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Theorem lcmfunsnlem 16679
Description: Lemma for lcmfdvds 16680 and lcmfunsn 16682. 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 16675 and lcmfunsnlem2 16678 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 4008 . . . 4 (𝑥 = ∅ → (𝑥 ⊆ ℤ ↔ ∅ ⊆ ℤ))
2 raleq 3322 . . . . . . 7 (𝑥 = ∅ → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚 ∈ ∅ 𝑚𝑘))
3 fveq2 6905 . . . . . . . 8 (𝑥 = ∅ → (lcm𝑥) = (lcm‘∅))
43breq1d 5152 . . . . . . 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 4160 . . . . . . . 8 (𝑥 = ∅ → (𝑥 ∪ {𝑛}) = (∅ ∪ {𝑛}))
87fveq2d 6909 . . . . . . 7 (𝑥 = ∅ → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(∅ ∪ {𝑛})))
93oveq1d 7447 . . . . . . 7 (𝑥 = ∅ → ((lcm𝑥) lcm 𝑛) = ((lcm‘∅) lcm 𝑛))
108, 9eqeq12d 2752 . . . . . 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 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 4008 . . . 4 (𝑥 = 𝑦 → (𝑥 ⊆ ℤ ↔ 𝑦 ⊆ ℤ))
15 raleq 3322 . . . . . . 7 (𝑥 = 𝑦 → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚𝑦 𝑚𝑘))
16 fveq2 6905 . . . . . . . 8 (𝑥 = 𝑦 → (lcm𝑥) = (lcm𝑦))
1716breq1d 5152 . . . . . . 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 4160 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥 ∪ {𝑛}) = (𝑦 ∪ {𝑛}))
2120fveq2d 6909 . . . . . . 7 (𝑥 = 𝑦 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑦 ∪ {𝑛})))
2216oveq1d 7447 . . . . . . 7 (𝑥 = 𝑦 → ((lcm𝑥) lcm 𝑛) = ((lcm𝑦) lcm 𝑛))
2321, 22eqeq12d 2752 . . . . . 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 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 4008 . . . 4 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ⊆ ℤ ↔ (𝑦 ∪ {𝑧}) ⊆ ℤ))
28 raleq 3322 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚 ∈ (𝑦 ∪ {𝑧})𝑚𝑘))
29 fveq2 6905 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm𝑥) = (lcm‘(𝑦 ∪ {𝑧})))
3029breq1d 5152 . . . . . . 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 4160 . . . . . . . 8 (𝑥 = (𝑦 ∪ {𝑧}) → (𝑥 ∪ {𝑛}) = ((𝑦 ∪ {𝑧}) ∪ {𝑛}))
3433fveq2d 6909 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘((𝑦 ∪ {𝑧}) ∪ {𝑛})))
3529oveq1d 7447 . . . . . . 7 (𝑥 = (𝑦 ∪ {𝑧}) → ((lcm𝑥) lcm 𝑛) = ((lcm‘(𝑦 ∪ {𝑧})) lcm 𝑛))
3634, 35eqeq12d 2752 . . . . . 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 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 4008 . . . 4 (𝑥 = 𝑌 → (𝑥 ⊆ ℤ ↔ 𝑌 ⊆ ℤ))
41 raleq 3322 . . . . . . 7 (𝑥 = 𝑌 → (∀𝑚𝑥 𝑚𝑘 ↔ ∀𝑚𝑌 𝑚𝑘))
42 fveq2 6905 . . . . . . . 8 (𝑥 = 𝑌 → (lcm𝑥) = (lcm𝑌))
4342breq1d 5152 . . . . . . 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 4160 . . . . . . . 8 (𝑥 = 𝑌 → (𝑥 ∪ {𝑛}) = (𝑌 ∪ {𝑛}))
4746fveq2d 6909 . . . . . . 7 (𝑥 = 𝑌 → (lcm‘(𝑥 ∪ {𝑛})) = (lcm‘(𝑌 ∪ {𝑛})))
4842oveq1d 7447 . . . . . . 7 (𝑥 = 𝑌 → ((lcm𝑥) lcm 𝑛) = ((lcm𝑌) lcm 𝑛))
4947, 48eqeq12d 2752 . . . . . 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 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 16672 . . . . . . . 8 (lcm‘∅) = 1
54 1dvds 16309 . . . . . . . 8 (𝑘 ∈ ℤ → 1 ∥ 𝑘)
5553, 54eqbrtrid 5177 . . . . . . 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 3145 . . . 4 (∅ ⊆ ℤ → ∀𝑘 ∈ ℤ (∀𝑚 ∈ ∅ 𝑚𝑘 → (lcm‘∅) ∥ 𝑘))
59 uncom 4157 . . . . . . . . . 10 (∅ ∪ {𝑛}) = ({𝑛} ∪ ∅)
60 un0 4393 . . . . . . . . . 10 ({𝑛} ∪ ∅) = {𝑛}
6159, 60eqtri 2764 . . . . . . . . 9 (∅ ∪ {𝑛}) = {𝑛}
6261a1i 11 . . . . . . . 8 (𝑛 ∈ ℤ → (∅ ∪ {𝑛}) = {𝑛})
6362fveq2d 6909 . . . . . . 7 (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = (lcm‘{𝑛}))
64 lcmfsn 16673 . . . . . . 7 (𝑛 ∈ ℤ → (lcm‘{𝑛}) = (abs‘𝑛))
6553a1i 11 . . . . . . . . 9 (𝑛 ∈ ℤ → (lcm‘∅) = 1)
6665oveq1d 7447 . . . . . . . 8 (𝑛 ∈ ℤ → ((lcm‘∅) lcm 𝑛) = (1 lcm 𝑛))
67 1z 12649 . . . . . . . . 9 1 ∈ ℤ
68 lcmcom 16631 . . . . . . . . 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 16648 . . . . . . . 8 (𝑛 ∈ ℤ → (𝑛 lcm 1) = (abs‘𝑛))
7166, 69, 703eqtrrd 2781 . . . . . . 7 (𝑛 ∈ ℤ → (abs‘𝑛) = ((lcm‘∅) lcm 𝑛))
7263, 64, 713eqtrd 2780 . . . . . 6 (𝑛 ∈ ℤ → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
7372adantl 481 . . . . 5 ((∅ ⊆ ℤ ∧ 𝑛 ∈ ℤ) → (lcm‘(∅ ∪ {𝑛})) = ((lcm‘∅) lcm 𝑛))
7473ralrimiva 3145 . . . 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 4189 . . . . . . . 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 3483 . . . . . . . . . . 11 𝑧 ∈ V
8180snss 4784 . . . . . . . . . 10 (𝑧 ∈ ℤ ↔ {𝑧} ⊆ ℤ)
82 lcmfunsnlem1 16675 . . . . . . . . . . . 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 16678 . . . . . . . . . . . 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 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 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 9205 . 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 1539  wcel 2107  wral 3060  cun 3948  wss 3950  c0 4332  {csn 4625   class class class wbr 5142  cfv 6560  (class class class)co 7432  Fincfn 8986  1c1 11157  cz 12615  abscabs 15274  cdvds 16291   lcm clcm 16626  lcmclcmf 16627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-inf 9484  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-fl 13833  df-mod 13911  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-prod 15941  df-dvds 16292  df-gcd 16533  df-lcm 16628  df-lcmf 16629
This theorem is referenced by:  lcmfdvds  16680  lcmfunsn  16682
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