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Theorem lcmneg 12057
Description: Negating one operand of the lcm operator does not alter the result. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Assertion
Ref Expression
lcmneg ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (𝑀 lcm 𝑁))

Proof of Theorem lcmneg
StepHypRef Expression
1 lcm0val 12048 . . . . . . . 8 (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0)
2 znegcl 9273 . . . . . . . . 9 (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
3 lcm0val 12048 . . . . . . . . 9 (-𝑁 ∈ ℤ → (-𝑁 lcm 0) = 0)
42, 3syl 14 . . . . . . . 8 (𝑁 ∈ ℤ → (-𝑁 lcm 0) = 0)
51, 4eqtr4d 2213 . . . . . . 7 (𝑁 ∈ ℤ → (𝑁 lcm 0) = (-𝑁 lcm 0))
65ad2antlr 489 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑁 lcm 0) = (-𝑁 lcm 0))
7 oveq2 5877 . . . . . . . 8 (𝑀 = 0 → (𝑁 lcm 𝑀) = (𝑁 lcm 0))
8 oveq2 5877 . . . . . . . 8 (𝑀 = 0 → (-𝑁 lcm 𝑀) = (-𝑁 lcm 0))
97, 8eqeq12d 2192 . . . . . . 7 (𝑀 = 0 → ((𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀) ↔ (𝑁 lcm 0) = (-𝑁 lcm 0)))
109adantl 277 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀) ↔ (𝑁 lcm 0) = (-𝑁 lcm 0)))
116, 10mpbird 167 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀))
12 lcmcom 12047 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑁 lcm 𝑀))
13 lcmcom 12047 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (-𝑁 lcm 𝑀))
142, 13sylan2 286 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (-𝑁 lcm 𝑀))
1512, 14eqeq12d 2192 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀)))
1615adantr 276 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀)))
1711, 16mpbird 167 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
18 neg0 8193 . . . . . . . 8 -0 = 0
1918oveq2i 5880 . . . . . . 7 (𝑀 lcm -0) = (𝑀 lcm 0)
2019eqcomi 2181 . . . . . 6 (𝑀 lcm 0) = (𝑀 lcm -0)
21 oveq2 5877 . . . . . 6 (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm 0))
22 negeq 8140 . . . . . . 7 (𝑁 = 0 → -𝑁 = -0)
2322oveq2d 5885 . . . . . 6 (𝑁 = 0 → (𝑀 lcm -𝑁) = (𝑀 lcm -0))
2420, 21, 233eqtr4a 2236 . . . . 5 (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
2524adantl 277 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
2617, 25jaodan 797 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
27 dvdslcm 12052 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁)))
282, 27sylan2 286 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁)))
29 simpr 110 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
30 lcmcl 12055 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℕ0)
312, 30sylan2 286 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℕ0)
3231nn0zd 9362 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℤ)
33 negdvdsb 11798 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ (𝑀 lcm -𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 lcm -𝑁) ↔ -𝑁 ∥ (𝑀 lcm -𝑁)))
3429, 32, 33syl2anc 411 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∥ (𝑀 lcm -𝑁) ↔ -𝑁 ∥ (𝑀 lcm -𝑁)))
3534anbi2d 464 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)) ↔ (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁))))
3628, 35mpbird 167 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)))
3736adantr 276 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)))
38 zcn 9247 . . . . . . . . . . . . 13 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3938negeq0d 8250 . . . . . . . . . . . 12 (𝑁 ∈ ℤ → (𝑁 = 0 ↔ -𝑁 = 0))
4039orbi2d 790 . . . . . . . . . . 11 (𝑁 ∈ ℤ → ((𝑀 = 0 ∨ 𝑁 = 0) ↔ (𝑀 = 0 ∨ -𝑁 = 0)))
4140notbid 667 . . . . . . . . . 10 (𝑁 ∈ ℤ → (¬ (𝑀 = 0 ∨ 𝑁 = 0) ↔ ¬ (𝑀 = 0 ∨ -𝑁 = 0)))
4241biimpa 296 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ -𝑁 = 0))
4342adantll 476 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ -𝑁 = 0))
44 lcmn0cl 12051 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
452, 44sylanl2 403 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
4643, 45syldan 282 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
47 simpl 109 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
48 3anass 982 . . . . . . 7 (((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ ((𝑀 lcm -𝑁) ∈ ℕ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
4946, 47, 48sylanbrc 417 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
50 simpr 110 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ 𝑁 = 0))
51 lcmledvds 12053 . . . . . 6 ((((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)) → (𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁)))
5249, 50, 51syl2anc 411 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)) → (𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁)))
5337, 52mpd 13 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁))
54 dvdslcm 12052 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
5554adantr 276 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
56 simplr 528 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℤ)
57 lcmn0cl 12051 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℕ)
5857nnzd 9363 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℤ)
59 negdvdsb 11798 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 lcm 𝑁) ↔ -𝑁 ∥ (𝑀 lcm 𝑁)))
6056, 58, 59syl2anc 411 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑁 ∥ (𝑀 lcm 𝑁) ↔ -𝑁 ∥ (𝑀 lcm 𝑁)))
6160anbi2d 464 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)) ↔ (𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁))))
62 lcmledvds 12053 . . . . . . . . . 10 ((((𝑀 lcm 𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁)))
6362ex 115 . . . . . . . . 9 (((𝑀 lcm 𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∨ -𝑁 = 0) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
642, 63syl3an3 1273 . . . . . . . 8 (((𝑀 lcm 𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∨ -𝑁 = 0) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
65643expib 1206 . . . . . . 7 ((𝑀 lcm 𝑁) ∈ ℕ → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∨ -𝑁 = 0) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁)))))
6657, 47, 43, 65syl3c 63 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ -𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁)))
6761, 66sylbid 150 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁)))
6855, 67mpd 13 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))
69 lcmcl 12055 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0)
7069nn0red 9219 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℝ)
7130nn0red 9219 . . . . . . 7 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℝ)
722, 71sylan2 286 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℝ)
7370, 72letri3d 8063 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ ((𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁) ∧ (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
7473adantr 276 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ ((𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁) ∧ (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
7553, 68, 74mpbir2and 944 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
76 lcmmndc 12045 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0))
77 exmiddc 836 . . . 4 (DECID (𝑀 = 0 ∨ 𝑁 = 0) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0)))
7876, 77syl 14 . . 3 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∨ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∨ 𝑁 = 0)))
7926, 75, 78mpjaodan 798 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
8079eqcomd 2183 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (𝑀 lcm 𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wcel 2148   class class class wbr 4000  (class class class)co 5869  cr 7801  0cc0 7802  cle 7983  -cneg 8119  cn 8908  0cn0 9165  cz 9242  cdvds 11778   lcm clcm 12043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-lcm 12044
This theorem is referenced by:  neglcm  12058  lcmabs  12059
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