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Theorem lcmneg 10663
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 10654 . . . . . . . 8 (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0)
2 znegcl 8515 . . . . . . . . 9 (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
3 lcm0val 10654 . . . . . . . . 9 (-𝑁 ∈ ℤ → (-𝑁 lcm 0) = 0)
42, 3syl 14 . . . . . . . 8 (𝑁 ∈ ℤ → (-𝑁 lcm 0) = 0)
51, 4eqtr4d 2118 . . . . . . 7 (𝑁 ∈ ℤ → (𝑁 lcm 0) = (-𝑁 lcm 0))
65ad2antlr 473 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑁 lcm 0) = (-𝑁 lcm 0))
7 oveq2 5571 . . . . . . . 8 (𝑀 = 0 → (𝑁 lcm 𝑀) = (𝑁 lcm 0))
8 oveq2 5571 . . . . . . . 8 (𝑀 = 0 → (-𝑁 lcm 𝑀) = (-𝑁 lcm 0))
97, 8eqeq12d 2097 . . . . . . 7 (𝑀 = 0 → ((𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀) ↔ (𝑁 lcm 0) = (-𝑁 lcm 0)))
109adantl 271 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀) ↔ (𝑁 lcm 0) = (-𝑁 lcm 0)))
116, 10mpbird 165 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀))
12 lcmcom 10653 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑁 lcm 𝑀))
13 lcmcom 10653 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (-𝑁 lcm 𝑀))
142, 13sylan2 280 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (-𝑁 lcm 𝑀))
1512, 14eqeq12d 2097 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀)))
1615adantr 270 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀)))
1711, 16mpbird 165 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
18 neg0 7473 . . . . . . . 8 -0 = 0
1918oveq2i 5574 . . . . . . 7 (𝑀 lcm -0) = (𝑀 lcm 0)
2019eqcomi 2087 . . . . . 6 (𝑀 lcm 0) = (𝑀 lcm -0)
21 oveq2 5571 . . . . . 6 (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm 0))
22 negeq 7420 . . . . . . 7 (𝑁 = 0 → -𝑁 = -0)
2322oveq2d 5579 . . . . . 6 (𝑁 = 0 → (𝑀 lcm -𝑁) = (𝑀 lcm -0))
2420, 21, 233eqtr4a 2141 . . . . 5 (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
2524adantl 271 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
2617, 25jaodan 744 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
27 dvdslcm 10658 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁)))
282, 27sylan2 280 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁)))
29 simpr 108 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
30 lcmcl 10661 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℕ0)
312, 30sylan2 280 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℕ0)
3231nn0zd 8600 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℤ)
33 negdvdsb 10419 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ (𝑀 lcm -𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 lcm -𝑁) ↔ -𝑁 ∥ (𝑀 lcm -𝑁)))
3429, 32, 33syl2anc 403 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∥ (𝑀 lcm -𝑁) ↔ -𝑁 ∥ (𝑀 lcm -𝑁)))
3534anbi2d 452 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)) ↔ (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁))))
3628, 35mpbird 165 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)))
3736adantr 270 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)))
38 zcn 8489 . . . . . . . . . . . . 13 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3938negeq0d 7530 . . . . . . . . . . . 12 (𝑁 ∈ ℤ → (𝑁 = 0 ↔ -𝑁 = 0))
4039orbi2d 737 . . . . . . . . . . 11 (𝑁 ∈ ℤ → ((𝑀 = 0 ∨ 𝑁 = 0) ↔ (𝑀 = 0 ∨ -𝑁 = 0)))
4140notbid 625 . . . . . . . . . 10 (𝑁 ∈ ℤ → (¬ (𝑀 = 0 ∨ 𝑁 = 0) ↔ ¬ (𝑀 = 0 ∨ -𝑁 = 0)))
4241biimpa 290 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ -𝑁 = 0))
4342adantll 460 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ -𝑁 = 0))
44 lcmn0cl 10657 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
452, 44sylanl2 395 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
4643, 45syldan 276 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
47 simpl 107 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
48 3anass 924 . . . . . . 7 (((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ ((𝑀 lcm -𝑁) ∈ ℕ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
4946, 47, 48sylanbrc 408 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
50 simpr 108 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ 𝑁 = 0))
51 lcmledvds 10659 . . . . . 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 403 . . . . 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 10658 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
5554adantr 270 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
56 simplr 497 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℤ)
57 lcmn0cl 10657 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℕ)
5857nnzd 8601 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℤ)
59 negdvdsb 10419 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 lcm 𝑁) ↔ -𝑁 ∥ (𝑀 lcm 𝑁)))
6056, 58, 59syl2anc 403 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑁 ∥ (𝑀 lcm 𝑁) ↔ -𝑁 ∥ (𝑀 lcm 𝑁)))
6160anbi2d 452 . . . . . 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 10659 . . . . . . . . . 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 113 . . . . . . . . 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 1205 . . . . . . . 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 1142 . . . . . . 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 62 . . . . . 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 148 . . . . 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 10661 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0)
7069nn0red 8461 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℝ)
7130nn0red 8461 . . . . . . 7 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℝ)
722, 71sylan2 280 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℝ)
7370, 72letri3d 7345 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ ((𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁) ∧ (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
7473adantr 270 . . . 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 886 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
76 lcmmndc 10651 . . . 4 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∨ 𝑁 = 0))
77 exmiddc 778 . . . 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 745 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
8079eqcomd 2088 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (𝑀 lcm 𝑁))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 776  w3a 920   = wceq 1285  wcel 1434   class class class wbr 3805  (class class class)co 5563  cr 7094  0cc0 7095  cle 7268  -cneg 7399  cn 8158  0cn0 8407  cz 8484  cdvds 10403   lcm clcm 10649
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208  ax-arch 7209  ax-caucvg 7210
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-isom 4961  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-frec 6060  df-sup 6491  df-inf 6492  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880  df-inn 8159  df-2 8217  df-3 8218  df-4 8219  df-n0 8408  df-z 8485  df-uz 8753  df-q 8838  df-rp 8868  df-fz 9158  df-fzo 9282  df-fl 9404  df-mod 9457  df-iseq 9574  df-iexp 9625  df-cj 9930  df-re 9931  df-im 9932  df-rsqrt 10085  df-abs 10086  df-dvds 10404  df-lcm 10650
This theorem is referenced by:  neglcm  10664  lcmabs  10665
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