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Theorem lcmftp 15630
Description: The least common multiple of a triple of integers is the least common multiple of the third integer and the least common multiple of the first two integers. Although there would be a shorter proof using lcmfunsn 15638, this explicit proof (not based on induction) should be kept. (Proof modification is discouraged.) (Contributed by AV, 23-Aug-2020.)
Assertion
Ref Expression
lcmftp ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))

Proof of Theorem lcmftp
Dummy variables 𝑘 𝑚 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0z 11635 . . . . . . 7 0 ∈ ℤ
2 eltpg 4383 . . . . . . 7 (0 ∈ ℤ → (0 ∈ {𝐴, 𝐵, 𝐶} ↔ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶)))
31, 2ax-mp 5 . . . . . 6 (0 ∈ {𝐴, 𝐵, 𝐶} ↔ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶))
43biimpri 219 . . . . 5 ((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → 0 ∈ {𝐴, 𝐵, 𝐶})
5 tpssi 4521 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → {𝐴, 𝐵, 𝐶} ⊆ ℤ)
64, 5anim12ci 607 . . . 4 (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ({𝐴, 𝐵, 𝐶} ⊆ ℤ ∧ 0 ∈ {𝐴, 𝐵, 𝐶}))
7 lcmf0val 15616 . . . 4 (({𝐴, 𝐵, 𝐶} ⊆ ℤ ∧ 0 ∈ {𝐴, 𝐵, 𝐶}) → (lcm‘{𝐴, 𝐵, 𝐶}) = 0)
86, 7syl 17 . . 3 (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (lcm‘{𝐴, 𝐵, 𝐶}) = 0)
9 0zd 11636 . . . . . . . . . 10 (𝐶 ∈ ℤ → 0 ∈ ℤ)
10 lcmcom 15587 . . . . . . . . . 10 ((0 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 lcm 𝐶) = (𝐶 lcm 0))
119, 10mpancom 679 . . . . . . . . 9 (𝐶 ∈ ℤ → (0 lcm 𝐶) = (𝐶 lcm 0))
12 lcm0val 15588 . . . . . . . . 9 (𝐶 ∈ ℤ → (𝐶 lcm 0) = 0)
1311, 12eqtrd 2799 . . . . . . . 8 (𝐶 ∈ ℤ → (0 lcm 𝐶) = 0)
1413eqcomd 2771 . . . . . . 7 (𝐶 ∈ ℤ → 0 = (0 lcm 𝐶))
15143ad2ant3 1165 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (0 lcm 𝐶))
1615adantl 473 . . . . 5 ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (0 lcm 𝐶))
17 0zd 11636 . . . . . . . . . . 11 (𝐵 ∈ ℤ → 0 ∈ ℤ)
18 lcmcom 15587 . . . . . . . . . . 11 ((0 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (0 lcm 𝐵) = (𝐵 lcm 0))
1917, 18mpancom 679 . . . . . . . . . 10 (𝐵 ∈ ℤ → (0 lcm 𝐵) = (𝐵 lcm 0))
20 lcm0val 15588 . . . . . . . . . 10 (𝐵 ∈ ℤ → (𝐵 lcm 0) = 0)
2119, 20eqtrd 2799 . . . . . . . . 9 (𝐵 ∈ ℤ → (0 lcm 𝐵) = 0)
2221eqcomd 2771 . . . . . . . 8 (𝐵 ∈ ℤ → 0 = (0 lcm 𝐵))
23223ad2ant2 1164 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (0 lcm 𝐵))
2423adantl 473 . . . . . 6 ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (0 lcm 𝐵))
2524oveq1d 6857 . . . . 5 ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = ((0 lcm 𝐵) lcm 𝐶))
26 oveq1 6849 . . . . . . 7 (0 = 𝐴 → (0 lcm 𝐵) = (𝐴 lcm 𝐵))
2726oveq1d 6857 . . . . . 6 (0 = 𝐴 → ((0 lcm 𝐵) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
2827adantr 472 . . . . 5 ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((0 lcm 𝐵) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
2916, 25, 283eqtrd 2803 . . . 4 ((0 = 𝐴 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
30 lcm0val 15588 . . . . . . . . 9 (𝐴 ∈ ℤ → (𝐴 lcm 0) = 0)
3130eqcomd 2771 . . . . . . . 8 (𝐴 ∈ ℤ → 0 = (𝐴 lcm 0))
32313ad2ant1 1163 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = (𝐴 lcm 0))
3332adantl 473 . . . . . 6 ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = (𝐴 lcm 0))
3433oveq1d 6857 . . . . 5 ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = ((𝐴 lcm 0) lcm 𝐶))
35133ad2ant3 1165 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 lcm 𝐶) = 0)
3635adantl 473 . . . . 5 ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (0 lcm 𝐶) = 0)
37 oveq2 6850 . . . . . . 7 (0 = 𝐵 → (𝐴 lcm 0) = (𝐴 lcm 𝐵))
3837adantr 472 . . . . . 6 ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 0) = (𝐴 lcm 𝐵))
3938oveq1d 6857 . . . . 5 ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 0) lcm 𝐶) = ((𝐴 lcm 𝐵) lcm 𝐶))
4034, 36, 393eqtr3d 2807 . . . 4 ((0 = 𝐵 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
41 lcmcl 15595 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℕ0)
4241nn0zd 11727 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℤ)
43 lcm0val 15588 . . . . . . . 8 ((𝐴 lcm 𝐵) ∈ ℤ → ((𝐴 lcm 𝐵) lcm 0) = 0)
4443eqcomd 2771 . . . . . . 7 ((𝐴 lcm 𝐵) ∈ ℤ → 0 = ((𝐴 lcm 𝐵) lcm 0))
4542, 44syl 17 . . . . . 6 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 0 = ((𝐴 lcm 𝐵) lcm 0))
46453adant3 1162 . . . . 5 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 0 = ((𝐴 lcm 𝐵) lcm 0))
47 oveq2 6850 . . . . 5 (0 = 𝐶 → ((𝐴 lcm 𝐵) lcm 0) = ((𝐴 lcm 𝐵) lcm 𝐶))
4846, 47sylan9eqr 2821 . . . 4 ((0 = 𝐶 ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
4929, 40, 483jaoian 1554 . . 3 (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 = ((𝐴 lcm 𝐵) lcm 𝐶))
508, 49eqtrd 2799 . 2 (((0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))
51423adant3 1162 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 lcm 𝐵) ∈ ℤ)
52 simp3 1168 . . . . . . . . . 10 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐶 ∈ ℤ)
5351, 52jca 507 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
5453adantl 473 . . . . . . . 8 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
55 dvdslcm 15592 . . . . . . . 8 (((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
5654, 55syl 17 . . . . . . 7 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
57 dvdslcm 15592 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)))
58573adant3 1162 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)))
59 simp1 1166 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐴 ∈ ℤ)
60 lcmcl 15595 . . . . . . . . . . . . . . . . . . . 20 (((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ0)
6153, 60syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ0)
6261nn0zd 11727 . . . . . . . . . . . . . . . . . 18 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ)
6359, 51, 623jca 1158 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
64 dvdstr 15303 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ) → ((𝐴 ∥ (𝐴 lcm 𝐵) ∧ (𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
6563, 64syl 17 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 ∥ (𝐴 lcm 𝐵) ∧ (𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
6665expd 404 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∥ (𝐴 lcm 𝐵) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
6766com12 32 . . . . . . . . . . . . . 14 (𝐴 ∥ (𝐴 lcm 𝐵) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
6867adantr 472 . . . . . . . . . . . . 13 ((𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
6958, 68mpcom 38 . . . . . . . . . . . 12 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
7069adantl 473 . . . . . . . . . . 11 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
7170com12 32 . . . . . . . . . 10 ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
7271adantr 472 . . . . . . . . 9 (((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
7372impcom 396 . . . . . . . 8 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))) → 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
74 simpr 477 . . . . . . . . . . . . . . 15 ((𝐴 ∥ (𝐴 lcm 𝐵) ∧ 𝐵 ∥ (𝐴 lcm 𝐵)) → 𝐵 ∥ (𝐴 lcm 𝐵))
7557, 74syl 17 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 𝐵 ∥ (𝐴 lcm 𝐵))
76753adant3 1162 . . . . . . . . . . . . 13 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐵 ∥ (𝐴 lcm 𝐵))
7776adantl 473 . . . . . . . . . . . 12 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∥ (𝐴 lcm 𝐵))
78 simp2 1167 . . . . . . . . . . . . . . 15 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → 𝐵 ∈ ℤ)
7978, 51, 623jca 1158 . . . . . . . . . . . . . 14 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐵 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
8079adantl 473 . . . . . . . . . . . . 13 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐵 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ))
81 dvdstr 15303 . . . . . . . . . . . . 13 ((𝐵 ∈ ℤ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℤ) → ((𝐵 ∥ (𝐴 lcm 𝐵) ∧ (𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
8280, 81syl 17 . . . . . . . . . . . 12 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐵 ∥ (𝐴 lcm 𝐵) ∧ (𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
8377, 82mpand 686 . . . . . . . . . . 11 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
8483com12 32 . . . . . . . . . 10 ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) → ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
8584adantr 472 . . . . . . . . 9 (((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
8685impcom 396 . . . . . . . 8 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))) → 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
87 simpr 477 . . . . . . . . 9 (((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)) → 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
8887adantl 473 . . . . . . . 8 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))) → 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
8973, 86, 883jca 1158 . . . . . . 7 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ ((𝐴 lcm 𝐵) ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))) → (𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
9056, 89mpdan 678 . . . . . 6 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
91 breq1 4812 . . . . . . . 8 (𝑚 = 𝐴 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
92 breq1 4812 . . . . . . . 8 (𝑚 = 𝐵 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
93 breq1 4812 . . . . . . . 8 (𝑚 = 𝐶 → (𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶)))
9491, 92, 93raltpg 4392 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ (𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
9594adantl 473 . . . . . 6 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ↔ (𝐴 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐵 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ 𝐶 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))))
9690, 95mpbird 248 . . . . 5 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶))
97 breq1 4812 . . . . . . . . 9 (𝑚 = 𝐴 → (𝑚𝑘𝐴𝑘))
98 breq1 4812 . . . . . . . . 9 (𝑚 = 𝐵 → (𝑚𝑘𝐵𝑘))
99 breq1 4812 . . . . . . . . 9 (𝑚 = 𝐶 → (𝑚𝑘𝐶𝑘))
10097, 98, 99raltpg 4392 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚𝑘 ↔ (𝐴𝑘𝐵𝑘𝐶𝑘)))
101100ad2antlr 718 . . . . . . 7 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚𝑘 ↔ (𝐴𝑘𝐵𝑘𝐶𝑘)))
102 simpr 477 . . . . . . . . . . 11 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℕ)
10351ad2antlr 718 . . . . . . . . . . 11 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝐴 lcm 𝐵) ∈ ℤ)
10452ad2antlr 718 . . . . . . . . . . 11 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → 𝐶 ∈ ℤ)
105102, 103, 1043jca 1158 . . . . . . . . . 10 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℕ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
106105adantr 472 . . . . . . . . 9 ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴𝑘𝐵𝑘𝐶𝑘)) → (𝑘 ∈ ℕ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ))
107 3ioran 1131 . . . . . . . . . . . . . . . . 17 (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ↔ (¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶))
108 eqcom 2772 . . . . . . . . . . . . . . . . . . . . . 22 (0 = 𝐴𝐴 = 0)
109108notbii 311 . . . . . . . . . . . . . . . . . . . . 21 (¬ 0 = 𝐴 ↔ ¬ 𝐴 = 0)
110 eqcom 2772 . . . . . . . . . . . . . . . . . . . . . 22 (0 = 𝐵𝐵 = 0)
111110notbii 311 . . . . . . . . . . . . . . . . . . . . 21 (¬ 0 = 𝐵 ↔ ¬ 𝐵 = 0)
112109, 111anbi12i 620 . . . . . . . . . . . . . . . . . . . 20 ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
113112biimpi 207 . . . . . . . . . . . . . . . . . . 19 ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
114 ioran 1006 . . . . . . . . . . . . . . . . . . 19 (¬ (𝐴 = 0 ∨ 𝐵 = 0) ↔ (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
115113, 114sylibr 225 . . . . . . . . . . . . . . . . . 18 ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
1161153adant3 1162 . . . . . . . . . . . . . . . . 17 ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
117107, 116sylbi 208 . . . . . . . . . . . . . . . 16 (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
118 id 22 . . . . . . . . . . . . . . . . 17 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
1191183adant3 1162 . . . . . . . . . . . . . . . 16 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
120117, 119anim12ci 607 . . . . . . . . . . . . . . 15 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)))
121 lcmn0cl 15591 . . . . . . . . . . . . . . 15 (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)) → (𝐴 lcm 𝐵) ∈ ℕ)
122120, 121syl 17 . . . . . . . . . . . . . 14 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 𝐵) ∈ ℕ)
123 nnne0 11310 . . . . . . . . . . . . . . 15 ((𝐴 lcm 𝐵) ∈ ℕ → (𝐴 lcm 𝐵) ≠ 0)
124123neneqd 2942 . . . . . . . . . . . . . 14 ((𝐴 lcm 𝐵) ∈ ℕ → ¬ (𝐴 lcm 𝐵) = 0)
125122, 124syl 17 . . . . . . . . . . . . 13 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ (𝐴 lcm 𝐵) = 0)
126 eqcom 2772 . . . . . . . . . . . . . . . . . 18 (0 = 𝐶𝐶 = 0)
127126notbii 311 . . . . . . . . . . . . . . . . 17 (¬ 0 = 𝐶 ↔ ¬ 𝐶 = 0)
128127biimpi 207 . . . . . . . . . . . . . . . 16 (¬ 0 = 𝐶 → ¬ 𝐶 = 0)
1291283ad2ant3 1165 . . . . . . . . . . . . . . 15 ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ 𝐶 = 0)
130107, 129sylbi 208 . . . . . . . . . . . . . 14 (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ 𝐶 = 0)
131130adantr 472 . . . . . . . . . . . . 13 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ 𝐶 = 0)
132125, 131jca 507 . . . . . . . . . . . 12 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
133132adantr 472 . . . . . . . . . . 11 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
134133adantr 472 . . . . . . . . . 10 ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴𝑘𝐵𝑘𝐶𝑘)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
135 ioran 1006 . . . . . . . . . 10 (¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0) ↔ (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
136134, 135sylibr 225 . . . . . . . . 9 ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴𝑘𝐵𝑘𝐶𝑘)) → ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0))
137119adantl 473 . . . . . . . . . . . . . . 15 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
138 nnz 11646 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → 𝑘 ∈ ℤ)
139137, 138anim12ci 607 . . . . . . . . . . . . . 14 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)))
140 3anass 1116 . . . . . . . . . . . . . 14 ((𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (𝑘 ∈ ℤ ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ)))
141139, 140sylibr 225 . . . . . . . . . . . . 13 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ))
142 lcmdvds 15602 . . . . . . . . . . . . 13 ((𝑘 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴𝑘𝐵𝑘) → (𝐴 lcm 𝐵) ∥ 𝑘))
143141, 142syl 17 . . . . . . . . . . . 12 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → ((𝐴𝑘𝐵𝑘) → (𝐴 lcm 𝐵) ∥ 𝑘))
144143com12 32 . . . . . . . . . . 11 ((𝐴𝑘𝐵𝑘) → (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝐴 lcm 𝐵) ∥ 𝑘))
1451443adant3 1162 . . . . . . . . . 10 ((𝐴𝑘𝐵𝑘𝐶𝑘) → (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (𝐴 lcm 𝐵) ∥ 𝑘))
146145impcom 396 . . . . . . . . 9 ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴𝑘𝐵𝑘𝐶𝑘)) → (𝐴 lcm 𝐵) ∥ 𝑘)
147 simp3 1168 . . . . . . . . . 10 ((𝐴𝑘𝐵𝑘𝐶𝑘) → 𝐶𝑘)
148147adantl 473 . . . . . . . . 9 ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴𝑘𝐵𝑘𝐶𝑘)) → 𝐶𝑘)
149 lcmledvds 15593 . . . . . . . . . 10 (((𝑘 ∈ ℕ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0)) → (((𝐴 lcm 𝐵) ∥ 𝑘𝐶𝑘) → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))
150149imp 395 . . . . . . . . 9 ((((𝑘 ∈ ℕ ∧ (𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0)) ∧ ((𝐴 lcm 𝐵) ∥ 𝑘𝐶𝑘)) → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘)
151106, 136, 146, 148, 150syl22anc 867 . . . . . . . 8 ((((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) ∧ (𝐴𝑘𝐵𝑘𝐶𝑘)) → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘)
152151ex 401 . . . . . . 7 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → ((𝐴𝑘𝐵𝑘𝐶𝑘) → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))
153101, 152sylbid 231 . . . . . 6 (((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) ∧ 𝑘 ∈ ℕ) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))
154153ralrimiva 3113 . . . . 5 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ∀𝑘 ∈ ℕ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))
15596, 154jca 507 . . . 4 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ ∀𝑘 ∈ ℕ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘)))
156109biimpi 207 . . . . . . . . . . . . . . . 16 (¬ 0 = 𝐴 → ¬ 𝐴 = 0)
157111biimpi 207 . . . . . . . . . . . . . . . 16 (¬ 0 = 𝐵 → ¬ 𝐵 = 0)
158156, 157anim12i 606 . . . . . . . . . . . . . . 15 ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → (¬ 𝐴 = 0 ∧ ¬ 𝐵 = 0))
159158, 114sylibr 225 . . . . . . . . . . . . . 14 ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
1601593adant3 1162 . . . . . . . . . . . . 13 ((¬ 0 = 𝐴 ∧ ¬ 0 = 𝐵 ∧ ¬ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
161107, 160sylbi 208 . . . . . . . . . . . 12 (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ (𝐴 = 0 ∨ 𝐵 = 0))
162161, 119anim12ci 607 . . . . . . . . . . 11 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝐴 = 0 ∨ 𝐵 = 0)))
163162, 121syl 17 . . . . . . . . . 10 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (𝐴 lcm 𝐵) ∈ ℕ)
164163, 124syl 17 . . . . . . . . 9 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ (𝐴 lcm 𝐵) = 0)
165164, 131jca 507 . . . . . . . 8 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (¬ (𝐴 lcm 𝐵) = 0 ∧ ¬ 𝐶 = 0))
166165, 135sylibr 225 . . . . . . 7 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0))
16754, 166jca 507 . . . . . 6 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0)))
168 lcmn0cl 15591 . . . . . 6 ((((𝐴 lcm 𝐵) ∈ ℤ ∧ 𝐶 ∈ ℤ) ∧ ¬ ((𝐴 lcm 𝐵) = 0 ∨ 𝐶 = 0)) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ)
169167, 168syl 17 . . . . 5 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ)
1705adantl 473 . . . . 5 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → {𝐴, 𝐵, 𝐶} ⊆ ℤ)
171 tpfi 8443 . . . . . 6 {𝐴, 𝐵, 𝐶} ∈ Fin
172171a1i 11 . . . . 5 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → {𝐴, 𝐵, 𝐶} ∈ Fin)
1733a1i 11 . . . . . . . . 9 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 ∈ {𝐴, 𝐵, 𝐶} ↔ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶)))
174173biimpd 220 . . . . . . . 8 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (0 ∈ {𝐴, 𝐵, 𝐶} → (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶)))
175174con3d 149 . . . . . . 7 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) → ¬ 0 ∈ {𝐴, 𝐵, 𝐶}))
176175impcom 396 . . . . . 6 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ¬ 0 ∈ {𝐴, 𝐵, 𝐶})
177 df-nel 3041 . . . . . 6 (0 ∉ {𝐴, 𝐵, 𝐶} ↔ ¬ 0 ∈ {𝐴, 𝐵, 𝐶})
178176, 177sylibr 225 . . . . 5 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → 0 ∉ {𝐴, 𝐵, 𝐶})
179 lcmf 15627 . . . . 5 ((((𝐴 lcm 𝐵) lcm 𝐶) ∈ ℕ ∧ ({𝐴, 𝐵, 𝐶} ⊆ ℤ ∧ {𝐴, 𝐵, 𝐶} ∈ Fin ∧ 0 ∉ {𝐴, 𝐵, 𝐶})) → (((𝐴 lcm 𝐵) lcm 𝐶) = (lcm‘{𝐴, 𝐵, 𝐶}) ↔ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ ∀𝑘 ∈ ℕ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))))
180169, 170, 172, 178, 179syl13anc 1491 . . . 4 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (((𝐴 lcm 𝐵) lcm 𝐶) = (lcm‘{𝐴, 𝐵, 𝐶}) ↔ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚 ∥ ((𝐴 lcm 𝐵) lcm 𝐶) ∧ ∀𝑘 ∈ ℕ (∀𝑚 ∈ {𝐴, 𝐵, 𝐶}𝑚𝑘 → ((𝐴 lcm 𝐵) lcm 𝐶) ≤ 𝑘))))
181155, 180mpbird 248 . . 3 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → ((𝐴 lcm 𝐵) lcm 𝐶) = (lcm‘{𝐴, 𝐵, 𝐶}))
182181eqcomd 2771 . 2 ((¬ (0 = 𝐴 ∨ 0 = 𝐵 ∨ 0 = 𝐶) ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ)) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))
18350, 182pm2.61ian 846 1 ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (lcm‘{𝐴, 𝐵, 𝐶}) = ((𝐴 lcm 𝐵) lcm 𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3o 1106  w3a 1107   = wceq 1652  wcel 2155  wnel 3040  wral 3055  wss 3732  {ctp 4338   class class class wbr 4809  cfv 6068  (class class class)co 6842  Fincfn 8160  0cc0 10189  cle 10329  cn 11274  0cn0 11538  cz 11624  cdvds 15265   lcm clcm 15582  lcmclcmf 15583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-n0 11539  df-z 11625  df-uz 11887  df-rp 12029  df-fz 12534  df-fzo 12674  df-fl 12801  df-mod 12877  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-prod 14919  df-dvds 15266  df-gcd 15498  df-lcm 15584  df-lcmf 15585
This theorem is referenced by:  lcmf2a3a4e12  15641
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