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Theorem lcmfunsnlem 16580
Description: Lemma for lcmfdvds 16581 and lcmfunsn 16583. 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 16576 and lcmfunsnlem2 16579 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 4007 . . . 4 (𝑥 = ∅ → (𝑥 ⊆ ℤ ↔ ∅ ⊆ ℤ))
2 raleq 3322 . . . . . . 7 (𝑥 = ∅ → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚 ∈ ∅ 𝑚𝑘))
3 fveq2 6891 . . . . . . . 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 3177 . . . . 5 (𝑥 = ∅ → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘)))
7 uneq1 4156 . . . . . . . 8 (𝑥 = ∅ → (𝑥 ∪ {𝑛}) = (∅ ∪ {𝑛}))
87fveq2d 6895 . . . . . . 7 (𝑥 = ∅ → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(∅ ∪ {𝑛})))
93oveq1d 7426 . . . . . . 7 (𝑥 = ∅ → ((lcm𝑥) lcm 𝑛) = ((lcm‘∅) lcm 𝑛))
108, 9eqeq12d 2748 . . . . . 6 (𝑥 = ∅ → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
1110ralbidv 3177 . . . . 5 (𝑥 = ∅ → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
126, 11anbi12d 631 . . . 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 4007 . . . 4 (𝑥 = 𝑦 → (𝑥 ⊆ ℤ ↔ 𝑦 ⊆ ℤ))
15 raleq 3322 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚𝑦 𝑚𝑘))
16 fveq2 6891 . . . . . . . 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 3177 . . . . 5 (𝑥 = 𝑦 → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚𝑦 𝑚𝑘 → (lcm𝑦) ∥ 𝑘)))
20 uneq1 4156 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 ∪ {𝑛}) = (𝑦 ∪ {𝑛}))
2120fveq2d 6895 . . . . . . 7 (𝑥 = 𝑦 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑛})))
2216oveq1d 7426 . . . . . . 7 (𝑥 = 𝑦 → ((lcm𝑥) lcm 𝑛) = ((lcm𝑦) lcm 𝑛))
2321, 22eqeq12d 2748 . . . . . 6 (𝑥 = 𝑦 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))
2423ralbidv 3177 . . . . 5 (𝑥 = 𝑦 → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(𝑦 ∪ {𝑛})) = ((lcm𝑦) lcm 𝑛)))
2519, 24anbi12d 631 . . . 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 4007 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℤ ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ))
28 raleq 3322 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘))
29 fveq2 6891 . . . . . . . 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 3177 . . . . 5 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘 → (lcm‘(𝑦 ∪ {𝑧})) ∥ 𝑘)))
33 uneq1 4156 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
3433fveq2d 6895 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})))
3529oveq1d 7426 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm𝑥) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
3634, 35eqeq12d 2748 . . . . . 6 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛)))
3736ralbidv 3177 . . . . 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 631 . . . 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 4007 . . . 4 (𝑥 = 𝑌 → (𝑥 ⊆ ℤ ↔ 𝑌 ⊆ ℤ))
41 raleq 3322 . . . . . . 7 (𝑥 = 𝑌 → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚𝑌 𝑚𝑘))
42 fveq2 6891 . . . . . . . 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 3177 . . . . 5 (𝑥 = 𝑌 → (∀𝑘 ∈ ℤ (∀𝑚𝑥 𝑚𝑘 → (lcm𝑥) ∥ 𝑘) ↔ ∀𝑘 ∈ ℤ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘)))
46 uneq1 4156 . . . . . . . 8 (𝑥 = 𝑌 → (𝑥 ∪ {𝑛}) = (𝑌 ∪ {𝑛}))
4746fveq2d 6895 . . . . . . 7 (𝑥 = 𝑌 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑌 ∪ {𝑛})))
4842oveq1d 7426 . . . . . . 7 (𝑥 = 𝑌 → ((lcm𝑥) lcm 𝑛) = ((lcm𝑌) lcm 𝑛))
4947, 48eqeq12d 2748 . . . . . 6 (𝑥 = 𝑌 → ((lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛)))
5049ralbidv 3177 . . . . 5 (𝑥 = 𝑌 → (∀𝑛 ∈ ℤ (lcm‘(𝑥 ∪ {𝑛})) = ((lcm𝑥) lcm 𝑛) ↔ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛)))
5145, 50anbi12d 631 . . . 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 16573 . . . . . . . 8 (lcm‘∅) = 1
54 1dvds 16216 . . . . . . . 8 (𝑘 ∈ ℤ → 1 ∥ 𝑘)
5553, 54eqbrtrid 5183 . . . . . . 7 (𝑘 ∈ ℤ → (lcm‘∅) ∥ 𝑘)
5655a1d 25 . . . . . 6 (𝑘 ∈ ℤ → (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘))
5756adantl 482 . . . . 5 ((∅ ⊆ ℤ ∧ 𝑘 ∈ ℤ) → (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘))
5857ralrimiva 3146 . . . 4 (∅ ⊆ ℤ → ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘))
59 uncom 4153 . . . . . . . . . 10 (∅ ∪ {𝑛}) = ({𝑛} ∪ ∅)
60 un0 4390 . . . . . . . . . 10 ({𝑛} ∪ ∅) = {𝑛}
6159, 60eqtri 2760 . . . . . . . . 9 (∅ ∪ {𝑛}) = {𝑛}
6261a1i 11 . . . . . . . 8 (𝑛 ∈ ℤ → (∅ ∪ {𝑛}) = {𝑛})
6362fveq2d 6895 . . . . . . 7 (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = (lcm‘{𝑛}))
64 lcmfsn 16574 . . . . . . 7 (𝑛 ∈ ℤ → (lcm‘{𝑛}) = (abs‘𝑛))
6553a1i 11 . . . . . . . . 9 (𝑛 ∈ ℤ → (lcm‘∅) = 1)
6665oveq1d 7426 . . . . . . . 8 (𝑛 ∈ ℤ → ((lcm‘∅) lcm 𝑛) = (1 lcm 𝑛))
67 1z 12594 . . . . . . . . 9 1 ∈ ℤ
68 lcmcom 16532 . . . . . . . . 9 ((1 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (1 lcm 𝑛) = (𝑛 lcm 1))
6967, 68mpan 688 . . . . . . . 8 (𝑛 ∈ ℤ → (1 lcm 𝑛) = (𝑛 lcm 1))
70 lcm1 16549 . . . . . . . 8 (𝑛 ∈ ℤ → (𝑛 lcm 1) = (abs‘𝑛))
7166, 69, 703eqtrrd 2777 . . . . . . 7 (𝑛 ∈ ℤ → (abs‘𝑛) = ((lcm‘∅) lcm 𝑛))
7263, 64, 713eqtrd 2776 . . . . . 6 (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
7372adantl 482 . . . . 5 ((∅ ⊆ ℤ ∧ 𝑛 ∈ ℤ) → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
7473ralrimiva 3146 . . . 4 (∅ ⊆ ℤ → ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
7558, 74jca 512 . . 3 (∅ ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛)))
76 unss 4184 . . . . . . . 8 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ)
77 simpl 483 . . . . . . . 8 ((𝑦 ⊆ ℤ ∧ {𝑧} ⊆ ℤ) → 𝑦 ⊆ ℤ)
7876, 77sylbir 234 . . . . . . 7 ((𝑦 ∪ {𝑧}) ⊆ ℤ → 𝑦 ⊆ ℤ)
7978adantl 482 . . . . . 6 ((𝑦 ∈ Fin ∧ (𝑦 ∪ {𝑧}) ⊆ ℤ) → 𝑦 ⊆ ℤ)
80 vex 3478 . . . . . . . . . . 11 𝑧 ∈ V
8180snss 4789 . . . . . . . . . 10 (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
82 lcmfunsnlem1 16576 . . . . . . . . . . . 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 16579 . . . . . . . . . . . 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 512 . . . . . . . . . . 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 1352 . . . . . . . . . 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 234 . . . . . . . . 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 408 . . . . . . . 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 234 . . . . . . 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 408 . . . . . 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 413 . . . 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 9166 . 2 (𝑌 ∈ Fin → (𝑌 ⊆ ℤ → (∀𝑘 ∈ ℤ (∀𝑚𝑌 𝑚𝑘 → (lcm𝑌) ∥ 𝑘) ∧ ∀𝑛 ∈ ℤ (lcm‘(𝑌 ∪ {𝑛})) = ((lcm𝑌) lcm 𝑛))))
9493impcom 408 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 396  w3a 1087   = wceq 1541  wcel 2106  wral 3061  cun 3946  wss 3948  c0 4322  {csn 4628   class class class wbr 5148  cfv 6543  (class class class)co 7411  Fincfn 8941  1c1 11113  cz 12560  abscabs 15183  cdvds 16199   lcm clcm 16527  lcmclcmf 16528
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-div 11874  df-nn 12215  df-2 12277  df-3 12278  df-n0 12475  df-z 12561  df-uz 12825  df-rp 12977  df-fz 13487  df-fzo 13630  df-fl 13759  df-mod 13837  df-seq 13969  df-exp 14030  df-hash 14293  df-cj 15048  df-re 15049  df-im 15050  df-sqrt 15184  df-abs 15185  df-clim 15434  df-prod 15852  df-dvds 16200  df-gcd 16438  df-lcm 16529  df-lcmf 16530
This theorem is referenced by:  lcmfdvds  16581  lcmfunsn  16583
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