛) = ((lcm𝑦) lcm 𝑧))
108, 9eqeq12d 2752 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
1110rspcv 3577 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ℤ → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
125, 11syl 17 . . . . . . . . . . . . . . 15 (((𝑖𝑦𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
13 ssel 3937 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ⊆ ℤ → (𝑖𝑦𝑖 ∈ ℤ))
14133ad2ant2 1134 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑖𝑦𝑖 ∈ ℤ))
1514adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑖𝑦𝑖 ∈ ℤ))
1615impcom 408 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∈ ℤ)
17 lcmfcl 16504 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
1817nn0zd 12525 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
19183adant1 1130 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
2019adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (lcm𝑦) ∈ ℤ)
2120adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (lcm𝑦) ∈ ℤ)
22 lcmcl 16477 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℕ0)
233, 22sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℕ0)
2423nn0zd 12525 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℤ)
2524adantl 482 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (𝑧 lcm 𝑛) ∈ ℤ)
26 lcmcl 16477 . . . . . . . . . . . . . . . . . . . . . . 23 (((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → ((lcm𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℕ0)
2721, 25, 26syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℕ0)
2827nn0zd 12525 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℤ)
29 breq1 5108 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑖 → (𝑘 ∥ (lcm𝑦) ↔ 𝑖 ∥ (lcm𝑦)))
3029rspcv 3577 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖𝑦 → (∀𝑘𝑦 𝑘 ∥ (lcm𝑦) → 𝑖 ∥ (lcm𝑦)))
31 dvdslcmf 16507 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ∀𝑘𝑦 𝑘 ∥ (lcm𝑦))
32313adant1 1130 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ∀𝑘𝑦 𝑘 ∥ (lcm𝑦))
3332adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ∀𝑘𝑦 𝑘 ∥ (lcm𝑦))
3430, 33impel 506 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (lcm𝑦))
3520, 24jca 512 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ))
3635adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ))
37 dvdslcm 16474 . . . . . . . . . . . . . . . . . . . . . . 23 (((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → ((lcm𝑦) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛)) ∧ (𝑧 lcm 𝑛) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛))))
3837simpld 495 . . . . . . . . . . . . . . . . . . . . . 22 (((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → (lcm𝑦) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
3936, 38syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (lcm𝑦) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
4016, 21, 28, 34, 39dvdstrd 16177 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
414adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑧 ∈ ℤ)
42 simprr 771 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
43 lcmass 16490 . . . . . . . . . . . . . . . . . . . . 21 (((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) = ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
4421, 41, 42, 43syl3anc 1371 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) = ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
4540, 44breqtrrd 5133 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
4645ex 413 . . . . . . . . . . . . . . . . . 18 (𝑖𝑦 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
47 elsni 4603 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ {𝑧} → 𝑖 = 𝑧)
48173adant1 1130 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
4948nn0zd 12525 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
50 lcmcl 16477 . . . . . . . . . . . . . . . . . . . . . . . 24 (((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((lcm𝑦) lcm 𝑧) ∈ ℕ0)
5149, 3, 50syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm𝑦) lcm 𝑧) ∈ ℕ0)
5251nn0zd 12525 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm𝑦) lcm 𝑧) ∈ ℤ)
5352adantr 481 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) lcm 𝑧) ∈ ℤ)
54 lcmcl 16477 . . . . . . . . . . . . . . . . . . . . . . 23 ((((lcm𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) ∈ ℕ0)
5552, 54sylan 580 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) ∈ ℕ0)
5655nn0zd 12525 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) ∈ ℤ)
5719, 3jca 512 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
5857adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
59 dvdslcm 16474 . . . . . . . . . . . . . . . . . . . . . . 23 (((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((lcm𝑦) ∥ ((lcm𝑦) lcm 𝑧) ∧ 𝑧 ∥ ((lcm𝑦) lcm 𝑧)))
6059simprd 496 . . . . . . . . . . . . . . . . . . . . . 22 (((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → 𝑧 ∥ ((lcm𝑦) lcm 𝑧))
6158, 60syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∥ ((lcm𝑦) lcm 𝑧))
62 dvdslcm 16474 . . . . . . . . . . . . . . . . . . . . . . 23 ((((lcm𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛) ∧ 𝑛 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
6362simpld 495 . . . . . . . . . . . . . . . . . . . . . 22 ((((lcm𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) lcm 𝑧) ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
6452, 63sylan 580 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) lcm 𝑧) ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
654, 53, 56, 61, 64dvdstrd 16177 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
66 breq1 5108 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑧 → (𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛) ↔ 𝑧 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
6765, 66syl5ibr 245 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑧 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
6847, 67syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ {𝑧} → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
6946, 68jaoi 855 . . . . . . . . . . . . . . . . 17 ((𝑖𝑦𝑖 ∈ {𝑧}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
7069imp 407 . . . . . . . . . . . . . . . 16 (((𝑖𝑦𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
71 oveq1 7364 . . . . . . . . . . . . . . . . 17 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (((lcm𝑦) lcm 𝑧) lcm 𝑛))
7271breq2d 5117 . . . . . . . . . . . . . . . 16 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → (𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ↔ 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
7370, 72syl5ibrcom 246 . . . . . . . . . . . . . . 15 (((𝑖𝑦𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
7412, 73syld 47 . . . . . . . . . . . . . 14 (((𝑖𝑦𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
7574ex 413 . . . . . . . . . . . . 13 ((𝑖𝑦𝑖 ∈ {𝑧}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
762, 75sylbi 216 . . . . . . . . . . . 12 (𝑖 ∈ (𝑦 ∪ {𝑧}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
77 elsni 4603 . . . . . . . . . . . . 13 (𝑖 ∈ {𝑛} → 𝑖 = 𝑛)
78 simp2 1137 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
79 snssi 4768 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
80793ad2ant1 1133 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
8178, 80unssd 4146 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
82 simp3 1138 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
83 snfi 8988 . . . . . . . . . . . . . . . . . . . . . 22 {𝑧} ∈ Fin
84 unfi 9116 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
8582, 83, 84sylancl 586 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
86 lcmfcl 16504 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ0)
8781, 85, 86syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ0)
8887nn0zd 12525 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
8988anim1i 615 . . . . . . . . . . . . . . . . . 18 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
9089adantr 481 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
91 dvdslcm 16474 . . . . . . . . . . . . . . . . 17 (((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
9290, 91syl 17 . . . . . . . . . . . . . . . 16 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
9392simprd 496 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
94 breq1 5108 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → (𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ↔ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
9593, 94syl5ibr 245 . . . . . . . . . . . . . 14 (𝑖 = 𝑛 → ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
9695expd 416 . . . . . . . . . . . . 13 (𝑖 = 𝑛 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
9777, 96syl 17 . . . . . . . . . . . 12 (𝑖 ∈ {𝑛} → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
9876, 97jaoi 855 . . . . . . . . . . 11 ((𝑖 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑖 ∈ {𝑛}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
991, 98sylbi 216 . . . . . . . . . 10 (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
10099com13 88 . . . . . . . . 9 (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
101100expd 416 . . . . . . . 8 (∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) → ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 ∈ ℤ → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
102101adantl 482 . . . . . . 7 ((∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 ∈ ℤ → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
103102impcom 408 . . . . . 6 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → (𝑛 ∈ ℤ → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
104103impcom 408 . . . . 5 ((𝑛 ∈ ℤ ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))) → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
105104adantl 482 . . . 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))))) → (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
106105ralrimiv 3142 . . 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))))) → ∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
107 lcmfunsnlem2lem1 16514 . . 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))
10889adantr 481 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
10981adantr 481 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
11085adantr 481 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑦 ∪ {𝑧}) ∈ Fin)
111 df-nel 3050 . . . . . . . . . . . . . . . . . . . 20 (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
112111biimpi 215 . . . . . . . . . . . . . . . . . . 19 (0 ∉ 𝑦 → ¬ 0 ∈ 𝑦)
1131123ad2ant1 1133 . . . . . . . . . . . . . . . . . 18 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ 𝑦)
114 elsni 4603 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ {𝑧} → 0 = 𝑧)
115114eqcomd 2742 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ {𝑧} → 𝑧 = 0)
116115necon3ai 2968 . . . . . . . . . . . . . . . . . . 19 (𝑧 ≠ 0 → ¬ 0 ∈ {𝑧})
1171163ad2ant2 1134 . . . . . . . . . . . . . . . . . 18 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ {𝑧})
118 ioran 982 . . . . . . . . . . . . . . . . . 18 (¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
119113, 117, 118sylanbrc 583 . . . . . . . . . . . . . . . . 17 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
120 elun 4108 . . . . . . . . . . . . . . . . 17 (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
121119, 120sylnibr 328 . . . . . . . . . . . . . . . 16 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ (𝑦 ∪ {𝑧}))
122 df-nel 3050 . . . . . . . . . . . . . . . 16 (0 ∉ (𝑦 ∪ {𝑧}) ↔ ¬ 0 ∈ (𝑦 ∪ {𝑧}))
123121, 122sylibr 233 . . . . . . . . . . . . . . 15 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ (𝑦 ∪ {𝑧}))
124 lcmfn0cl 16502 . . . . . . . . . . . . . . 15 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
125109, 110, 123, 124syl2an3an 1422 . . . . . . . . . . . . . 14 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
126125nnne0d 12203 . . . . . . . . . . . . 13 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (lcm‘(𝑦 ∪ {𝑧})) ≠ 0)
127126neneqd 2948 . . . . . . . . . . . 12 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ (lcm‘(𝑦 ∪ {𝑧})) = 0)
128 neneq 2949 . . . . . . . . . . . . . 14 (𝑛 ≠ 0 → ¬ 𝑛 = 0)
1291283ad2ant3 1135 . . . . . . . . . . . . 13 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 𝑛 = 0)
130129adantl 482 . . . . . . . . . . . 12 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ 𝑛 = 0)
131 ioran 982 . . . . . . . . . . . 12 (¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0) ↔ (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
132127, 130, 131sylanbrc 583 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))
133 lcmn0cl 16473 . . . . . . . . . . 11 ((((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ)
134108, 132, 133syl2anc 584 . . . . . . . . . 10 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ)
135 snssi 4768 . . . . . . . . . . . . . 14 (𝑛 ∈ ℤ → {𝑛} ⊆ ℤ)
136135adantl 482 . . . . . . . . . . . . 13 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → {𝑛} ⊆ ℤ)
137109, 136unssd 4146 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
138137adantr 481 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
13983, 84mpan2 689 . . . . . . . . . . . . . . 15 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
140 snfi 8988 . . . . . . . . . . . . . . 15 {𝑛} ∈ Fin
141 unfi 9116 . . . . . . . . . . . . . . 15 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ {𝑛} ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
142139, 140, 141sylancl 586 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
1431423ad2ant3 1135 . . . . . . . . . . . . 13 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
144143adantr 481 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
145144adantr 481 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
146 elun 4108 . . . . . . . . . . . . . . . 16 (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
147 nnel 3058 . . . . . . . . . . . . . . . . . . . . 21 (¬ 0 ∉ 𝑦 ↔ 0 ∈ 𝑦)
148147biimpri 227 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ 𝑦 → ¬ 0 ∉ 𝑦)
1491483mix1d 1336 . . . . . . . . . . . . . . . . . . 19 (0 ∈ 𝑦 → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
150 nne 2947 . . . . . . . . . . . . . . . . . . . . 21 𝑧 ≠ 0 ↔ 𝑧 = 0)
151115, 150sylibr 233 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ {𝑧} → ¬ 𝑧 ≠ 0)
1521513mix2d 1337 . . . . . . . . . . . . . . . . . . 19 (0 ∈ {𝑧} → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
153149, 152jaoi 855 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
154120, 153sylbi 216 . . . . . . . . . . . . . . . . 17 (0 ∈ (𝑦 ∪ {𝑧}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
155 elsni 4603 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ {𝑛} → 0 = 𝑛)
156155eqcomd 2742 . . . . . . . . . . . . . . . . . . 19 (0 ∈ {𝑛} → 𝑛 = 0)
157 nne 2947 . . . . . . . . . . . . . . . . . . 19 𝑛 ≠ 0 ↔ 𝑛 = 0)
158156, 157sylibr 233 . . . . . . . . . . . . . . . . . 18 (0 ∈ {𝑛} → ¬ 𝑛 ≠ 0)
1591583mix3d 1338 . . . . . . . . . . . . . . . . 17 (0 ∈ {𝑛} → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
160154, 159jaoi 855 . . . . . . . . . . . . . . . 16 ((0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
161146, 160sylbi 216 . . . . . . . . . . . . . . 15 (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
162 3ianor 1107 . . . . . . . . . . . . . . 15 (¬ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ↔ (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
163161, 162sylibr 233 . . . . . . . . . . . . . 14 (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → ¬ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0))
164163con2i 139 . . . . . . . . . . . . 13 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
165 df-nel 3050 . . . . . . . . . . . . 13 (0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ ¬ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
166164, 165sylibr 233 . . . . . . . . . . . 12 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
167166adantl 482 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
168138, 145, 1673jca 1128 . . . . . . . . . 10 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛})))
169134, 168jca 512 . . . . . . . . 9 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))))
170169ex 413 . . . . . . . 8 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛})))))
171170ex 413 . . . . . . 7 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑛 ∈ ℤ → ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))))))
172171adantr 481 . . . . . 6 (((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) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))))))
173172impcom 408 . . . . 5 ((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) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛})))))
174173impcom 408 . . . 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))))) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))))
175 lcmf 16509 . . . 4 ((((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ ∧ (((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ ∧ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin ∧ 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) ↔ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ ∀𝑘 ∈ ℕ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))))
176174, 175syl 17 . . 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))))) → (((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) ↔ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ ∀𝑘 ∈ ℕ (∀𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛})𝑖𝑘 → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ≤ 𝑘))))
177106, 107, 176mpbir2and 711 . 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))))) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})))
178177eqcomd 2742 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))))) → (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845  w3o 1086  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:  lcmfunsnlem2  16516
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