Theorem lcmftp 15349

Description: The least common multiple of a triple of integers is the least common multiple of the third integer and the the least common multiple of the first two integers. Although there would be a shorter proof using lcmfunsn 15357, 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 e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> (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.

Step	Hyp	Ref	Expression
1		0z 11388	. . . . . . 7 |- 0 e. ZZ
2		eltpg 4227	. . . . . . 7 |- ( 0 e. ZZ -> ( 0 e. { A , B , C } <-> ( 0 = A \/ 0 = B \/ 0 = C ) ) )
3	1, 2	ax-mp 5	. . . . . 6 |- ( 0 e. { A , B , C } <-> ( 0 = A \/ 0 = B \/ 0 = C ) )
4	3	biimpri 218	. . . . 5 |- ( ( 0 = A \/ 0 = B \/ 0 = C ) -> 0 e. { A , B , C } )
5		tpssi 4369	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> { A , B , C } C_ ZZ )
6	4, 5	anim12ci 591	. . . 4 |- ( ( ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( { A , B , C } C_ ZZ /\ 0 e. { A , B , C } ) )
7		lcmf0val 15335	. . . 4 |- ( ( { A , B , C } C_ ZZ /\ 0 e. { A , B , C } ) -> (lcm ` { A , B , C } ) = 0 )
8	6, 7	syl 17	. . 3 |- ( ( ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> (lcm ` { A , B , C } ) = 0 )
9		0zd 11389	. . . . . . . . . 10 |- ( C e. ZZ -> 0 e. ZZ )
10		lcmcom 15306	. . . . . . . . . 10 |- ( ( 0 e. ZZ /\ C e. ZZ ) -> ( 0 lcm C ) = ( C lcm 0 ) )
11	9, 10	mpancom 703	. . . . . . . . 9 |- ( C e. ZZ -> ( 0 lcm C ) = ( C lcm 0 ) )
12		lcm0val 15307	. . . . . . . . 9 |- ( C e. ZZ -> ( C lcm 0 ) = 0 )
13	11, 12	eqtrd 2656	. . . . . . . 8 |- ( C e. ZZ -> ( 0 lcm C ) = 0 )
14	13	eqcomd 2628	. . . . . . 7 |- ( C e. ZZ -> 0 = ( 0 lcm C ) )
15	14	3ad2ant3 1084	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> 0 = ( 0 lcm C ) )
16	15	adantl 482	. . . . 5 |- ( ( 0 = A /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> 0 = ( 0 lcm C ) )
17		0zd 11389	. . . . . . . . . . 11 |- ( B e. ZZ -> 0 e. ZZ )
18		lcmcom 15306	. . . . . . . . . . 11 |- ( ( 0 e. ZZ /\ B e. ZZ ) -> ( 0 lcm B ) = ( B lcm 0 ) )
19	17, 18	mpancom 703	. . . . . . . . . 10 |- ( B e. ZZ -> ( 0 lcm B ) = ( B lcm 0 ) )
20		lcm0val 15307	. . . . . . . . . 10 |- ( B e. ZZ -> ( B lcm 0 ) = 0 )
21	19, 20	eqtrd 2656	. . . . . . . . 9 |- ( B e. ZZ -> ( 0 lcm B ) = 0 )
22	21	eqcomd 2628	. . . . . . . 8 |- ( B e. ZZ -> 0 = ( 0 lcm B ) )
23	22	3ad2ant2 1083	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> 0 = ( 0 lcm B ) )
24	23	adantl 482	. . . . . 6 |- ( ( 0 = A /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> 0 = ( 0 lcm B ) )
25	24	oveq1d 6665	. . . . 5 |- ( ( 0 = A /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( 0 lcm C ) = ( ( 0 lcm B ) lcm C ) )
26		oveq1 6657	. . . . . . 7 |- ( 0 = A -> ( 0 lcm B ) = ( A lcm B ) )
27	26	oveq1d 6665	. . . . . 6 |- ( 0 = A -> ( ( 0 lcm B ) lcm C ) = ( ( A lcm B ) lcm C ) )
28	27	adantr 481	. . . . 5 |- ( ( 0 = A /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( 0 lcm B ) lcm C ) = ( ( A lcm B ) lcm C ) )
29	16, 25, 28	3eqtrd 2660	. . . 4 |- ( ( 0 = A /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> 0 = ( ( A lcm B ) lcm C ) )
30		lcm0val 15307	. . . . . . . . 9 |- ( A e. ZZ -> ( A lcm 0 ) = 0 )
31	30	eqcomd 2628	. . . . . . . 8 |- ( A e. ZZ -> 0 = ( A lcm 0 ) )
32	31	3ad2ant1 1082	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> 0 = ( A lcm 0 ) )
33	32	adantl 482	. . . . . 6 |- ( ( 0 = B /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> 0 = ( A lcm 0 ) )
34	33	oveq1d 6665	. . . . 5 |- ( ( 0 = B /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( 0 lcm C ) = ( ( A lcm 0 ) lcm C ) )
35	13	3ad2ant3 1084	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( 0 lcm C ) = 0 )
36	35	adantl 482	. . . . 5 |- ( ( 0 = B /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( 0 lcm C ) = 0 )
37		oveq2 6658	. . . . . . 7 |- ( 0 = B -> ( A lcm 0 ) = ( A lcm B ) )
38	37	adantr 481	. . . . . 6 |- ( ( 0 = B /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( A lcm 0 ) = ( A lcm B ) )
39	38	oveq1d 6665	. . . . 5 |- ( ( 0 = B /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( A lcm 0 ) lcm C ) = ( ( A lcm B ) lcm C ) )
40	34, 36, 39	3eqtr3d 2664	. . . 4 |- ( ( 0 = B /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> 0 = ( ( A lcm B ) lcm C ) )
41		lcmcl 15314	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A lcm B ) e. NN0 )
42	41	nn0zd 11480	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A lcm B ) e. ZZ )
43		lcm0val 15307	. . . . . . . 8 |- ( ( A lcm B ) e. ZZ -> ( ( A lcm B ) lcm 0 ) = 0 )
44	43	eqcomd 2628	. . . . . . 7 |- ( ( A lcm B ) e. ZZ -> 0 = ( ( A lcm B ) lcm 0 ) )
45	42, 44	syl 17	. . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> 0 = ( ( A lcm B ) lcm 0 ) )
46	45	3adant3 1081	. . . . 5 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> 0 = ( ( A lcm B ) lcm 0 ) )
47		oveq2 6658	. . . . 5 |- ( 0 = C -> ( ( A lcm B ) lcm 0 ) = ( ( A lcm B ) lcm C ) )
48	46, 47	sylan9eqr 2678	. . . 4 |- ( ( 0 = C /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> 0 = ( ( A lcm B ) lcm C ) )
49	29, 40, 48	3jaoian 1393	. . 3 |- ( ( ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> 0 = ( ( A lcm B ) lcm C ) )
50	8, 49	eqtrd 2656	. 2 |- ( ( ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> (lcm ` { A , B , C } ) = ( ( A lcm B ) lcm C ) )
51	42	3adant3 1081	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A lcm B ) e. ZZ )
52		simp3 1063	. . . . . . . . . 10 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> C e. ZZ )
53	51, 52	jca 554	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A lcm B ) e. ZZ /\ C e. ZZ ) )
54	53	adantl 482	. . . . . . . 8 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( A lcm B ) e. ZZ /\ C e. ZZ ) )
55		dvdslcm 15311	. . . . . . . 8 |- ( ( ( A lcm B ) e. ZZ /\ C e. ZZ ) -> ( ( A lcm B ) || ( ( A lcm B ) lcm C ) /\ C || ( ( A lcm B ) lcm C ) ) )
56	54, 55	syl 17	. . . . . . 7 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( A lcm B ) || ( ( A lcm B ) lcm C ) /\ C || ( ( A lcm B ) lcm C ) ) )
57		dvdslcm 15311	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A || ( A lcm B ) /\ B || ( A lcm B ) ) )
58	57	3adant3 1081	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A || ( A lcm B ) /\ B || ( A lcm B ) ) )
59		simp1 1061	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> A e. ZZ )
60		lcmcl 15314	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( A lcm B ) e. ZZ /\ C e. ZZ ) -> ( ( A lcm B ) lcm C ) e. NN0 )
61	53, 60	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A lcm B ) lcm C ) e. NN0 )
62	61	nn0zd 11480	. . . . . . . . . . . . . . . . . 18 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A lcm B ) lcm C ) e. ZZ )
63	59, 51, 62	3jca 1242	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A e. ZZ /\ ( A lcm B ) e. ZZ /\ ( ( A lcm B ) lcm C ) e. ZZ ) )
64		dvdstr 15018	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ ( A lcm B ) e. ZZ /\ ( ( A lcm B ) lcm C ) e. ZZ ) -> ( ( A || ( A lcm B ) /\ ( A lcm B ) || ( ( A lcm B ) lcm C ) ) -> A || ( ( A lcm B ) lcm C ) ) )
65	63, 64	syl 17	. . . . . . . . . . . . . . . 16 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A || ( A lcm B ) /\ ( A lcm B ) || ( ( A lcm B ) lcm C ) ) -> A || ( ( A lcm B ) lcm C ) ) )
66	65	expd 452	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A || ( A lcm B ) -> ( ( A lcm B ) || ( ( A lcm B ) lcm C ) -> A || ( ( A lcm B ) lcm C ) ) ) )
67	66	com12 32	. . . . . . . . . . . . . 14 |- ( A || ( A lcm B ) -> ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A lcm B ) || ( ( A lcm B ) lcm C ) -> A || ( ( A lcm B ) lcm C ) ) ) )
68	67	adantr 481	. . . . . . . . . . . . 13 |- ( ( A || ( A lcm B ) /\ B || ( A lcm B ) ) -> ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A lcm B ) || ( ( A lcm B ) lcm C ) -> A || ( ( A lcm B ) lcm C ) ) ) )
69	58, 68	mpcom 38	. . . . . . . . . . . 12 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A lcm B ) || ( ( A lcm B ) lcm C ) -> A || ( ( A lcm B ) lcm C ) ) )
70	69	adantl 482	. . . . . . . . . . 11 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( A lcm B ) || ( ( A lcm B ) lcm C ) -> A || ( ( A lcm B ) lcm C ) ) )
71	70	com12 32	. . . . . . . . . 10 |- ( ( A lcm B ) || ( ( A lcm B ) lcm C ) -> ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> A || ( ( A lcm B ) lcm C ) ) )
72	71	adantr 481	. . . . . . . . 9 |- ( ( ( A lcm B ) || ( ( A lcm B ) lcm C ) /\ C || ( ( A lcm B ) lcm C ) ) -> ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> A || ( ( A lcm B ) lcm C ) ) )
73	72	impcom 446	. . . . . . . 8 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ ( ( 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 ) )
75	57, 74	syl 17	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ B e. ZZ ) -> B || ( A lcm B ) )
76	75	3adant3 1081	. . . . . . . . . . . . 13 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> B || ( A lcm B ) )
77	76	adantl 482	. . . . . . . . . . . 12 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> B || ( A lcm B ) )
78		simp2 1062	. . . . . . . . . . . . . . 15 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> B e. ZZ )
79	78, 51, 62	3jca 1242	. . . . . . . . . . . . . 14 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( B e. ZZ /\ ( A lcm B ) e. ZZ /\ ( ( A lcm B ) lcm C ) e. ZZ ) )
80	79	adantl 482	. . . . . . . . . . . . 13 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( B e. ZZ /\ ( A lcm B ) e. ZZ /\ ( ( A lcm B ) lcm C ) e. ZZ ) )
81		dvdstr 15018	. . . . . . . . . . . . 13 |- ( ( B e. ZZ /\ ( A lcm B ) e. ZZ /\ ( ( A lcm B ) lcm C ) e. ZZ ) -> ( ( B || ( A lcm B ) /\ ( A lcm B ) || ( ( A lcm B ) lcm C ) ) -> B || ( ( A lcm B ) lcm C ) ) )
82	80, 81	syl 17	. . . . . . . . . . . 12 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( B || ( A lcm B ) /\ ( A lcm B ) || ( ( A lcm B ) lcm C ) ) -> B || ( ( A lcm B ) lcm C ) ) )
83	77, 82	mpand 711	. . . . . . . . . . 11 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( A lcm B ) || ( ( A lcm B ) lcm C ) -> B || ( ( A lcm B ) lcm C ) ) )
84	83	com12 32	. . . . . . . . . 10 |- ( ( A lcm B ) || ( ( A lcm B ) lcm C ) -> ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> B || ( ( A lcm B ) lcm C ) ) )
85	84	adantr 481	. . . . . . . . 9 |- ( ( ( A lcm B ) || ( ( A lcm B ) lcm C ) /\ C || ( ( A lcm B ) lcm C ) ) -> ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> B || ( ( A lcm B ) lcm C ) ) )
86	85	impcom 446	. . . . . . . 8 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ ( ( 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 ) )
88	87	adantl 482	. . . . . . . 8 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ ( ( A lcm B ) || ( ( A lcm B ) lcm C ) /\ C || ( ( A lcm B ) lcm C ) ) ) -> C || ( ( A lcm B ) lcm C ) )
89	73, 86, 88	3jca 1242	. . . . . . 7 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ ( ( 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 ) ) )
90	56, 89	mpdan 702	. . . . . 6 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( A || ( ( A lcm B ) lcm C ) /\ B || ( ( A lcm B ) lcm C ) /\ C || ( ( A lcm B ) lcm C ) ) )
91		breq1 4656	. . . . . . . 8 |- ( m = A -> ( m || ( ( A lcm B ) lcm C ) <-> A || ( ( A lcm B ) lcm C ) ) )
92		breq1 4656	. . . . . . . 8 |- ( m = B -> ( m || ( ( A lcm B ) lcm C ) <-> B || ( ( A lcm B ) lcm C ) ) )
93		breq1 4656	. . . . . . . 8 |- ( m = C -> ( m || ( ( A lcm B ) lcm C ) <-> C || ( ( A lcm B ) lcm C ) ) )
94	91, 92, 93	raltpg 4236	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A. m e. { 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 ) ) ) )
95	94	adantl 482	. . . . . 6 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( A. m e. { 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 ) ) ) )
96	90, 95	mpbird 247	. . . . 5 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> A. m e. { A , B , C } m || ( ( A lcm B ) lcm C ) )
97		breq1 4656	. . . . . . . . 9 |- ( m = A -> ( m || k <-> A || k ) )
98		breq1 4656	. . . . . . . . 9 |- ( m = B -> ( m || k <-> B || k ) )
99		breq1 4656	. . . . . . . . 9 |- ( m = C -> ( m || k <-> C || k ) )
100	97, 98, 99	raltpg 4236	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A. m e. { A , B , C } m || k <-> ( A || k /\ B || k /\ C || k ) ) )
101	100	ad2antlr 763	. . . . . . 7 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( A. m e. { A , B , C } m || k <-> ( A || k /\ B || k /\ C || k ) ) )
102		simpr 477	. . . . . . . . . . 11 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> k e. NN )
103	51	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( A lcm B ) e. ZZ )
104	52	ad2antlr 763	. . . . . . . . . . 11 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> C e. ZZ )
105	102, 103, 104	3jca 1242	. . . . . . . . . 10 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( k e. NN /\ ( A lcm B ) e. ZZ /\ C e. ZZ ) )
106	105	adantr 481	. . . . . . . . 9 |- ( ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) /\ ( A || k /\ B || k /\ C || k ) ) -> ( k e. NN /\ ( A lcm B ) e. ZZ /\ C e. ZZ ) )
107		3ioran 1056	. . . . . . . . . . . . . . . . 17 |- ( -. ( 0 = A \/ 0 = B \/ 0 = C ) <-> ( -. 0 = A /\ -. 0 = B /\ -. 0 = C ) )
108		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 = A <-> A = 0 )
109	108	notbii 310	. . . . . . . . . . . . . . . . . . . . 21 |- ( -. 0 = A <-> -. A = 0 )
110		eqcom 2629	. . . . . . . . . . . . . . . . . . . . . 22 |- ( 0 = B <-> B = 0 )
111	110	notbii 310	. . . . . . . . . . . . . . . . . . . . 21 |- ( -. 0 = B <-> -. B = 0 )
112	109, 111	anbi12i 733	. . . . . . . . . . . . . . . . . . . 20 |- ( ( -. 0 = A /\ -. 0 = B ) <-> ( -. A = 0 /\ -. B = 0 ) )
113	112	biimpi 206	. . . . . . . . . . . . . . . . . . 19 |- ( ( -. 0 = A /\ -. 0 = B ) -> ( -. A = 0 /\ -. B = 0 ) )
114		ioran 511	. . . . . . . . . . . . . . . . . . 19 |- ( -. ( A = 0 \/ B = 0 ) <-> ( -. A = 0 /\ -. B = 0 ) )
115	113, 114	sylibr 224	. . . . . . . . . . . . . . . . . 18 |- ( ( -. 0 = A /\ -. 0 = B ) -> -. ( A = 0 \/ B = 0 ) )
116	115	3adant3 1081	. . . . . . . . . . . . . . . . 17 |- ( ( -. 0 = A /\ -. 0 = B /\ -. 0 = C ) -> -. ( A = 0 \/ B = 0 ) )
117	107, 116	sylbi 207	. . . . . . . . . . . . . . . 16 |- ( -. ( 0 = A \/ 0 = B \/ 0 = C ) -> -. ( A = 0 \/ B = 0 ) )
118		id 22	. . . . . . . . . . . . . . . . 17 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A e. ZZ /\ B e. ZZ ) )
119	118	3adant3 1081	. . . . . . . . . . . . . . . 16 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A e. ZZ /\ B e. ZZ ) )
120	117, 119	anim12ci 591	. . . . . . . . . . . . . . 15 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 \/ B = 0 ) ) )
121		lcmn0cl 15310	. . . . . . . . . . . . . . 15 |- ( ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 \/ B = 0 ) ) -> ( A lcm B ) e. NN )
122	120, 121	syl 17	. . . . . . . . . . . . . 14 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( A lcm B ) e. NN )
123		nnne0 11053	. . . . . . . . . . . . . . 15 |- ( ( A lcm B ) e. NN -> ( A lcm B ) =/= 0 )
124	123	neneqd 2799	. . . . . . . . . . . . . 14 |- ( ( A lcm B ) e. NN -> -. ( A lcm B ) = 0 )
125	122, 124	syl 17	. . . . . . . . . . . . 13 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> -. ( A lcm B ) = 0 )
126		eqcom 2629	. . . . . . . . . . . . . . . . . 18 |- ( 0 = C <-> C = 0 )
127	126	notbii 310	. . . . . . . . . . . . . . . . 17 |- ( -. 0 = C <-> -. C = 0 )
128	127	biimpi 206	. . . . . . . . . . . . . . . 16 |- ( -. 0 = C -> -. C = 0 )
129	128	3ad2ant3 1084	. . . . . . . . . . . . . . 15 |- ( ( -. 0 = A /\ -. 0 = B /\ -. 0 = C ) -> -. C = 0 )
130	107, 129	sylbi 207	. . . . . . . . . . . . . 14 |- ( -. ( 0 = A \/ 0 = B \/ 0 = C ) -> -. C = 0 )
131	130	adantr 481	. . . . . . . . . . . . 13 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> -. C = 0 )
132	125, 131	jca 554	. . . . . . . . . . . 12 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( -. ( A lcm B ) = 0 /\ -. C = 0 ) )
133	132	adantr 481	. . . . . . . . . . 11 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( -. ( A lcm B ) = 0 /\ -. C = 0 ) )
134	133	adantr 481	. . . . . . . . . 10 |- ( ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) /\ ( A || k /\ B || k /\ C || k ) ) -> ( -. ( A lcm B ) = 0 /\ -. C = 0 ) )
135		ioran 511	. . . . . . . . . 10 |- ( -. ( ( A lcm B ) = 0 \/ C = 0 ) <-> ( -. ( A lcm B ) = 0 /\ -. C = 0 ) )
136	134, 135	sylibr 224	. . . . . . . . 9 |- ( ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) /\ ( A || k /\ B || k /\ C || k ) ) -> -. ( ( A lcm B ) = 0 \/ C = 0 ) )
137	119	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( A e. ZZ /\ B e. ZZ ) )
138		nnz 11399	. . . . . . . . . . . . . . 15 |- ( k e. NN -> k e. ZZ )
139	137, 138	anim12ci 591	. . . . . . . . . . . . . 14 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( k e. ZZ /\ ( A e. ZZ /\ B e. ZZ ) ) )
140		3anass 1042	. . . . . . . . . . . . . 14 |- ( ( k e. ZZ /\ A e. ZZ /\ B e. ZZ ) <-> ( k e. ZZ /\ ( A e. ZZ /\ B e. ZZ ) ) )
141	139, 140	sylibr 224	. . . . . . . . . . . . 13 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( k e. ZZ /\ A e. ZZ /\ B e. ZZ ) )
142		lcmdvds 15321	. . . . . . . . . . . . 13 |- ( ( k e. ZZ /\ A e. ZZ /\ B e. ZZ ) -> ( ( A || k /\ B || k ) -> ( A lcm B ) || k ) )
143	141, 142	syl 17	. . . . . . . . . . . 12 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( ( A || k /\ B || k ) -> ( A lcm B ) || k ) )
144	143	com12 32	. . . . . . . . . . 11 |- ( ( A || k /\ B || k ) -> ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( A lcm B ) || k ) )
145	144	3adant3 1081	. . . . . . . . . 10 |- ( ( A || k /\ B || k /\ C || k ) -> ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( A lcm B ) || k ) )
146	145	impcom 446	. . . . . . . . 9 |- ( ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) /\ ( A || k /\ B || k /\ C || k ) ) -> ( A lcm B ) || k )
147		simp3 1063	. . . . . . . . . 10 |- ( ( A || k /\ B || k /\ C || k ) -> C || k )
148	147	adantl 482	. . . . . . . . 9 |- ( ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) /\ ( A || k /\ B || k /\ C || k ) ) -> C || k )
149		lcmledvds 15312	. . . . . . . . . 10 |- ( ( ( k e. NN /\ ( A lcm B ) e. ZZ /\ C e. ZZ ) /\ -. ( ( A lcm B ) = 0 \/ C = 0 ) ) -> ( ( ( A lcm B ) || k /\ C || k ) -> ( ( A lcm B ) lcm C ) <_ k ) )
150	149	imp 445	. . . . . . . . 9 |- ( ( ( ( k e. NN /\ ( A lcm B ) e. ZZ /\ C e. ZZ ) /\ -. ( ( A lcm B ) = 0 \/ C = 0 ) ) /\ ( ( A lcm B ) || k /\ C || k ) ) -> ( ( A lcm B ) lcm C ) <_ k )
151	106, 136, 146, 148, 150	syl22anc 1327	. . . . . . . 8 |- ( ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) /\ ( A || k /\ B || k /\ C || k ) ) -> ( ( A lcm B ) lcm C ) <_ k )
152	151	ex 450	. . . . . . 7 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( ( A || k /\ B || k /\ C || k ) -> ( ( A lcm B ) lcm C ) <_ k ) )
153	101, 152	sylbid 230	. . . . . 6 |- ( ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) /\ k e. NN ) -> ( A. m e. { A , B , C } m || k -> ( ( A lcm B ) lcm C ) <_ k ) )
154	153	ralrimiva 2966	. . . . 5 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> A. k e. NN ( A. m e. { A , B , C } m || k -> ( ( A lcm B ) lcm C ) <_ k ) )
155	96, 154	jca 554	. . . 4 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( A. m e. { A , B , C } m || ( ( A lcm B ) lcm C ) /\ A. k e. NN ( A. m e. { A , B , C } m || k -> ( ( A lcm B ) lcm C ) <_ k ) ) )
156	109	biimpi 206	. . . . . . . . . . . . . . . 16 |- ( -. 0 = A -> -. A = 0 )
157	111	biimpi 206	. . . . . . . . . . . . . . . 16 |- ( -. 0 = B -> -. B = 0 )
158	156, 157	anim12i 590	. . . . . . . . . . . . . . 15 |- ( ( -. 0 = A /\ -. 0 = B ) -> ( -. A = 0 /\ -. B = 0 ) )
159	158, 114	sylibr 224	. . . . . . . . . . . . . 14 |- ( ( -. 0 = A /\ -. 0 = B ) -> -. ( A = 0 \/ B = 0 ) )
160	159	3adant3 1081	. . . . . . . . . . . . 13 |- ( ( -. 0 = A /\ -. 0 = B /\ -. 0 = C ) -> -. ( A = 0 \/ B = 0 ) )
161	107, 160	sylbi 207	. . . . . . . . . . . 12 |- ( -. ( 0 = A \/ 0 = B \/ 0 = C ) -> -. ( A = 0 \/ B = 0 ) )
162	161, 119	anim12ci 591	. . . . . . . . . . 11 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( A e. ZZ /\ B e. ZZ ) /\ -. ( A = 0 \/ B = 0 ) ) )
163	162, 121	syl 17	. . . . . . . . . 10 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( A lcm B ) e. NN )
164	163, 124	syl 17	. . . . . . . . 9 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> -. ( A lcm B ) = 0 )
165	164, 131	jca 554	. . . . . . . 8 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( -. ( A lcm B ) = 0 /\ -. C = 0 ) )
166	165, 135	sylibr 224	. . . . . . 7 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> -. ( ( A lcm B ) = 0 \/ C = 0 ) )
167	54, 166	jca 554	. . . . . 6 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( ( A lcm B ) e. ZZ /\ C e. ZZ ) /\ -. ( ( A lcm B ) = 0 \/ C = 0 ) ) )
168		lcmn0cl 15310	. . . . . 6 |- ( ( ( ( A lcm B ) e. ZZ /\ C e. ZZ ) /\ -. ( ( A lcm B ) = 0 \/ C = 0 ) ) -> ( ( A lcm B ) lcm C ) e. NN )
169	167, 168	syl 17	. . . . 5 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( A lcm B ) lcm C ) e. NN )
170	5	adantl 482	. . . . 5 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> { A , B , C } C_ ZZ )
171		tpfi 8236	. . . . . 6 |- { A , B , C } e. Fin
172	171	a1i 11	. . . . 5 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> { A , B , C } e. Fin )
173	3	a1i 11	. . . . . . . . 9 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( 0 e. { A , B , C } <-> ( 0 = A \/ 0 = B \/ 0 = C ) ) )
174	173	biimpd 219	. . . . . . . 8 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( 0 e. { A , B , C } -> ( 0 = A \/ 0 = B \/ 0 = C ) ) )
175	174	con3d 148	. . . . . . 7 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( -. ( 0 = A \/ 0 = B \/ 0 = C ) -> -. 0 e. { A , B , C } ) )
176	175	impcom 446	. . . . . 6 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> -. 0 e. { A , B , C } )
177		df-nel 2898	. . . . . 6 |- ( 0 e/ { A , B , C } <-> -. 0 e. { A , B , C } )
178	176, 177	sylibr 224	. . . . 5 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> 0 e/ { A , B , C } )
179		lcmf 15346	. . . . 5 |- ( ( ( ( A lcm B ) lcm C ) e. NN /\ ( { A , B , C } C_ ZZ /\ { A , B , C } e. Fin /\ 0 e/ { A , B , C } ) ) -> ( ( ( A lcm B ) lcm C ) = (lcm ` { A , B , C } ) <-> ( A. m e. { A , B , C } m || ( ( A lcm B ) lcm C ) /\ A. k e. NN ( A. m e. { A , B , C } m || k -> ( ( A lcm B ) lcm C ) <_ k ) ) ) )
180	169, 170, 172, 178, 179	syl13anc 1328	. . . 4 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( ( A lcm B ) lcm C ) = (lcm ` { A , B , C } ) <-> ( A. m e. { A , B , C } m || ( ( A lcm B ) lcm C ) /\ A. k e. NN ( A. m e. { A , B , C } m || k -> ( ( A lcm B ) lcm C ) <_ k ) ) ) )
181	155, 180	mpbird 247	. . 3 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> ( ( A lcm B ) lcm C ) = (lcm ` { A , B , C } ) )
182	181	eqcomd 2628	. 2 |- ( ( -. ( 0 = A \/ 0 = B \/ 0 = C ) /\ ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) ) -> (lcm ` { A , B , C } ) = ( ( A lcm B ) lcm C ) )
183	50, 182	pm2.61ian 831	1 |- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> (lcm ` { A , B , C } ) = ( ( A lcm B ) lcm C ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   \/ w3o 1036   /\ w3a 1037   = wceq 1483   e. wcel 1990   e/ wnel 2897   A.wral 2912   C_ wss 3574   {ctp 4181   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Fincfn 7955   0cc0 9936   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   || cdvds 14983   lcm clcm 15301  lcmclcmf 15302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636  df-dvds 14984  df-gcd 15217  df-lcm 15303  df-lcmf 15304

This theorem is referenced by:  lcmf2a3a4e12  15360