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Theorem lcmneg 16579
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 16570 . . . . . . . 8 (𝑁 ∈ ℤ → (𝑁 lcm 0) = 0)
2 znegcl 12633 . . . . . . . . 9 (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
3 lcm0val 16570 . . . . . . . . 9 (-𝑁 ∈ ℤ → (-𝑁 lcm 0) = 0)
42, 3syl 17 . . . . . . . 8 (𝑁 ∈ ℤ → (-𝑁 lcm 0) = 0)
51, 4eqtr4d 2770 . . . . . . 7 (𝑁 ∈ ℤ → (𝑁 lcm 0) = (-𝑁 lcm 0))
65ad2antlr 725 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑁 lcm 0) = (-𝑁 lcm 0))
7 oveq2 7432 . . . . . . . 8 (𝑀 = 0 → (𝑁 lcm 𝑀) = (𝑁 lcm 0))
8 oveq2 7432 . . . . . . . 8 (𝑀 = 0 → (-𝑁 lcm 𝑀) = (-𝑁 lcm 0))
97, 8eqeq12d 2743 . . . . . . 7 (𝑀 = 0 → ((𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀) ↔ (𝑁 lcm 0) = (-𝑁 lcm 0)))
109adantl 480 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀) ↔ (𝑁 lcm 0) = (-𝑁 lcm 0)))
116, 10mpbird 256 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀))
12 lcmcom 16569 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑁 lcm 𝑀))
13 lcmcom 16569 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (-𝑁 lcm 𝑀))
142, 13sylan2 591 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (-𝑁 lcm 𝑀))
1512, 14eqeq12d 2743 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀)))
1615adantr 479 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ (𝑁 lcm 𝑀) = (-𝑁 lcm 𝑀)))
1711, 16mpbird 256 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑀 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
18 neg0 11542 . . . . . . . 8 -0 = 0
1918oveq2i 7435 . . . . . . 7 (𝑀 lcm -0) = (𝑀 lcm 0)
2019eqcomi 2736 . . . . . 6 (𝑀 lcm 0) = (𝑀 lcm -0)
21 oveq2 7432 . . . . . 6 (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm 0))
22 negeq 11488 . . . . . . 7 (𝑁 = 0 → -𝑁 = -0)
2322oveq2d 7440 . . . . . 6 (𝑁 = 0 → (𝑀 lcm -𝑁) = (𝑀 lcm -0))
2420, 21, 233eqtr4a 2793 . . . . 5 (𝑁 = 0 → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
2524adantl 480 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑁 = 0) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
2617, 25jaodan 955 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
27 dvdslcm 16574 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁)))
282, 27sylan2 591 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁)))
29 simpr 483 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
30 lcmcl 16577 . . . . . . . . . . 11 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℕ0)
312, 30sylan2 591 . . . . . . . . . 10 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℕ0)
3231nn0zd 12620 . . . . . . . . 9 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℤ)
33 negdvdsb 16255 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ (𝑀 lcm -𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 lcm -𝑁) ↔ -𝑁 ∥ (𝑀 lcm -𝑁)))
3429, 32, 33syl2anc 582 . . . . . . . 8 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∥ (𝑀 lcm -𝑁) ↔ -𝑁 ∥ (𝑀 lcm -𝑁)))
3534anbi2d 628 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)) ↔ (𝑀 ∥ (𝑀 lcm -𝑁) ∧ -𝑁 ∥ (𝑀 lcm -𝑁))))
3628, 35mpbird 256 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)))
3736adantr 479 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm -𝑁) ∧ 𝑁 ∥ (𝑀 lcm -𝑁)))
38 zcn 12599 . . . . . . . . . . . . 13 (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3938negeq0d 11599 . . . . . . . . . . . 12 (𝑁 ∈ ℤ → (𝑁 = 0 ↔ -𝑁 = 0))
4039orbi2d 913 . . . . . . . . . . 11 (𝑁 ∈ ℤ → ((𝑀 = 0 ∨ 𝑁 = 0) ↔ (𝑀 = 0 ∨ -𝑁 = 0)))
4140notbid 317 . . . . . . . . . 10 (𝑁 ∈ ℤ → (¬ (𝑀 = 0 ∨ 𝑁 = 0) ↔ ¬ (𝑀 = 0 ∨ -𝑁 = 0)))
4241biimpa 475 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ -𝑁 = 0))
4342adantll 712 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ -𝑁 = 0))
44 lcmn0cl 16573 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
452, 44sylanl2 679 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ -𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
4643, 45syldan 589 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm -𝑁) ∈ ℕ)
47 simpl 481 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
48 3anass 1092 . . . . . . 7 (((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ ((𝑀 lcm -𝑁) ∈ ℕ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
4946, 47, 48sylanbrc 581 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ((𝑀 lcm -𝑁) ∈ ℕ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ))
50 simpr 483 . . . . . 6 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → ¬ (𝑀 = 0 ∨ 𝑁 = 0))
51 lcmledvds 16575 . . . . . 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 582 . . . . 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 15 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁))
54 dvdslcm 16574 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
5554adantr 479 . . . . 5 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
56 simplr 767 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → 𝑁 ∈ ℤ)
57 lcmn0cl 16573 . . . . . . . . 9 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℕ)
5857nnzd 12621 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) ∈ ℤ)
59 negdvdsb 16255 . . . . . . . 8 ((𝑁 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ) → (𝑁 ∥ (𝑀 lcm 𝑁) ↔ -𝑁 ∥ (𝑀 lcm 𝑁)))
6056, 58, 59syl2anc 582 . . . . . . 7 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑁 ∥ (𝑀 lcm 𝑁) ↔ -𝑁 ∥ (𝑀 lcm 𝑁)))
6160anbi2d 628 . . . . . 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 16575 . . . . . . . . . 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 411 . . . . . . . . 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 1162 . . . . . . . 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 1119 . . . . . . 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 66 . . . . . 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 239 . . . . 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 15 . . . 4 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))
69 lcmcl 16577 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0)
7069nn0red 12569 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℝ)
7130nn0red 12569 . . . . . . 7 ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℝ)
722, 71sylan2 591 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) ∈ ℝ)
7370, 72letri3d 11392 . . . . 5 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) = (𝑀 lcm -𝑁) ↔ ((𝑀 lcm 𝑁) ≤ (𝑀 lcm -𝑁) ∧ (𝑀 lcm -𝑁) ≤ (𝑀 lcm 𝑁))))
7473adantr 479 . . . 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 711 . . 3 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∨ 𝑁 = 0)) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
7626, 75pm2.61dan 811 . 2 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) = (𝑀 lcm -𝑁))
7776eqcomd 2733 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm -𝑁) = (𝑀 lcm 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3a 1084   = wceq 1533  wcel 2098   class class class wbr 5150  (class class class)co 7424  cr 11143  0cc0 11144  cle 11285  -cneg 11481  cn 12248  0cn0 12508  cz 12594  cdvds 16236   lcm clcm 16564
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221  ax-pre-sup 11222
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-sup 9471  df-inf 9472  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-div 11908  df-nn 12249  df-2 12311  df-3 12312  df-n0 12509  df-z 12595  df-uz 12859  df-rp 13013  df-seq 14005  df-exp 14065  df-cj 15084  df-re 15085  df-im 15086  df-sqrt 15220  df-abs 15221  df-dvds 16237  df-lcm 16566
This theorem is referenced by:  neglcm  16580  lcmabs  16581
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