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Theorem lcmfunsnlem2lem2 15837
Description: Lemma 2 for lcmfunsnlem2 15838. (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 4008 . . . . . . . . . . 11 (𝑖 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (𝑖 ∈ (𝑦 ∪ {𝑧}) ∨ 𝑖 ∈ {𝑛}))
2 elun 4008 . . . . . . . . . . . . 13 (𝑖 ∈ (𝑦 ∪ {𝑧}) ↔ (𝑖𝑦𝑖 ∈ {𝑧}))
3 simp1 1116 . . . . . . . . . . . . . . . . . 18 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑧 ∈ ℤ)
43adantr 473 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∈ ℤ)
54adantl 474 . . . . . . . . . . . . . . . 16 (((𝑖𝑦𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑧 ∈ ℤ)
6 sneq 4445 . . . . . . . . . . . . . . . . . . . 20 (𝑛 = 𝑧 → {𝑛} = {𝑧})
76uneq2d 4022 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝑦 ∪ {𝑛}) = (𝑦 ∪ {𝑧}))
87fveq2d 6500 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 → (lcm‘(𝑦 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑧})))
9 oveq2 6982 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 → ((lcm𝑦) lcm 𝑛) = ((lcm𝑦) lcm 𝑧))
108, 9eqeq12d 2787 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 → ((lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧)))
1110rspcv 3525 . . . . . . . . . . . . . . . 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 breq1 4928 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑖 → (𝑘 ∥ (lcm𝑦) ↔ 𝑖 ∥ (lcm𝑦)))
1413rspcv 3525 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖𝑦 → (∀𝑘𝑦 𝑘 ∥ (lcm𝑦) → 𝑖 ∥ (lcm𝑦)))
15 dvdslcmf 15829 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ∀𝑘𝑦 𝑘 ∥ (lcm𝑦))
16153adant1 1110 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ∀𝑘𝑦 𝑘 ∥ (lcm𝑦))
1716adantr 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ∀𝑘𝑦 𝑘 ∥ (lcm𝑦))
1814, 17impel 498 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (lcm𝑦))
19 lcmfcl 15826 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
2019nn0zd 11896 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
21203adant1 1110 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
2221adantr 473 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (lcm𝑦) ∈ ℤ)
23 lcmcl 15799 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℕ0)
243, 23sylan 572 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℕ0)
2524nn0zd 11896 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑧 lcm 𝑛) ∈ ℤ)
2622, 25jca 504 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ))
2726adantl 474 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ))
28 dvdslcm 15796 . . . . . . . . . . . . . . . . . . . . . . 23 (((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → ((lcm𝑦) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛)) ∧ (𝑧 lcm 𝑛) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛))))
2928simpld 487 . . . . . . . . . . . . . . . . . . . . . 22 (((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → (lcm𝑦) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
3027, 29syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (lcm𝑦) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
31 ssel 3846 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ⊆ ℤ → (𝑖𝑦𝑖 ∈ ℤ))
32313ad2ant2 1114 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑖𝑦𝑖 ∈ ℤ))
3332adantr 473 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑖𝑦𝑖 ∈ ℤ))
3433impcom 399 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∈ ℤ)
3522adantl 474 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (lcm𝑦) ∈ ℤ)
3625adantl 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (𝑧 lcm 𝑛) ∈ ℤ)
37 lcmcl 15799 . . . . . . . . . . . . . . . . . . . . . . . 24 (((lcm𝑦) ∈ ℤ ∧ (𝑧 lcm 𝑛) ∈ ℤ) → ((lcm𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℕ0)
3835, 36, 37syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℕ0)
3938nn0zd 11896 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℤ)
40 dvdstr 15504 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ ℤ ∧ (lcm𝑦) ∈ ℤ ∧ ((lcm𝑦) lcm (𝑧 lcm 𝑛)) ∈ ℤ) → ((𝑖 ∥ (lcm𝑦) ∧ (lcm𝑦) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛))) → 𝑖 ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛))))
4134, 35, 39, 40syl3anc 1351 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((𝑖 ∥ (lcm𝑦) ∧ (lcm𝑦) ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛))) → 𝑖 ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛))))
4218, 30, 41mp2and 686 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
434adantl 474 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑧 ∈ ℤ)
44 simprr 760 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑛 ∈ ℤ)
45 lcmass 15812 . . . . . . . . . . . . . . . . . . . . 21 (((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) = ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
4635, 43, 44, 45syl3anc 1351 . . . . . . . . . . . . . . . . . . . 20 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) = ((lcm𝑦) lcm (𝑧 lcm 𝑛)))
4742, 46breqtrrd 4953 . . . . . . . . . . . . . . . . . . 19 ((𝑖𝑦 ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
4847ex 405 . . . . . . . . . . . . . . . . . 18 (𝑖𝑦 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
49 elsni 4452 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ {𝑧} → 𝑖 = 𝑧)
5021, 3jca 504 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
5150adantr 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ))
52 dvdslcm 15796 . . . . . . . . . . . . . . . . . . . . . . 23 (((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((lcm𝑦) ∥ ((lcm𝑦) lcm 𝑧) ∧ 𝑧 ∥ ((lcm𝑦) lcm 𝑧)))
5352simprd 488 . . . . . . . . . . . . . . . . . . . . . 22 (((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → 𝑧 ∥ ((lcm𝑦) lcm 𝑧))
5451, 53syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∥ ((lcm𝑦) lcm 𝑧))
55193adant1 1110 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℕ0)
5655nn0zd 11896 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm𝑦) ∈ ℤ)
57 lcmcl 15799 . . . . . . . . . . . . . . . . . . . . . . . 24 (((lcm𝑦) ∈ ℤ ∧ 𝑧 ∈ ℤ) → ((lcm𝑦) lcm 𝑧) ∈ ℕ0)
5856, 3, 57syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm𝑦) lcm 𝑧) ∈ ℕ0)
5958nn0zd 11896 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((lcm𝑦) lcm 𝑧) ∈ ℤ)
60 dvdslcm 15796 . . . . . . . . . . . . . . . . . . . . . . 23 ((((lcm𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛) ∧ 𝑛 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
6160simpld 487 . . . . . . . . . . . . . . . . . . . . . 22 ((((lcm𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) lcm 𝑧) ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
6259, 61sylan 572 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) lcm 𝑧) ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
6359adantr 473 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm𝑦) lcm 𝑧) ∈ ℤ)
64 lcmcl 15799 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((lcm𝑦) lcm 𝑧) ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) ∈ ℕ0)
6559, 64sylan 572 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) ∈ ℕ0)
6665nn0zd 11896 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (((lcm𝑦) lcm 𝑧) lcm 𝑛) ∈ ℤ)
67 dvdstr 15504 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℤ ∧ ((lcm𝑦) lcm 𝑧) ∈ ℤ ∧ (((lcm𝑦) lcm 𝑧) lcm 𝑛) ∈ ℤ) → ((𝑧 ∥ ((lcm𝑦) lcm 𝑧) ∧ ((lcm𝑦) lcm 𝑧) ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)) → 𝑧 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
684, 63, 66, 67syl3anc 1351 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑧 ∥ ((lcm𝑦) lcm 𝑧) ∧ ((lcm𝑦) lcm 𝑧) ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)) → 𝑧 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
6954, 62, 68mp2and 686 . . . . . . . . . . . . . . . . . . . 20 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑧 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
70 breq1 4928 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑧 → (𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛) ↔ 𝑧 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
7169, 70syl5ibr 238 . . . . . . . . . . . . . . . . . . 19 (𝑖 = 𝑧 → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
7249, 71syl 17 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ {𝑧} → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
7348, 72jaoi 843 . . . . . . . . . . . . . . . . 17 ((𝑖𝑦𝑖 ∈ {𝑧}) → (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
7473imp 398 . . . . . . . . . . . . . . . 16 (((𝑖𝑦𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛))
75 oveq1 6981 . . . . . . . . . . . . . . . . 17 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) = (((lcm𝑦) lcm 𝑧) lcm 𝑛))
7675breq2d 4937 . . . . . . . . . . . . . . . 16 ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → (𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ↔ 𝑖 ∥ (((lcm𝑦) lcm 𝑧) lcm 𝑛)))
7774, 76syl5ibrcom 239 . . . . . . . . . . . . . . 15 (((𝑖𝑦𝑖 ∈ {𝑧}) ∧ ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ)) → ((lcm‘(𝑦 ∪ {𝑧})) = ((lcm𝑦) lcm 𝑧) → 𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
7812, 77syld 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)))
7978ex 405 . . . . . . . . . . . . 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))))
802, 79sylbi 209 . . . . . . . . . . . 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))))
81 elsni 4452 . . . . . . . . . . . . 13 (𝑖 ∈ {𝑛} → 𝑖 = 𝑛)
82 simp2 1117 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ⊆ ℤ)
83 snssi 4611 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℤ → {𝑧} ⊆ ℤ)
84833ad2ant1 1113 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → {𝑧} ⊆ ℤ)
8582, 84unssd 4044 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
86 simp3 1118 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → 𝑦 ∈ Fin)
87 snfi 8389 . . . . . . . . . . . . . . . . . . . . . 22 {𝑧} ∈ Fin
88 unfi 8578 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑦 ∈ Fin ∧ {𝑧} ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
8986, 87, 88sylancl 577 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (𝑦 ∪ {𝑧}) ∈ Fin)
90 lcmfcl 15826 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ0)
9185, 89, 90syl2anc 576 . . . . . . . . . . . . . . . . . . . 20 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ0)
9291nn0zd 11896 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ)
9392anim1i 605 . . . . . . . . . . . . . . . . . 18 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
9493adantr 473 . . . . . . . . . . . . . . . . 17 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
95 dvdslcm 15796 . . . . . . . . . . . . . . . . 17 (((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) → ((lcm‘(𝑦 ∪ {𝑧})) ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∧ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
9694, 95syl 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)))
9796simprd 488 . . . . . . . . . . . . . . 15 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
98 breq1 4928 . . . . . . . . . . . . . . 15 (𝑖 = 𝑛 → (𝑖 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ↔ 𝑛 ∥ ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
9997, 98syl5ibr 238 . . . . . . . . . . . . . 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)))
10099expd 408 . . . . . . . . . . . . 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))))
10181, 100syl 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))))
10280, 101jaoi 843 . . . . . . . . . . 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))))
1031, 102sylbi 209 . . . . . . . . . 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))))
104103com13 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))))
105104expd 408 . . . . . . . 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)))))
106105adantl 474 . . . . . . 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)))))
107106impcom 399 . . . . . 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))))
108107impcom 399 . . . . 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)))
109108adantl 474 . . . 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)))
110109ralrimiv 3125 . . 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))
111 lcmfunsnlem2lem1 15836 . . 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))
11293adantr 473 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ))
11385adantr 473 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑦 ∪ {𝑧}) ⊆ ℤ)
11489adantr 473 . . . . . . . . . . . . . . 15 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → (𝑦 ∪ {𝑧}) ∈ Fin)
115 df-nel 3068 . . . . . . . . . . . . . . . . . . . 20 (0 ∉ 𝑦 ↔ ¬ 0 ∈ 𝑦)
116115biimpi 208 . . . . . . . . . . . . . . . . . . 19 (0 ∉ 𝑦 → ¬ 0 ∈ 𝑦)
1171163ad2ant1 1113 . . . . . . . . . . . . . . . . . 18 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ 𝑦)
118 elsni 4452 . . . . . . . . . . . . . . . . . . . . 21 (0 ∈ {𝑧} → 0 = 𝑧)
119118eqcomd 2778 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ {𝑧} → 𝑧 = 0)
120119necon3ai 2986 . . . . . . . . . . . . . . . . . . 19 (𝑧 ≠ 0 → ¬ 0 ∈ {𝑧})
1211203ad2ant2 1114 . . . . . . . . . . . . . . . . . 18 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ {𝑧})
122 ioran 966 . . . . . . . . . . . . . . . . . 18 (¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) ↔ (¬ 0 ∈ 𝑦 ∧ ¬ 0 ∈ {𝑧}))
123117, 121, 122sylanbrc 575 . . . . . . . . . . . . . . . . 17 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
124 elun 4008 . . . . . . . . . . . . . . . . 17 (0 ∈ (𝑦 ∪ {𝑧}) ↔ (0 ∈ 𝑦 ∨ 0 ∈ {𝑧}))
125123, 124sylnibr 321 . . . . . . . . . . . . . . . 16 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ (𝑦 ∪ {𝑧}))
126 df-nel 3068 . . . . . . . . . . . . . . . 16 (0 ∉ (𝑦 ∪ {𝑧}) ↔ ¬ 0 ∈ (𝑦 ∪ {𝑧}))
127125, 126sylibr 226 . . . . . . . . . . . . . . 15 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ (𝑦 ∪ {𝑧}))
128 lcmfn0cl 15824 . . . . . . . . . . . . . . 15 (((𝑦 ∪ {𝑧}) ⊆ ℤ ∧ (𝑦 ∪ {𝑧}) ∈ Fin ∧ 0 ∉ (𝑦 ∪ {𝑧})) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
129113, 114, 127, 128syl2an3an 1402 . . . . . . . . . . . . . 14 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (lcm‘(𝑦 ∪ {𝑧})) ∈ ℕ)
130129nnne0d 11488 . . . . . . . . . . . . 13 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → (lcm‘(𝑦 ∪ {𝑧})) ≠ 0)
131130neneqd 2966 . . . . . . . . . . . 12 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ (lcm‘(𝑦 ∪ {𝑧})) = 0)
132 neneq 2967 . . . . . . . . . . . . . 14 (𝑛 ≠ 0 → ¬ 𝑛 = 0)
1331323ad2ant3 1115 . . . . . . . . . . . . 13 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 𝑛 = 0)
134133adantl 474 . . . . . . . . . . . 12 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ 𝑛 = 0)
135 ioran 966 . . . . . . . . . . . 12 (¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0) ↔ (¬ (lcm‘(𝑦 ∪ {𝑧})) = 0 ∧ ¬ 𝑛 = 0))
136131, 134, 135sylanbrc 575 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0))
137 lcmn0cl 15795 . . . . . . . . . . 11 ((((lcm‘(𝑦 ∪ {𝑧})) ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ ¬ ((lcm‘(𝑦 ∪ {𝑧})) = 0 ∨ 𝑛 = 0)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ)
138112, 136, 137syl2anc 576 . . . . . . . . . 10 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛) ∈ ℕ)
139 snssi 4611 . . . . . . . . . . . . . 14 (𝑛 ∈ ℤ → {𝑛} ⊆ ℤ)
140139adantl 474 . . . . . . . . . . . . 13 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → {𝑛} ⊆ ℤ)
141113, 140unssd 4044 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
142141adantr 473 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ⊆ ℤ)
14387, 88mpan2 678 . . . . . . . . . . . . . . 15 (𝑦 ∈ Fin → (𝑦 ∪ {𝑧}) ∈ Fin)
144 snfi 8389 . . . . . . . . . . . . . . 15 {𝑛} ∈ Fin
145 unfi 8578 . . . . . . . . . . . . . . 15 (((𝑦 ∪ {𝑧}) ∈ Fin ∧ {𝑛} ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
146143, 144, 145sylancl 577 . . . . . . . . . . . . . 14 (𝑦 ∈ Fin → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
1471463ad2ant3 1115 . . . . . . . . . . . . 13 ((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
148147adantr 473 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
149148adantr 473 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ∈ Fin)
150 elun 4008 . . . . . . . . . . . . . . . 16 (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ (0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}))
151 nnel 3076 . . . . . . . . . . . . . . . . . . . . 21 (¬ 0 ∉ 𝑦 ↔ 0 ∈ 𝑦)
152151biimpri 220 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ 𝑦 → ¬ 0 ∉ 𝑦)
1531523mix1d 1316 . . . . . . . . . . . . . . . . . . 19 (0 ∈ 𝑦 → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
154 nne 2965 . . . . . . . . . . . . . . . . . . . . 21 𝑧 ≠ 0 ↔ 𝑧 = 0)
155119, 154sylibr 226 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ {𝑧} → ¬ 𝑧 ≠ 0)
1561553mix2d 1317 . . . . . . . . . . . . . . . . . . 19 (0 ∈ {𝑧} → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
157153, 156jaoi 843 . . . . . . . . . . . . . . . . . 18 ((0 ∈ 𝑦 ∨ 0 ∈ {𝑧}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
158124, 157sylbi 209 . . . . . . . . . . . . . . . . 17 (0 ∈ (𝑦 ∪ {𝑧}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
159 elsni 4452 . . . . . . . . . . . . . . . . . . . 20 (0 ∈ {𝑛} → 0 = 𝑛)
160159eqcomd 2778 . . . . . . . . . . . . . . . . . . 19 (0 ∈ {𝑛} → 𝑛 = 0)
161 nne 2965 . . . . . . . . . . . . . . . . . . 19 𝑛 ≠ 0 ↔ 𝑛 = 0)
162160, 161sylibr 226 . . . . . . . . . . . . . . . . . 18 (0 ∈ {𝑛} → ¬ 𝑛 ≠ 0)
1631623mix3d 1318 . . . . . . . . . . . . . . . . 17 (0 ∈ {𝑛} → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
164158, 163jaoi 843 . . . . . . . . . . . . . . . 16 ((0 ∈ (𝑦 ∪ {𝑧}) ∨ 0 ∈ {𝑛}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
165150, 164sylbi 209 . . . . . . . . . . . . . . 15 (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
166 3ianor 1087 . . . . . . . . . . . . . . 15 (¬ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) ↔ (¬ 0 ∉ 𝑦 ∨ ¬ 𝑧 ≠ 0 ∨ ¬ 𝑛 ≠ 0))
167165, 166sylibr 226 . . . . . . . . . . . . . 14 (0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) → ¬ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0))
168167con2i 137 . . . . . . . . . . . . 13 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → ¬ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
169 df-nel 3068 . . . . . . . . . . . . 13 (0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}) ↔ ¬ 0 ∈ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
170168, 169sylibr 226 . . . . . . . . . . . 12 ((0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0) → 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
171170adantl 474 . . . . . . . . . . 11 ((((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ 𝑛 ∈ ℤ) ∧ (0 ∉ 𝑦𝑧 ≠ 0 ∧ 𝑛 ≠ 0)) → 0 ∉ ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
172142, 149, 1713jca 1108 . . . . . . . . . 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})))
173138, 172jca 504 . . . . . . . . 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}))))
174173ex 405 . . . . . . . 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})))))
175174ex 405 . . . . . . 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}))))))
176175adantr 473 . . . . . 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}))))))
177176impcom 399 . . . . 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})))))
178177impcom 399 . . . 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}))))
179 lcmf 15831 . . . 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))))
180178, 179syl 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))))
181110, 111, 180mpbir2and 700 . 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})))
182181eqcomd 2778 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 198  wa 387  wo 833  w3o 1067  w3a 1068   = wceq 1507  wcel 2050  wne 2961  wnel 3067  wral 3082  cun 3821  wss 3823  {csn 4435   class class class wbr 4925  cfv 6185  (class class class)co 6974  Fincfn 8304  0cc0 10333  cle 10473  cn 11437  0cn0 11705  cz 11791  cdvds 15465   lcm clcm 15786  lcmclcmf 15787
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277  ax-inf2 8896  ax-cnex 10389  ax-resscn 10390  ax-1cn 10391  ax-icn 10392  ax-addcl 10393  ax-addrcl 10394  ax-mulcl 10395  ax-mulrcl 10396  ax-mulcom 10397  ax-addass 10398  ax-mulass 10399  ax-distr 10400  ax-i2m1 10401  ax-1ne0 10402  ax-1rid 10403  ax-rnegex 10404  ax-rrecex 10405  ax-cnre 10406  ax-pre-lttri 10407  ax-pre-lttrn 10408  ax-pre-ltadd 10409  ax-pre-mulgt0 10410  ax-pre-sup 10411
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-se 5363  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-isom 6194  df-riota 6935  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-oadd 7907  df-er 8087  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-sup 8699  df-inf 8700  df-oi 8767  df-card 9160  df-pnf 10474  df-mnf 10475  df-xr 10476  df-ltxr 10477  df-le 10478  df-sub 10670  df-neg 10671  df-div 11097  df-nn 11438  df-2 11501  df-3 11502  df-n0 11706  df-z 11792  df-uz 12057  df-rp 12203  df-fz 12707  df-fzo 12848  df-fl 12975  df-mod 13051  df-seq 13183  df-exp 13243  df-hash 13504  df-cj 14317  df-re 14318  df-im 14319  df-sqrt 14453  df-abs 14454  df-clim 14704  df-prod 15118  df-dvds 15466  df-gcd 15702  df-lcm 15788  df-lcmf 15789
This theorem is referenced by:  lcmfunsnlem2  15838
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