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Theorem lcmfunsnlem 16675
Description: Lemma for lcmfdvds 16676 and lcmfunsn 16678. These two theorems must be proven simultaneously by induction on the cardinality of a finite set 𝑌, because they depend on each other. This can be seen by the two parts lcmfunsnlem1 16671 and lcmfunsnlem2 16674 of the induction step, each of them using both induction hypotheses. (Contributed by AV, 26-Aug-2020.)
Assertion
Ref Expression
lcmfunsnlem ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛)))
Distinct variable group:   𝑘,𝑛,𝑚,𝑌

Proof of Theorem lcmfunsnlem
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 4021 . . . 4 (𝑥 = ∅ → (𝑥 ⊆ ℤ ↔ ∅ ⊆ ℤ))
2 raleq 3321 . . . . . . 7 (𝑥 = ∅ → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚 ∈ ∅ 𝑚𝑘))
3 fveq2 6907 . . . . . . . 8 (𝑥 = ∅ → (lcm𝑥) = (lcm‘∅))
43breq1d 5158 . . . . . . 7 (𝑥 = ∅ → ((lcm𝑥) ∥ 𝑘 ↔ (lcm‘∅) ∥ 𝑘))
52, 4imbi12d 344 . . . . . 6 (𝑥 = ∅ → ((∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘)))
65ralbidv 3176 . . . . 5 (𝑥 = ∅ → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘)))
7 uneq1 4171 . . . . . . . 8 (𝑥 = ∅ → (𝑥 ∪ {𝑛}) = (∅ ∪ {𝑛}))
87fveq2d 6911 . . . . . . 7 (𝑥 = ∅ → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(∅ ∪ {𝑛})))
93oveq1d 7446 . . . . . . 7 (𝑥 = ∅ → ((lcm𝑥) lcm 𝑛) = ((lcm‘∅) lcm 𝑛))
108, 9eqeq12d 2751 . . . . . 6 (𝑥 = ∅ → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
1110ralbidv 3176 . . . . 5 (𝑥 = ∅ → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
126, 11anbi12d 632 . . . 4 (𝑥 = ∅ → ((∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛)) ↔ (∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))))
131, 12imbi12d 344 . . 3 (𝑥 = ∅ → ((𝑥 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛))) ↔ (∅ ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))))
14 sseq1 4021 . . . 4 (𝑥 = 𝑦 → (𝑥 ⊆ ℤ ↔ 𝑦 ⊆ ℤ))
15 raleq 3321 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚𝑦 𝑚𝑘))
16 fveq2 6907 . . . . . . . 8 (𝑥 = 𝑦 → (lcm𝑥) = (lcm𝑦))
1716breq1d 5158 . . . . . . 7 (𝑥 = 𝑦 → ((lcm𝑥) ∥ 𝑘 ↔ (lcm𝑦) ∥ 𝑘))
1815, 17imbi12d 344 . . . . . 6 (𝑥 = 𝑦 → ((∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
1918ralbidv 3176 . . . . 5 (𝑥 = 𝑦 → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
20 uneq1 4171 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 ∪ {𝑛}) = (𝑦 ∪ {𝑛}))
2120fveq2d 6911 . . . . . . 7 (𝑥 = 𝑦 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑛})))
2216oveq1d 7446 . . . . . . 7 (𝑥 = 𝑦 → ((lcm𝑥) lcm 𝑛) = ((lcm𝑦) lcm 𝑛))
2321, 22eqeq12d 2751 . . . . . 6 (𝑥 = 𝑦 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))
2423ralbidv 3176 . . . . 5 (𝑥 = 𝑦 → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))
2519, 24anbi12d 632 . . . 4 (𝑥 = 𝑦 → ((∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛)) ↔ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))))
2614, 25imbi12d 344 . . 3 (𝑥 = 𝑦 → ((𝑥 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛))) ↔ (𝑦 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))))
27 sseq1 4021 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℤ ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ))
28 raleq 3321 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘))
29 fveq2 6907 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
3029breq1d 5158 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm𝑥) ∥ 𝑘 ↔ (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
3128, 30imbi12d 344 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
3231ralbidv 3176 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
33 uneq1 4171 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
3433fveq2d 6911 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})))
3529oveq1d 7446 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm𝑥) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
3634, 35eqeq12d 2751 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
3736ralbidv 3176 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
3832, 37anbi12d 632 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → ((∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛)) ↔ (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
3927, 38imbi12d 344 . . 3 (𝑥 = (𝑦 ∪ {𝑧}) → ((𝑥 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛))) ↔ ((𝑦 ∪ {𝑧}) ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
40 sseq1 4021 . . . 4 (𝑥 = 𝑌 → (𝑥 ⊆ ℤ ↔ 𝑌 ⊆ ℤ))
41 raleq 3321 . . . . . . 7 (𝑥 = 𝑌 → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚𝑌 𝑚𝑘))
42 fveq2 6907 . . . . . . . 8 (𝑥 = 𝑌 → (lcm𝑥) = (lcm𝑌))
4342breq1d 5158 . . . . . . 7 (𝑥 = 𝑌 → ((lcm𝑥) ∥ 𝑘 ↔ (lcm𝑌) ∥ 𝑘))
4441, 43imbi12d 344 . . . . . 6 (𝑥 = 𝑌 → ((∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘)))
4544ralbidv 3176 . . . . 5 (𝑥 = 𝑌 → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘)))
46 uneq1 4171 . . . . . . . 8 (𝑥 = 𝑌 → (𝑥 ∪ {𝑛}) = (𝑌 ∪ {𝑛}))
4746fveq2d 6911 . . . . . . 7 (𝑥 = 𝑌 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑌 ∪ {𝑛})))
4842oveq1d 7446 . . . . . . 7 (𝑥 = 𝑌 → ((lcm𝑥) lcm 𝑛) = ((lcm𝑌) lcm 𝑛))
4947, 48eqeq12d 2751 . . . . . 6 (𝑥 = 𝑌 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛)))
5049ralbidv 3176 . . . . 5 (𝑥 = 𝑌 → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛)))
5145, 50anbi12d 632 . . . 4 (𝑥 = 𝑌 → ((∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛)) ↔ (∀𝑘 ∈ ℤ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛))))
5240, 51imbi12d 344 . . 3 (𝑥 = 𝑌 → ((𝑥 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛))) ↔ (𝑌 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛)))))
53 lcmf0 16668 . . . . . . . 8 (lcm‘∅) = 1
54 1dvds 16305 . . . . . . . 8 (𝑘 ∈ ℤ → 1 ∥ 𝑘)
5553, 54eqbrtrid 5183 . . . . . . 7 (𝑘 ∈ ℤ → (lcm‘∅) ∥ 𝑘)
5655a1d 25 . . . . . 6 (𝑘 ∈ ℤ → (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘))
5756adantl 481 . . . . 5 ((∅ ⊆ ℤ ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘))
5857ralrimiva 3144 . . . 4 (∅ ⊆ ℤ → ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘))
59 uncom 4168 . . . . . . . . . 10 (∅ ∪ {𝑛}) = ({𝑛} ∪ ∅)
60 un0 4400 . . . . . . . . . 10 ({𝑛} ∪ ∅) = {𝑛}
6159, 60eqtri 2763 . . . . . . . . 9 (∅ ∪ {𝑛}) = {𝑛}
6261a1i 11 . . . . . . . 8 (𝑛 ∈ ℤ → (∅ ∪ {𝑛}) = {𝑛})
6362fveq2d 6911 . . . . . . 7 (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = (lcm‘{𝑛}))
64 lcmfsn 16669 . . . . . . 7 (𝑛 ∈ ℤ → (lcm‘{𝑛}) = (abs‘𝑛))
6553a1i 11 . . . . . . . . 9 (𝑛 ∈ ℤ → (lcm‘∅) = 1)
6665oveq1d 7446 . . . . . . . 8 (𝑛 ∈ ℤ → ((lcm‘∅) lcm 𝑛) = (1 lcm 𝑛))
67 1z 12645 . . . . . . . . 9 1 ∈ ℤ
68 lcmcom 16627 . . . . . . . . 9 ((1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (1 lcm 𝑛) = (𝑛 lcm 1))
6967, 68mpan 690 . . . . . . . 8 (𝑛 ∈ ℤ → (1 lcm 𝑛) = (𝑛 lcm 1))
70 lcm1 16644 . . . . . . . 8 (𝑛 ∈ ℤ → (𝑛 lcm 1) = (abs‘𝑛))
7166, 69, 703eqtrrd 2780 . . . . . . 7 (𝑛 ∈ ℤ → (abs‘𝑛) = ((lcm‘∅) lcm 𝑛))
7263, 64, 713eqtrd 2779 . . . . . 6 (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
7372adantl 481 . . . . 5 ((∅ ⊆ ℤ ∧ 𝑛 ∈ ℤ) → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
7473ralrimiva 3144 . . . 4 (∅ ⊆ ℤ → ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
7558, 74jca 511 . . 3 (∅ ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
76 unss 4200 . . . . . . . 8 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
77 simpl 482 . . . . . . . 8 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
7876, 77sylbir 235 . . . . . . 7 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
7978adantl 481 . . . . . 6 ((𝑦 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ ℤ) → 𝑦 ⊆ ℤ)
80 vex 3482 . . . . . . . . . . 11 𝑧 ∈ V
8180snss 4790 . . . . . . . . . 10 (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
82 lcmfunsnlem1 16671 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → ∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘))
83 lcmfunsnlem2 16674 . . . . . . . . . . . 12 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
8482, 83jca 511 . . . . . . . . . . 11 (((𝑧 ∈ ℤ ∧ 𝑦 ⊆ ℤ ∧ 𝑦 ∈ Fin) ∧ (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
85843exp1 1351 . . . . . . . . . 10 (𝑧 ∈ ℤ → (𝑦 ⊆ ℤ → (𝑦 ∈ Fin → ((∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))))
8681, 85sylbir 235 . . . . . . . . 9 ({𝑧} ⊆ ℤ → (𝑦 ⊆ ℤ → (𝑦 ∈ Fin → ((∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))))
8786impcom 407 . . . . . . . 8 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → (𝑦 ∈ Fin → ((∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
8876, 87sylbir 235 . . . . . . 7 ((𝑦 ∪ {𝑧}) ⊆ ℤ → (𝑦 ∈ Fin → ((∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
8988impcom 407 . . . . . 6 ((𝑦 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ ℤ) → ((∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
9079, 89embantd 59 . . . . 5 ((𝑦 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ ℤ) → ((𝑦 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))))
9190ex 412 . . . 4 (𝑦 ∈ Fin → ((𝑦 ∪ {𝑧}) ⊆ ℤ → ((𝑦 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
9291com23 86 . . 3 (𝑦 ∈ Fin → ((𝑦 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛))) → ((𝑦 ∪ {𝑧}) ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))))
9313, 26, 39, 52, 75, 92findcard2 9203 . 2 (𝑌 ∈ Fin → (𝑌 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛))))
9493impcom 407 1 ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (∀𝑘 ∈ ℤ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cun 3961  wss 3963  c0 4339  {csn 4631   class class class wbr 5148  cfv 6563  (class class class)co 7431  Fincfn 8984  1c1 11154  cz 12611  abscabs 15270  cdvds 16287   lcm clcm 16622  lcmclcmf 16623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-clim 15521  df-prod 15937  df-dvds 16288  df-gcd 16529  df-lcm 16624  df-lcmf 16625
This theorem is referenced by:  lcmfdvds  16676  lcmfunsn  16678
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