Theorem hgt750lema 30735

Description: An upper bound on the contribution of the non-prime terms in the Statement 7.50 of [Helfgott] p. 69. (Contributed by Thierry Arnoux, 1-Jan-2022.)

Hypotheses

Ref	Expression
hgt750leme.o	|- O = { z e. ZZ | -. 2 || z }
hgt750leme.n	|- ( ph -> N e. NN )
hgt750lemb.2	|- ( ph -> 2 <_ N )
hgt750lemb.a	|- A = { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) }
hgt750lema.f	|- F = ( d e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } |-> ( d o. if ( a = 0 , ( _I |` ( 0..^ 3 ) ) , ( (pmTrsp ` ( 0..^ 3 ) ) ` { a , 0 } ) ) ) )

Assertion

Ref	Expression
hgt750lema	|- ( ph -> sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) <_ ( 3 x. sum_ n e. A ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) ) )

Distinct variable groups:   z, O   A, c, d, n   N, c, n   ph, c, n   n, F   N, a, d, c, n   O, a, c, d, n   ph, a, d

Allowed substitution hints:   ph( z)   A( z, a)   F( z, a, c, d)   N( z)

Proof of Theorem hgt750lema

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzofi 12773	. . . 4 |- ( 0..^ 3 ) e. Fin
2	1	a1i 11	. . 3 |- ( ph -> ( 0..^ 3 ) e. Fin )
3		hgt750leme.n	. . . . . . 7 |- ( ph -> N e. NN )
4	3	nnnn0d 11351	. . . . . 6 |- ( ph -> N e. NN0 )
5		3nn0 11310	. . . . . . 7 |- 3 e. NN0
6	5	a1i 11	. . . . . 6 |- ( ph -> 3 e. NN0 )
7		ssid 3624	. . . . . . 7 |- NN C_ NN
8	7	a1i 11	. . . . . 6 |- ( ph -> NN C_ NN )
9	4, 6, 8	reprfi2 30701	. . . . 5 |- ( ph -> ( NN (repr ` 3 ) N ) e. Fin )
10		ssrab2 3687	. . . . . 6 |- { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } C_ ( NN (repr ` 3 ) N )
11	10	a1i 11	. . . . 5 |- ( ph -> { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } C_ ( NN (repr ` 3 ) N ) )
12	9, 11	ssfid 8183	. . . 4 |- ( ph -> { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } e. Fin )
13	12	adantr 481	. . 3 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } e. Fin )
14		vmaf 24845	. . . . . 6 |- Λ : NN --> RR
15	14	a1i 11	. . . . 5 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> Λ : NN --> RR )
16	7	a1i 11	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> NN C_ NN )
17	4	nn0zd 11480	. . . . . . . 8 |- ( ph -> N e. ZZ )
18	17	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> N e. ZZ )
19	5	a1i 11	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> 3 e. NN0 )
20		simpr 477	. . . . . . . 8 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } )
21	10, 20	sseldi 3601	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> n e. ( NN (repr ` 3 ) N ) )
22	16, 18, 19, 21	reprf 30690	. . . . . 6 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> n : ( 0..^ 3 ) --> NN )
23		c0ex 10034	. . . . . . . . 9 |- 0 e. _V
24	23	tpid1 4303	. . . . . . . 8 |- 0 e. { 0 , 1 , 2 }
25		fzo0to3tp 12554	. . . . . . . 8 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
26	24, 25	eleqtrri 2700	. . . . . . 7 |- 0 e. ( 0..^ 3 )
27	26	a1i 11	. . . . . 6 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> 0 e. ( 0..^ 3 ) )
28	22, 27	ffvelrnd 6360	. . . . 5 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> ( n ` 0 ) e. NN )
29	15, 28	ffvelrnd 6360	. . . 4 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> (Λ ` ( n ` 0 ) ) e. RR )
30		1ex 10035	. . . . . . . . . 10 |- 1 e. _V
31	30	tpid2 4304	. . . . . . . . 9 |- 1 e. { 0 , 1 , 2 }
32	31, 25	eleqtrri 2700	. . . . . . . 8 |- 1 e. ( 0..^ 3 )
33	32	a1i 11	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> 1 e. ( 0..^ 3 ) )
34	22, 33	ffvelrnd 6360	. . . . . 6 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> ( n ` 1 ) e. NN )
35	15, 34	ffvelrnd 6360	. . . . 5 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> (Λ ` ( n ` 1 ) ) e. RR )
36		2ex 11092	. . . . . . . . . 10 |- 2 e. _V
37	36	tpid3 4307	. . . . . . . . 9 |- 2 e. { 0 , 1 , 2 }
38	37, 25	eleqtrri 2700	. . . . . . . 8 |- 2 e. ( 0..^ 3 )
39	38	a1i 11	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> 2 e. ( 0..^ 3 ) )
40	22, 39	ffvelrnd 6360	. . . . . 6 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> ( n ` 2 ) e. NN )
41	15, 40	ffvelrnd 6360	. . . . 5 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> (Λ ` ( n ` 2 ) ) e. RR )
42	35, 41	remulcld 10070	. . . 4 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) e. RR )
43	29, 42	remulcld 10070	. . 3 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) e. RR )
44		vmage0 24847	. . . . 5 |- ( ( n ` 0 ) e. NN -> 0 <_ (Λ ` ( n ` 0 ) ) )
45	28, 44	syl 17	. . . 4 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> 0 <_ (Λ ` ( n ` 0 ) ) )
46		vmage0 24847	. . . . . 6 |- ( ( n ` 1 ) e. NN -> 0 <_ (Λ ` ( n ` 1 ) ) )
47	34, 46	syl 17	. . . . 5 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> 0 <_ (Λ ` ( n ` 1 ) ) )
48		vmage0 24847	. . . . . 6 |- ( ( n ` 2 ) e. NN -> 0 <_ (Λ ` ( n ` 2 ) ) )
49	40, 48	syl 17	. . . . 5 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> 0 <_ (Λ ` ( n ` 2 ) ) )
50	35, 41, 47, 49	mulge0d 10604	. . . 4 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> 0 <_ ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) )
51	29, 42, 45, 50	mulge0d 10604	. . 3 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> 0 <_ ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) )
52	2, 13, 43, 51	fsumiunle 29575	. 2 |- ( ph -> sum_ n e. U_ a e. ( 0..^ 3 ) { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) <_ sum_ a e. ( 0..^ 3 ) sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) )
53		eqid 2622	. . . 4 |- { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } = { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) }
54		inss2 3834	. . . . . 6 |- ( O i^i Prime ) C_ Prime
55		prmssnn 15390	. . . . . 6 |- Prime C_ NN
56	54, 55	sstri 3612	. . . . 5 |- ( O i^i Prime ) C_ NN
57	56	a1i 11	. . . 4 |- ( ph -> ( O i^i Prime ) C_ NN )
58	53, 8, 57, 4, 6	reprdifc 30705	. . 3 |- ( ph -> ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) = U_ a e. ( 0..^ 3 ) { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } )
59	58	sumeq1d 14431	. 2 |- ( ph -> sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) = sum_ n e. U_ a e. ( 0..^ 3 ) { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) )
60		ssrab2 3687	. . . . . . . 8 |- { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } C_ ( NN (repr ` 3 ) N )
61	60	a1i 11	. . . . . . 7 |- ( ph -> { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } C_ ( NN (repr ` 3 ) N ) )
62	9, 61	ssfid 8183	. . . . . 6 |- ( ph -> { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } e. Fin )
63	14	a1i 11	. . . . . . . 8 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> Λ : NN --> RR )
64	7	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> NN C_ NN )
65	17	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> N e. ZZ )
66	5	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> 3 e. NN0 )
67	61	sselda 3603	. . . . . . . . . 10 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> n e. ( NN (repr ` 3 ) N ) )
68	64, 65, 66, 67	reprf 30690	. . . . . . . . 9 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> n : ( 0..^ 3 ) --> NN )
69	26	a1i 11	. . . . . . . . 9 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> 0 e. ( 0..^ 3 ) )
70	68, 69	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> ( n ` 0 ) e. NN )
71	63, 70	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> (Λ ` ( n ` 0 ) ) e. RR )
72	32	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> 1 e. ( 0..^ 3 ) )
73	68, 72	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> ( n ` 1 ) e. NN )
74	63, 73	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> (Λ ` ( n ` 1 ) ) e. RR )
75	38	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> 2 e. ( 0..^ 3 ) )
76	68, 75	ffvelrnd 6360	. . . . . . . . 9 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> ( n ` 2 ) e. NN )
77	63, 76	ffvelrnd 6360	. . . . . . . 8 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> (Λ ` ( n ` 2 ) ) e. RR )
78	74, 77	remulcld 10070	. . . . . . 7 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) e. RR )
79	71, 78	remulcld 10070	. . . . . 6 |- ( ( ph /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) e. RR )
80	62, 79	fsumrecl 14465	. . . . 5 |- ( ph -> sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) e. RR )
81	80	recnd 10068	. . . 4 |- ( ph -> sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) e. CC )
82		fsumconst 14522	. . . 4 |- ( ( ( 0..^ 3 ) e. Fin /\ sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) e. CC ) -> sum_ a e. ( 0..^ 3 ) sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) = ( ( # ` ( 0..^ 3 ) ) x. sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) ) )
83	2, 81, 82	syl2anc 693	. . 3 |- ( ph -> sum_ a e. ( 0..^ 3 ) sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) = ( ( # ` ( 0..^ 3 ) ) x. sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) ) )
84		fveq1 6190	. . . . . . . 8 |- ( n = ( F ` e ) -> ( n ` 0 ) = ( ( F ` e ) ` 0 ) )
85	84	fveq2d 6195	. . . . . . 7 |- ( n = ( F ` e ) -> (Λ ` ( n ` 0 ) ) = (Λ ` ( ( F ` e ) ` 0 ) ) )
86		fveq1 6190	. . . . . . . . 9 |- ( n = ( F ` e ) -> ( n ` 1 ) = ( ( F ` e ) ` 1 ) )
87	86	fveq2d 6195	. . . . . . . 8 |- ( n = ( F ` e ) -> (Λ ` ( n ` 1 ) ) = (Λ ` ( ( F ` e ) ` 1 ) ) )
88		fveq1 6190	. . . . . . . . 9 |- ( n = ( F ` e ) -> ( n ` 2 ) = ( ( F ` e ) ` 2 ) )
89	88	fveq2d 6195	. . . . . . . 8 |- ( n = ( F ` e ) -> (Λ ` ( n ` 2 ) ) = (Λ ` ( ( F ` e ) ` 2 ) ) )
90	87, 89	oveq12d 6668	. . . . . . 7 |- ( n = ( F ` e ) -> ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) = ( (Λ ` ( ( F ` e ) ` 1 ) ) x. (Λ ` ( ( F ` e ) ` 2 ) ) ) )
91	85, 90	oveq12d 6668	. . . . . 6 |- ( n = ( F ` e ) -> ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) = ( (Λ ` ( ( F ` e ) ` 0 ) ) x. ( (Λ ` ( ( F ` e ) ` 1 ) ) x. (Λ ` ( ( F ` e ) ` 2 ) ) ) ) )
92		3nn 11186	. . . . . . . . . 10 |- 3 e. NN
93	92	a1i 11	. . . . . . . . 9 |- ( ph -> 3 e. NN )
94	93	ralrimivw 2967	. . . . . . . 8 |- ( ph -> A. a e. ( 0..^ 3 ) 3 e. NN )
95	94	r19.21bi 2932	. . . . . . 7 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> 3 e. NN )
96	17	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> N e. ZZ )
97	7	a1i 11	. . . . . . 7 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> NN C_ NN )
98		simpr 477	. . . . . . 7 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> a e. ( 0..^ 3 ) )
99		fveq1 6190	. . . . . . . . . 10 |- ( c = d -> ( c ` 0 ) = ( d ` 0 ) )
100	99	eleq1d 2686	. . . . . . . . 9 |- ( c = d -> ( ( c ` 0 ) e. ( O i^i Prime ) <-> ( d ` 0 ) e. ( O i^i Prime ) ) )
101	100	notbid 308	. . . . . . . 8 |- ( c = d -> ( -. ( c ` 0 ) e. ( O i^i Prime ) <-> -. ( d ` 0 ) e. ( O i^i Prime ) ) )
102	101	cbvrabv 3199	. . . . . . 7 |- { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } = { d e. ( NN (repr ` 3 ) N ) | -. ( d ` 0 ) e. ( O i^i Prime ) }
103		fveq1 6190	. . . . . . . . . 10 |- ( c = d -> ( c ` a ) = ( d ` a ) )
104	103	eleq1d 2686	. . . . . . . . 9 |- ( c = d -> ( ( c ` a ) e. ( O i^i Prime ) <-> ( d ` a ) e. ( O i^i Prime ) ) )
105	104	notbid 308	. . . . . . . 8 |- ( c = d -> ( -. ( c ` a ) e. ( O i^i Prime ) <-> -. ( d ` a ) e. ( O i^i Prime ) ) )
106	105	cbvrabv 3199	. . . . . . 7 |- { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } = { d e. ( NN (repr ` 3 ) N ) | -. ( d ` a ) e. ( O i^i Prime ) }
107		eqid 2622	. . . . . . 7 |- if ( a = 0 , ( _I |` ( 0..^ 3 ) ) , ( (pmTrsp ` ( 0..^ 3 ) ) ` { a , 0 } ) ) = if ( a = 0 , ( _I |` ( 0..^ 3 ) ) , ( (pmTrsp ` ( 0..^ 3 ) ) ` { a , 0 } ) )
108		hgt750lema.f	. . . . . . 7 |- F = ( d e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } |-> ( d o. if ( a = 0 , ( _I |` ( 0..^ 3 ) ) , ( (pmTrsp ` ( 0..^ 3 ) ) ` { a , 0 } ) ) ) )
109	95, 96, 97, 98, 102, 106, 107, 108	reprpmtf1o 30704	. . . . . 6 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> F : { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } -1-1-onto-> { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } )
110		eqidd 2623	. . . . . 6 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ e e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> ( F ` e ) = ( F ` e ) )
111	79	adantlr 751	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) e. RR )
112	111	recnd 10068	. . . . . 6 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ) -> ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) e. CC )
113	91, 13, 109, 110, 112	fsumf1o 14454	. . . . 5 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) = sum_ e e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( ( F ` e ) ` 0 ) ) x. ( (Λ ` ( ( F ` e ) ` 1 ) ) x. (Λ ` ( ( F ` e ) ` 2 ) ) ) ) )
114		fveq2 6191	. . . . . . . . . 10 |- ( e = n -> ( F ` e ) = ( F ` n ) )
115	114	fveq1d 6193	. . . . . . . . 9 |- ( e = n -> ( ( F ` e ) ` 0 ) = ( ( F ` n ) ` 0 ) )
116	115	fveq2d 6195	. . . . . . . 8 |- ( e = n -> (Λ ` ( ( F ` e ) ` 0 ) ) = (Λ ` ( ( F ` n ) ` 0 ) ) )
117	114	fveq1d 6193	. . . . . . . . . 10 |- ( e = n -> ( ( F ` e ) ` 1 ) = ( ( F ` n ) ` 1 ) )
118	117	fveq2d 6195	. . . . . . . . 9 |- ( e = n -> (Λ ` ( ( F ` e ) ` 1 ) ) = (Λ ` ( ( F ` n ) ` 1 ) ) )
119	114	fveq1d 6193	. . . . . . . . . 10 |- ( e = n -> ( ( F ` e ) ` 2 ) = ( ( F ` n ) ` 2 ) )
120	119	fveq2d 6195	. . . . . . . . 9 |- ( e = n -> (Λ ` ( ( F ` e ) ` 2 ) ) = (Λ ` ( ( F ` n ) ` 2 ) ) )
121	118, 120	oveq12d 6668	. . . . . . . 8 |- ( e = n -> ( (Λ ` ( ( F ` e ) ` 1 ) ) x. (Λ ` ( ( F ` e ) ` 2 ) ) ) = ( (Λ ` ( ( F ` n ) ` 1 ) ) x. (Λ ` ( ( F ` n ) ` 2 ) ) ) )
122	116, 121	oveq12d 6668	. . . . . . 7 |- ( e = n -> ( (Λ ` ( ( F ` e ) ` 0 ) ) x. ( (Λ ` ( ( F ` e ) ` 1 ) ) x. (Λ ` ( ( F ` e ) ` 2 ) ) ) ) = ( (Λ ` ( ( F ` n ) ` 0 ) ) x. ( (Λ ` ( ( F ` n ) ` 1 ) ) x. (Λ ` ( ( F ` n ) ` 2 ) ) ) ) )
123	122	cbvsumv 14426	. . . . . 6 |- sum_ e e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( ( F ` e ) ` 0 ) ) x. ( (Λ ` ( ( F ` e ) ` 1 ) ) x. (Λ ` ( ( F ` e ) ` 2 ) ) ) ) = sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( ( F ` n ) ` 0 ) ) x. ( (Λ ` ( ( F ` n ) ` 1 ) ) x. (Λ ` ( ( F ` n ) ` 2 ) ) ) )
124	123	a1i 11	. . . . 5 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> sum_ e e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( ( F ` e ) ` 0 ) ) x. ( (Λ ` ( ( F ` e ) ` 1 ) ) x. (Λ ` ( ( F ` e ) ` 2 ) ) ) ) = sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( ( F ` n ) ` 0 ) ) x. ( (Λ ` ( ( F ` n ) ` 1 ) ) x. (Λ ` ( ( F ` n ) ` 2 ) ) ) ) )
125		ovexd 6680	. . . . . . . 8 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> ( 0..^ 3 ) e. _V )
126	98	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> a e. ( 0..^ 3 ) )
127	125, 126, 27, 107	pmtridf1o 29856	. . . . . . 7 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> if ( a = 0 , ( _I |` ( 0..^ 3 ) ) , ( (pmTrsp ` ( 0..^ 3 ) ) ` { a , 0 } ) ) : ( 0..^ 3 ) -1-1-onto-> ( 0..^ 3 ) )
128	108, 127, 22, 15, 20	hgt750lemg 30732	. . . . . 6 |- ( ( ( ph /\ a e. ( 0..^ 3 ) ) /\ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ) -> ( (Λ ` ( ( F ` n ) ` 0 ) ) x. ( (Λ ` ( ( F ` n ) ` 1 ) ) x. (Λ ` ( ( F ` n ) ` 2 ) ) ) ) = ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) )
129	128	sumeq2dv 14433	. . . . 5 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( ( F ` n ) ` 0 ) ) x. ( (Λ ` ( ( F ` n ) ` 1 ) ) x. (Λ ` ( ( F ` n ) ` 2 ) ) ) ) = sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) )
130	113, 124, 129	3eqtrrd 2661	. . . 4 |- ( ( ph /\ a e. ( 0..^ 3 ) ) -> sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) = sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) )
131	130	sumeq2dv 14433	. . 3 |- ( ph -> sum_ a e. ( 0..^ 3 ) sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) = sum_ a e. ( 0..^ 3 ) sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) )
132		hashfzo0 13217	. . . . . . 7 |- ( 3 e. NN0 -> ( # ` ( 0..^ 3 ) ) = 3 )
133	5, 132	ax-mp 5	. . . . . 6 |- ( # ` ( 0..^ 3 ) ) = 3
134	133	a1i 11	. . . . 5 |- ( ph -> ( # ` ( 0..^ 3 ) ) = 3 )
135	134	eqcomd 2628	. . . 4 |- ( ph -> 3 = ( # ` ( 0..^ 3 ) ) )
136		hgt750lemb.a	. . . . . 6 |- A = { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) }
137	136	a1i 11	. . . . 5 |- ( ph -> A = { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } )
138	137	sumeq1d 14431	. . . 4 |- ( ph -> sum_ n e. A ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) = sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) )
139	135, 138	oveq12d 6668	. . 3 |- ( ph -> ( 3 x. sum_ n e. A ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) ) = ( ( # ` ( 0..^ 3 ) ) x. sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` 0 ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) ) )
140	83, 131, 139	3eqtr4rd 2667	. 2 |- ( ph -> ( 3 x. sum_ n e. A ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) ) = sum_ a e. ( 0..^ 3 ) sum_ n e. { c e. ( NN (repr ` 3 ) N ) | -. ( c ` a ) e. ( O i^i Prime ) } ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) )
141	52, 59, 140	3brtr4d 4685	1 |- ( ph -> sum_ n e. ( ( NN (repr ` 3 ) N ) \ ( ( O i^i Prime ) (repr ` 3 ) N ) ) ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) <_ ( 3 x. sum_ n e. A ( (Λ ` ( n ` 0 ) ) x. ( (Λ ` ( n ` 1 ) ) x. (Λ ` ( n ` 2 ) ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ifcif 4086   {cpr 4179   {ctp 4181   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   |` cres 5116   o. ccom 5118   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   <_ cle 10075   NNcn 11020   2c2 11070   3c3 11071   NN0cn0 11292   ZZcz 11377  ..^cfzo 12465   #chash 13117   sum_csu 14416   || cdvds 14983   Primecprime 15385  pmTrspcpmtr 17861  Λcvma 24818  reprcrepr 30686

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-reg 8497  ax-inf2 8538  ax-ac2 9285  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  ax-addf 10015  ax-mulf 10016

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-iin 4523  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-r1 8627  df-rank 8628  df-card 8765  df-ac 8939  df-cda 8990  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-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-ioc 12180  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-prod 14636  df-ef 14798  df-sin 14800  df-cos 14801  df-pi 14803  df-dvds 14984  df-gcd 15217  df-prm 15386  df-pc 15542  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-pmtr 17862  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631  df-log 24303  df-vma 24824  df-repr 30687

This theorem is referenced by:  hgt750leme  30736