Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tgblthelfgott Structured version   Visualization version   GIF version
Theorem tgblthelfgott 46483
Description: The ternary Goldbach conjecture is valid for all odd numbers less than 8.8 x 10^30 (actually 8.875694 x 10^30, see section 1.2.2 in [Helfgott] p. 4, using bgoldbachlt 46481, ax-hgprmladder 46482 and bgoldbtbnd 46477. (Contributed by AV, 4-Aug-2020.) (Revised by AV, 9-Sep-2021.)
Assertion
Ref Expression
tgblthelfgott ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ 𝑁 ∈ GoldbachOdd )
Proof of Theorem tgblthelfgott
Dummy variables 𝑛 𝑑 𝑓 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hgprmladder 46482 . 2 βˆƒπ‘‘ ∈ (β„€β‰₯β€˜3)βˆƒπ‘“ ∈ (RePartβ€˜π‘‘)(((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))
2 1nn0 12488 . . . . . . . . . 10 1 ∈ β„•0
3 1nn 12223 . . . . . . . . . 10 1 ∈ β„•
42, 3decnncl 12697 . . . . . . . . 9 11 ∈ β„•
54nnzi 12586 . . . . . . . 8 11 ∈ β„€
6 8nn0 12495 . . . . . . . . . . 11 8 ∈ β„•0
7 8nn 12307 . . . . . . . . . . 11 8 ∈ β„•
86, 7decnncl 12697 . . . . . . . . . 10 88 ∈ β„•
9 10nn 12693 . . . . . . . . . . 11 10 ∈ β„•
10 2nn0 12489 . . . . . . . . . . . . 13 2 ∈ β„•0
11 9nn 12310 . . . . . . . . . . . . 13 9 ∈ β„•
1210, 11decnncl 12697 . . . . . . . . . . . 12 29 ∈ β„•
1312nnnn0i 12480 . . . . . . . . . . 11 29 ∈ β„•0
14 nnexpcl 14040 . . . . . . . . . . 11 ((10 ∈ β„• ∧ 29 ∈ β„•0) β†’ (10↑29) ∈ β„•)
159, 13, 14mp2an 691 . . . . . . . . . 10 (10↑29) ∈ β„•
168, 15nnmulcli 12237 . . . . . . . . 9 (88 Β· (10↑29)) ∈ β„•
1716nnzi 12586 . . . . . . . 8 (88 Β· (10↑29)) ∈ β„€
18 1re 11214 . . . . . . . . . . 11 1 ∈ ℝ
198nnrei 12221 . . . . . . . . . . 11 88 ∈ ℝ
2018, 19pm3.2i 472 . . . . . . . . . 10 (1 ∈ ℝ ∧ 88 ∈ ℝ)
21 0le1 11737 . . . . . . . . . . 11 0 ≀ 1
22 1lt10 12816 . . . . . . . . . . . 12 1 < 10
237, 6, 2, 22declti 12715 . . . . . . . . . . 11 1 < 88
2421, 23pm3.2i 472 . . . . . . . . . 10 (0 ≀ 1 ∧ 1 < 88)
25 nnexpcl 14040 . . . . . . . . . . . . 13 ((10 ∈ β„• ∧ 1 ∈ β„•0) β†’ (10↑1) ∈ β„•)
269, 2, 25mp2an 691 . . . . . . . . . . . 12 (10↑1) ∈ β„•
2726nnrei 12221 . . . . . . . . . . 11 (10↑1) ∈ ℝ
2815nnrei 12221 . . . . . . . . . . 11 (10↑29) ∈ ℝ
2927, 28pm3.2i 472 . . . . . . . . . 10 ((10↑1) ∈ ℝ ∧ (10↑29) ∈ ℝ)
30 0re 11216 . . . . . . . . . . . . 13 0 ∈ ℝ
31 10re 12696 . . . . . . . . . . . . 13 10 ∈ ℝ
32 10pos 12694 . . . . . . . . . . . . 13 0 < 10
3330, 31, 32ltleii 11337 . . . . . . . . . . . 12 0 ≀ 10
349nncni 12222 . . . . . . . . . . . . 13 10 ∈ β„‚
35 exp1 14033 . . . . . . . . . . . . 13 (10 ∈ β„‚ β†’ (10↑1) = 10)
3634, 35ax-mp 5 . . . . . . . . . . . 12 (10↑1) = 10
3733, 36breqtrri 5176 . . . . . . . . . . 11 0 ≀ (10↑1)
38 1z 12592 . . . . . . . . . . . . 13 1 ∈ β„€
3912nnzi 12586 . . . . . . . . . . . . 13 29 ∈ β„€
4031, 38, 393pm3.2i 1340 . . . . . . . . . . . 12 (10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 29 ∈ β„€)
41 2nn 12285 . . . . . . . . . . . . . 14 2 ∈ β„•
42 9nn0 12496 . . . . . . . . . . . . . 14 9 ∈ β„•0
4341, 42, 2, 22declti 12715 . . . . . . . . . . . . 13 1 < 29
4422, 43pm3.2i 472 . . . . . . . . . . . 12 (1 < 10 ∧ 1 < 29)
45 ltexp2a 14131 . . . . . . . . . . . 12 (((10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 29 ∈ β„€) ∧ (1 < 10 ∧ 1 < 29)) β†’ (10↑1) < (10↑29))
4640, 44, 45mp2an 691 . . . . . . . . . . 11 (10↑1) < (10↑29)
4737, 46pm3.2i 472 . . . . . . . . . 10 (0 ≀ (10↑1) ∧ (10↑1) < (10↑29))
48 ltmul12a 12070 . . . . . . . . . 10 ((((1 ∈ ℝ ∧ 88 ∈ ℝ) ∧ (0 ≀ 1 ∧ 1 < 88)) ∧ (((10↑1) ∈ ℝ ∧ (10↑29) ∈ ℝ) ∧ (0 ≀ (10↑1) ∧ (10↑1) < (10↑29)))) β†’ (1 Β· (10↑1)) < (88 Β· (10↑29)))
4920, 24, 29, 47, 48mp4an 692 . . . . . . . . 9 (1 Β· (10↑1)) < (88 Β· (10↑29))
5026nnzi 12586 . . . . . . . . . . . 12 (10↑1) ∈ β„€
51 zmulcl 12611 . . . . . . . . . . . 12 ((1 ∈ β„€ ∧ (10↑1) ∈ β„€) β†’ (1 Β· (10↑1)) ∈ β„€)
5238, 50, 51mp2an 691 . . . . . . . . . . 11 (1 Β· (10↑1)) ∈ β„€
53 zltp1le 12612 . . . . . . . . . . 11 (((1 Β· (10↑1)) ∈ β„€ ∧ (88 Β· (10↑29)) ∈ β„€) β†’ ((1 Β· (10↑1)) < (88 Β· (10↑29)) ↔ ((1 Β· (10↑1)) + 1) ≀ (88 Β· (10↑29))))
5452, 17, 53mp2an 691 . . . . . . . . . 10 ((1 Β· (10↑1)) < (88 Β· (10↑29)) ↔ ((1 Β· (10↑1)) + 1) ≀ (88 Β· (10↑29)))
55 1t10e1p1e11 46018 . . . . . . . . . . . 12 11 = ((1 Β· (10↑1)) + 1)
5655eqcomi 2742 . . . . . . . . . . 11 ((1 Β· (10↑1)) + 1) = 11
5756breq1i 5156 . . . . . . . . . 10 (((1 Β· (10↑1)) + 1) ≀ (88 Β· (10↑29)) ↔ 11 ≀ (88 Β· (10↑29)))
5854, 57bitri 275 . . . . . . . . 9 ((1 Β· (10↑1)) < (88 Β· (10↑29)) ↔ 11 ≀ (88 Β· (10↑29)))
5949, 58mpbi 229 . . . . . . . 8 11 ≀ (88 Β· (10↑29))
60 eluz2 12828 . . . . . . . 8 ((88 Β· (10↑29)) ∈ (β„€β‰₯β€˜11) ↔ (11 ∈ β„€ ∧ (88 Β· (10↑29)) ∈ β„€ ∧ 11 ≀ (88 Β· (10↑29))))
615, 17, 59, 60mpbir3an 1342 . . . . . . 7 (88 Β· (10↑29)) ∈ (β„€β‰₯β€˜11)
6261a1i 11 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ (88 Β· (10↑29)) ∈ (β„€β‰₯β€˜11))
63 4nn 12295 . . . . . . . . . 10 4 ∈ β„•
642, 7decnncl 12697 . . . . . . . . . . . 12 18 ∈ β„•
6564nnnn0i 12480 . . . . . . . . . . 11 18 ∈ β„•0
66 nnexpcl 14040 . . . . . . . . . . 11 ((10 ∈ β„• ∧ 18 ∈ β„•0) β†’ (10↑18) ∈ β„•)
679, 65, 66mp2an 691 . . . . . . . . . 10 (10↑18) ∈ β„•
6863, 67nnmulcli 12237 . . . . . . . . 9 (4 Β· (10↑18)) ∈ β„•
6968nnzi 12586 . . . . . . . 8 (4 Β· (10↑18)) ∈ β„€
70 4re 12296 . . . . . . . . . . 11 4 ∈ ℝ
7118, 70pm3.2i 472 . . . . . . . . . 10 (1 ∈ ℝ ∧ 4 ∈ ℝ)
72 1lt4 12388 . . . . . . . . . . 11 1 < 4
7321, 72pm3.2i 472 . . . . . . . . . 10 (0 ≀ 1 ∧ 1 < 4)
7467nnrei 12221 . . . . . . . . . . 11 (10↑18) ∈ ℝ
7527, 74pm3.2i 472 . . . . . . . . . 10 ((10↑1) ∈ ℝ ∧ (10↑18) ∈ ℝ)
7664nnzi 12586 . . . . . . . . . . . . 13 18 ∈ β„€
7731, 38, 763pm3.2i 1340 . . . . . . . . . . . 12 (10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 18 ∈ β„€)
783, 6, 2, 22declti 12715 . . . . . . . . . . . . 13 1 < 18
7922, 78pm3.2i 472 . . . . . . . . . . . 12 (1 < 10 ∧ 1 < 18)
80 ltexp2a 14131 . . . . . . . . . . . 12 (((10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 18 ∈ β„€) ∧ (1 < 10 ∧ 1 < 18)) β†’ (10↑1) < (10↑18))
8177, 79, 80mp2an 691 . . . . . . . . . . 11 (10↑1) < (10↑18)
8237, 81pm3.2i 472 . . . . . . . . . 10 (0 ≀ (10↑1) ∧ (10↑1) < (10↑18))
83 ltmul12a 12070 . . . . . . . . . 10 ((((1 ∈ ℝ ∧ 4 ∈ ℝ) ∧ (0 ≀ 1 ∧ 1 < 4)) ∧ (((10↑1) ∈ ℝ ∧ (10↑18) ∈ ℝ) ∧ (0 ≀ (10↑1) ∧ (10↑1) < (10↑18)))) β†’ (1 Β· (10↑1)) < (4 Β· (10↑18)))
8471, 73, 75, 82, 83mp4an 692 . . . . . . . . 9 (1 Β· (10↑1)) < (4 Β· (10↑18))
85 4z 12596 . . . . . . . . . . . 12 4 ∈ β„€
8667nnzi 12586 . . . . . . . . . . . 12 (10↑18) ∈ β„€
87 zmulcl 12611 . . . . . . . . . . . 12 ((4 ∈ β„€ ∧ (10↑18) ∈ β„€) β†’ (4 Β· (10↑18)) ∈ β„€)
8885, 86, 87mp2an 691 . . . . . . . . . . 11 (4 Β· (10↑18)) ∈ β„€
89 zltp1le 12612 . . . . . . . . . . 11 (((1 Β· (10↑1)) ∈ β„€ ∧ (4 Β· (10↑18)) ∈ β„€) β†’ ((1 Β· (10↑1)) < (4 Β· (10↑18)) ↔ ((1 Β· (10↑1)) + 1) ≀ (4 Β· (10↑18))))
9052, 88, 89mp2an 691 . . . . . . . . . 10 ((1 Β· (10↑1)) < (4 Β· (10↑18)) ↔ ((1 Β· (10↑1)) + 1) ≀ (4 Β· (10↑18)))
9156breq1i 5156 . . . . . . . . . 10 (((1 Β· (10↑1)) + 1) ≀ (4 Β· (10↑18)) ↔ 11 ≀ (4 Β· (10↑18)))
9290, 91bitri 275 . . . . . . . . 9 ((1 Β· (10↑1)) < (4 Β· (10↑18)) ↔ 11 ≀ (4 Β· (10↑18)))
9384, 92mpbi 229 . . . . . . . 8 11 ≀ (4 Β· (10↑18))
94 eluz2 12828 . . . . . . . 8 ((4 Β· (10↑18)) ∈ (β„€β‰₯β€˜11) ↔ (11 ∈ β„€ ∧ (4 Β· (10↑18)) ∈ β„€ ∧ 11 ≀ (4 Β· (10↑18))))
955, 69, 93, 94mpbir3an 1342 . . . . . . 7 (4 Β· (10↑18)) ∈ (β„€β‰₯β€˜11)
9695a1i 11 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ (4 Β· (10↑18)) ∈ (β„€β‰₯β€˜11))
97 simpl 484 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 𝑛 ∈ Even )
98 simprl 770 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 4 < 𝑛)
99 evenz 46298 . . . . . . . . . . . . . 14 (𝑛 ∈ Even β†’ 𝑛 ∈ β„€)
10099zred 12666 . . . . . . . . . . . . 13 (𝑛 ∈ Even β†’ 𝑛 ∈ ℝ)
10168nnrei 12221 . . . . . . . . . . . . 13 (4 Β· (10↑18)) ∈ ℝ
102 ltle 11302 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ ∧ (4 Β· (10↑18)) ∈ ℝ) β†’ (𝑛 < (4 Β· (10↑18)) β†’ 𝑛 ≀ (4 Β· (10↑18))))
103100, 101, 102sylancl 587 . . . . . . . . . . . 12 (𝑛 ∈ Even β†’ (𝑛 < (4 Β· (10↑18)) β†’ 𝑛 ≀ (4 Β· (10↑18))))
104103a1d 25 . . . . . . . . . . 11 (𝑛 ∈ Even β†’ (4 < 𝑛 β†’ (𝑛 < (4 Β· (10↑18)) β†’ 𝑛 ≀ (4 Β· (10↑18)))))
105104imp32 420 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 𝑛 ≀ (4 Β· (10↑18)))
106 ax-bgbltosilva 46478 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≀ (4 Β· (10↑18))) β†’ 𝑛 ∈ GoldbachEven )
10797, 98, 105, 106syl3anc 1372 . . . . . . . . 9 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 𝑛 ∈ GoldbachEven )
108107ex 414 . . . . . . . 8 (𝑛 ∈ Even β†’ ((4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18))) β†’ 𝑛 ∈ GoldbachEven ))
109108a1i 11 . . . . . . 7 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ (𝑛 ∈ Even β†’ ((4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18))) β†’ 𝑛 ∈ GoldbachEven )))
110109ralrimiv 3146 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ βˆ€π‘› ∈ Even ((4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18))) β†’ 𝑛 ∈ GoldbachEven ))
111 simpl 484 . . . . . . 7 ((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) β†’ 𝑑 ∈ (β„€β‰₯β€˜3))
112111ad2antrr 725 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ 𝑑 ∈ (β„€β‰₯β€˜3))
113 simpr 486 . . . . . . 7 ((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) β†’ 𝑓 ∈ (RePartβ€˜π‘‘))
114113ad2antrr 725 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ 𝑓 ∈ (RePartβ€˜π‘‘))
115 simpr 486 . . . . . . 7 ((((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)))) β†’ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))
116115ad2antlr 726 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))
117 simpl1 1192 . . . . . . 7 ((((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)))) β†’ (π‘“β€˜0) = 7)
118117ad2antlr 726 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ (π‘“β€˜0) = 7)
119 simpl2 1193 . . . . . . 7 ((((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)))) β†’ (π‘“β€˜1) = 13)
120119ad2antlr 726 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ (π‘“β€˜1) = 13)
1216, 11decnncl 12697 . . . . . . . . . . . . 13 89 ∈ β„•
122121nnrei 12221 . . . . . . . . . . . 12 89 ∈ ℝ
12315nngt0i 12251 . . . . . . . . . . . . 13 0 < (10↑29)
12428, 123pm3.2i 472 . . . . . . . . . . . 12 ((10↑29) ∈ ℝ ∧ 0 < (10↑29))
12519, 122, 1243pm3.2i 1340 . . . . . . . . . . 11 (88 ∈ ℝ ∧ 89 ∈ ℝ ∧ ((10↑29) ∈ ℝ ∧ 0 < (10↑29)))
126 8lt9 12411 . . . . . . . . . . . 12 8 < 9
1276, 6, 11, 126declt 12705 . . . . . . . . . . 11 88 < 89
128 ltmul1a 12063 . . . . . . . . . . 11 (((88 ∈ ℝ ∧ 89 ∈ ℝ ∧ ((10↑29) ∈ ℝ ∧ 0 < (10↑29))) ∧ 88 < 89) β†’ (88 Β· (10↑29)) < (89 Β· (10↑29)))
129125, 127, 128mp2an 691 . . . . . . . . . 10 (88 Β· (10↑29)) < (89 Β· (10↑29))
130 breq2 5153 . . . . . . . . . 10 ((π‘“β€˜π‘‘) = (89 Β· (10↑29)) β†’ ((88 Β· (10↑29)) < (π‘“β€˜π‘‘) ↔ (88 Β· (10↑29)) < (89 Β· (10↑29))))
131129, 130mpbiri 258 . . . . . . . . 9 ((π‘“β€˜π‘‘) = (89 Β· (10↑29)) β†’ (88 Β· (10↑29)) < (π‘“β€˜π‘‘))
1321313ad2ant3 1136 . . . . . . . 8 (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) β†’ (88 Β· (10↑29)) < (π‘“β€˜π‘‘))
133132adantr 482 . . . . . . 7 ((((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)))) β†’ (88 Β· (10↑29)) < (π‘“β€˜π‘‘))
134133ad2antlr 726 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ (88 Β· (10↑29)) < (π‘“β€˜π‘‘))
135121, 15nnmulcli 12237 . . . . . . . . . . 11 (89 Β· (10↑29)) ∈ β„•
136135nnrei 12221 . . . . . . . . . 10 (89 Β· (10↑29)) ∈ ℝ
137 eleq1 2822 . . . . . . . . . 10 ((π‘“β€˜π‘‘) = (89 Β· (10↑29)) β†’ ((π‘“β€˜π‘‘) ∈ ℝ ↔ (89 Β· (10↑29)) ∈ ℝ))
138136, 137mpbiri 258 . . . . . . . . 9 ((π‘“β€˜π‘‘) = (89 Β· (10↑29)) β†’ (π‘“β€˜π‘‘) ∈ ℝ)
1391383ad2ant3 1136 . . . . . . . 8 (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) β†’ (π‘“β€˜π‘‘) ∈ ℝ)
140139adantr 482 . . . . . . 7 ((((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)))) β†’ (π‘“β€˜π‘‘) ∈ ℝ)
141140ad2antlr 726 . . . . . 6 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ (π‘“β€˜π‘‘) ∈ ℝ)
14262, 96, 110, 112, 114, 116, 118, 120, 134, 141bgoldbtbnd 46477 . . . . 5 ((((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) ∧ (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–))))) ∧ (𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))) β†’ βˆ€π‘› ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) β†’ 𝑛 ∈ GoldbachOdd ))
143142exp31 421 . . . 4 ((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) β†’ ((((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)))) β†’ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ βˆ€π‘› ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) β†’ 𝑛 ∈ GoldbachOdd ))))
144143rexlimivv 3200 . . 3 (βˆƒπ‘‘ ∈ (β„€β‰₯β€˜3)βˆƒπ‘“ ∈ (RePartβ€˜π‘‘)(((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)))) β†’ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ βˆ€π‘› ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) β†’ 𝑛 ∈ GoldbachOdd )))
145 breq2 5153 . . . . . . . 8 (𝑛 = 𝑁 β†’ (7 < 𝑛 ↔ 7 < 𝑁))
146 breq1 5152 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝑛 < (88 Β· (10↑29)) ↔ 𝑁 < (88 Β· (10↑29))))
147145, 146anbi12d 632 . . . . . . 7 (𝑛 = 𝑁 β†’ ((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) ↔ (7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))))
148 eleq1 2822 . . . . . . 7 (𝑛 = 𝑁 β†’ (𝑛 ∈ GoldbachOdd ↔ 𝑁 ∈ GoldbachOdd ))
149147, 148imbi12d 345 . . . . . 6 (𝑛 = 𝑁 β†’ (((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) β†’ 𝑛 ∈ GoldbachOdd ) ↔ ((7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ 𝑁 ∈ GoldbachOdd )))
150149rspcv 3609 . . . . 5 (𝑁 ∈ Odd β†’ (βˆ€π‘› ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) β†’ 𝑛 ∈ GoldbachOdd ) β†’ ((7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ 𝑁 ∈ GoldbachOdd )))
151150com23 86 . . . 4 (𝑁 ∈ Odd β†’ ((7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ (βˆ€π‘› ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) β†’ 𝑛 ∈ GoldbachOdd ) β†’ 𝑁 ∈ GoldbachOdd )))
1521513impib 1117 . . 3 ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ (βˆ€π‘› ∈ Odd ((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) β†’ 𝑛 ∈ GoldbachOdd ) β†’ 𝑁 ∈ GoldbachOdd ))
153144, 152sylcom 30 . 2 (βˆƒπ‘‘ ∈ (β„€β‰₯β€˜3)βˆƒπ‘“ ∈ (RePartβ€˜π‘‘)(((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) ∧ βˆ€π‘– ∈ (0..^𝑑)((π‘“β€˜π‘–) ∈ (β„™ βˆ– {2}) ∧ ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)) < ((4 Β· (10↑18)) βˆ’ 4) ∧ 4 < ((π‘“β€˜(𝑖 + 1)) βˆ’ (π‘“β€˜π‘–)))) β†’ ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ 𝑁 ∈ GoldbachOdd ))
1541, 153ax-mp 5 1 ((𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29))) β†’ 𝑁 ∈ GoldbachOdd )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   βˆ– cdif 3946  {csn 4629   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  β„cr 11109  0cc0 11110  1c1 11111   + caddc 11113   Β· cmul 11115   < clt 11248   ≀ cle 11249   βˆ’ cmin 11444  β„•cn 12212  2c2 12267  3c3 12268  4c4 12269  7c7 12272  8c8 12273  9c9 12274  β„•0cn0 12472  β„€cz 12558  cdc 12677  β„€β‰₯cuz 12822  ..^cfzo 13627  β†‘cexp 14027  β„™cprime 16608  RePartciccp 46081   Even ceven 46292   Odd codd 46293   GoldbachEven cgbe 46413   GoldbachOdd cgbo 46415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-bgbltosilva 46478  ax-hgprmladder 46482
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-rp 12975  df-ico 13330  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-dvds 16198  df-prm 16609  df-iccp 46082  df-even 46294  df-odd 46295  df-gbe 46416  df-gbo 46418
This theorem is referenced by:  tgoldbachlt  46484
  Copyright terms: Public domain W3C validator