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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tgblthelfgott Structured version   Visualization version   GIF version
Theorem tgblthelfgott 46469
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 46467, ax-hgprmladder 46468 and bgoldbtbnd 46463. (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 46468 . 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 12484 . . . . . . . . . 10 1 ∈ β„•0
3 1nn 12219 . . . . . . . . . 10 1 ∈ β„•
42, 3decnncl 12693 . . . . . . . . 9 11 ∈ β„•
54nnzi 12582 . . . . . . . 8 11 ∈ β„€
6 8nn0 12491 . . . . . . . . . . 11 8 ∈ β„•0
7 8nn 12303 . . . . . . . . . . 11 8 ∈ β„•
86, 7decnncl 12693 . . . . . . . . . 10 88 ∈ β„•
9 10nn 12689 . . . . . . . . . . 11 10 ∈ β„•
10 2nn0 12485 . . . . . . . . . . . . 13 2 ∈ β„•0
11 9nn 12306 . . . . . . . . . . . . 13 9 ∈ β„•
1210, 11decnncl 12693 . . . . . . . . . . . 12 29 ∈ β„•
1312nnnn0i 12476 . . . . . . . . . . 11 29 ∈ β„•0
14 nnexpcl 14036 . . . . . . . . . . 11 ((10 ∈ β„• ∧ 29 ∈ β„•0) β†’ (10↑29) ∈ β„•)
159, 13, 14mp2an 690 . . . . . . . . . 10 (10↑29) ∈ β„•
168, 15nnmulcli 12233 . . . . . . . . 9 (88 Β· (10↑29)) ∈ β„•
1716nnzi 12582 . . . . . . . 8 (88 Β· (10↑29)) ∈ β„€
18 1re 11210 . . . . . . . . . . 11 1 ∈ ℝ
198nnrei 12217 . . . . . . . . . . 11 88 ∈ ℝ
2018, 19pm3.2i 471 . . . . . . . . . 10 (1 ∈ ℝ ∧ 88 ∈ ℝ)
21 0le1 11733 . . . . . . . . . . 11 0 ≀ 1
22 1lt10 12812 . . . . . . . . . . . 12 1 < 10
237, 6, 2, 22declti 12711 . . . . . . . . . . 11 1 < 88
2421, 23pm3.2i 471 . . . . . . . . . 10 (0 ≀ 1 ∧ 1 < 88)
25 nnexpcl 14036 . . . . . . . . . . . . 13 ((10 ∈ β„• ∧ 1 ∈ β„•0) β†’ (10↑1) ∈ β„•)
269, 2, 25mp2an 690 . . . . . . . . . . . 12 (10↑1) ∈ β„•
2726nnrei 12217 . . . . . . . . . . 11 (10↑1) ∈ ℝ
2815nnrei 12217 . . . . . . . . . . 11 (10↑29) ∈ ℝ
2927, 28pm3.2i 471 . . . . . . . . . 10 ((10↑1) ∈ ℝ ∧ (10↑29) ∈ ℝ)
30 0re 11212 . . . . . . . . . . . . 13 0 ∈ ℝ
31 10re 12692 . . . . . . . . . . . . 13 10 ∈ ℝ
32 10pos 12690 . . . . . . . . . . . . 13 0 < 10
3330, 31, 32ltleii 11333 . . . . . . . . . . . 12 0 ≀ 10
349nncni 12218 . . . . . . . . . . . . 13 10 ∈ β„‚
35 exp1 14029 . . . . . . . . . . . . 13 (10 ∈ β„‚ β†’ (10↑1) = 10)
3634, 35ax-mp 5 . . . . . . . . . . . 12 (10↑1) = 10
3733, 36breqtrri 5174 . . . . . . . . . . 11 0 ≀ (10↑1)
38 1z 12588 . . . . . . . . . . . . 13 1 ∈ β„€
3912nnzi 12582 . . . . . . . . . . . . 13 29 ∈ β„€
4031, 38, 393pm3.2i 1339 . . . . . . . . . . . 12 (10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 29 ∈ β„€)
41 2nn 12281 . . . . . . . . . . . . . 14 2 ∈ β„•
42 9nn0 12492 . . . . . . . . . . . . . 14 9 ∈ β„•0
4341, 42, 2, 22declti 12711 . . . . . . . . . . . . 13 1 < 29
4422, 43pm3.2i 471 . . . . . . . . . . . 12 (1 < 10 ∧ 1 < 29)
45 ltexp2a 14127 . . . . . . . . . . . 12 (((10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 29 ∈ β„€) ∧ (1 < 10 ∧ 1 < 29)) β†’ (10↑1) < (10↑29))
4640, 44, 45mp2an 690 . . . . . . . . . . 11 (10↑1) < (10↑29)
4737, 46pm3.2i 471 . . . . . . . . . 10 (0 ≀ (10↑1) ∧ (10↑1) < (10↑29))
48 ltmul12a 12066 . . . . . . . . . 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 691 . . . . . . . . 9 (1 Β· (10↑1)) < (88 Β· (10↑29))
5026nnzi 12582 . . . . . . . . . . . 12 (10↑1) ∈ β„€
51 zmulcl 12607 . . . . . . . . . . . 12 ((1 ∈ β„€ ∧ (10↑1) ∈ β„€) β†’ (1 Β· (10↑1)) ∈ β„€)
5238, 50, 51mp2an 690 . . . . . . . . . . 11 (1 Β· (10↑1)) ∈ β„€
53 zltp1le 12608 . . . . . . . . . . 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 690 . . . . . . . . . 10 ((1 Β· (10↑1)) < (88 Β· (10↑29)) ↔ ((1 Β· (10↑1)) + 1) ≀ (88 Β· (10↑29)))
55 1t10e1p1e11 46004 . . . . . . . . . . . 12 11 = ((1 Β· (10↑1)) + 1)
5655eqcomi 2741 . . . . . . . . . . 11 ((1 Β· (10↑1)) + 1) = 11
5756breq1i 5154 . . . . . . . . . 10 (((1 Β· (10↑1)) + 1) ≀ (88 Β· (10↑29)) ↔ 11 ≀ (88 Β· (10↑29)))
5854, 57bitri 274 . . . . . . . . 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 12824 . . . . . . . 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 1341 . . . . . . 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 12291 . . . . . . . . . 10 4 ∈ β„•
642, 7decnncl 12693 . . . . . . . . . . . 12 18 ∈ β„•
6564nnnn0i 12476 . . . . . . . . . . 11 18 ∈ β„•0
66 nnexpcl 14036 . . . . . . . . . . 11 ((10 ∈ β„• ∧ 18 ∈ β„•0) β†’ (10↑18) ∈ β„•)
679, 65, 66mp2an 690 . . . . . . . . . 10 (10↑18) ∈ β„•
6863, 67nnmulcli 12233 . . . . . . . . 9 (4 Β· (10↑18)) ∈ β„•
6968nnzi 12582 . . . . . . . 8 (4 Β· (10↑18)) ∈ β„€
70 4re 12292 . . . . . . . . . . 11 4 ∈ ℝ
7118, 70pm3.2i 471 . . . . . . . . . 10 (1 ∈ ℝ ∧ 4 ∈ ℝ)
72 1lt4 12384 . . . . . . . . . . 11 1 < 4
7321, 72pm3.2i 471 . . . . . . . . . 10 (0 ≀ 1 ∧ 1 < 4)
7467nnrei 12217 . . . . . . . . . . 11 (10↑18) ∈ ℝ
7527, 74pm3.2i 471 . . . . . . . . . 10 ((10↑1) ∈ ℝ ∧ (10↑18) ∈ ℝ)
7664nnzi 12582 . . . . . . . . . . . . 13 18 ∈ β„€
7731, 38, 763pm3.2i 1339 . . . . . . . . . . . 12 (10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 18 ∈ β„€)
783, 6, 2, 22declti 12711 . . . . . . . . . . . . 13 1 < 18
7922, 78pm3.2i 471 . . . . . . . . . . . 12 (1 < 10 ∧ 1 < 18)
80 ltexp2a 14127 . . . . . . . . . . . 12 (((10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 18 ∈ β„€) ∧ (1 < 10 ∧ 1 < 18)) β†’ (10↑1) < (10↑18))
8177, 79, 80mp2an 690 . . . . . . . . . . 11 (10↑1) < (10↑18)
8237, 81pm3.2i 471 . . . . . . . . . 10 (0 ≀ (10↑1) ∧ (10↑1) < (10↑18))
83 ltmul12a 12066 . . . . . . . . . 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 691 . . . . . . . . 9 (1 Β· (10↑1)) < (4 Β· (10↑18))
85 4z 12592 . . . . . . . . . . . 12 4 ∈ β„€
8667nnzi 12582 . . . . . . . . . . . 12 (10↑18) ∈ β„€
87 zmulcl 12607 . . . . . . . . . . . 12 ((4 ∈ β„€ ∧ (10↑18) ∈ β„€) β†’ (4 Β· (10↑18)) ∈ β„€)
8885, 86, 87mp2an 690 . . . . . . . . . . 11 (4 Β· (10↑18)) ∈ β„€
89 zltp1le 12608 . . . . . . . . . . 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 690 . . . . . . . . . 10 ((1 Β· (10↑1)) < (4 Β· (10↑18)) ↔ ((1 Β· (10↑1)) + 1) ≀ (4 Β· (10↑18)))
9156breq1i 5154 . . . . . . . . . 10 (((1 Β· (10↑1)) + 1) ≀ (4 Β· (10↑18)) ↔ 11 ≀ (4 Β· (10↑18)))
9290, 91bitri 274 . . . . . . . . 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 12824 . . . . . . . 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 1341 . . . . . . 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 483 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 𝑛 ∈ Even )
98 simprl 769 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 4 < 𝑛)
99 evenz 46284 . . . . . . . . . . . . . 14 (𝑛 ∈ Even β†’ 𝑛 ∈ β„€)
10099zred 12662 . . . . . . . . . . . . 13 (𝑛 ∈ Even β†’ 𝑛 ∈ ℝ)
10168nnrei 12217 . . . . . . . . . . . . 13 (4 Β· (10↑18)) ∈ ℝ
102 ltle 11298 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ ∧ (4 Β· (10↑18)) ∈ ℝ) β†’ (𝑛 < (4 Β· (10↑18)) β†’ 𝑛 ≀ (4 Β· (10↑18))))
103100, 101, 102sylancl 586 . . . . . . . . . . . 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 419 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 𝑛 ≀ (4 Β· (10↑18)))
106 ax-bgbltosilva 46464 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≀ (4 Β· (10↑18))) β†’ 𝑛 ∈ GoldbachEven )
10797, 98, 105, 106syl3anc 1371 . . . . . . . . 9 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 𝑛 ∈ GoldbachEven )
108107ex 413 . . . . . . . 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 3145 . . . . . 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 483 . . . . . . 7 ((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) β†’ 𝑑 ∈ (β„€β‰₯β€˜3))
112111ad2antrr 724 . . . . . 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 485 . . . . . . 7 ((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) β†’ 𝑓 ∈ (RePartβ€˜π‘‘))
114113ad2antrr 724 . . . . . 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 485 . . . . . . 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 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†’ 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 1191 . . . . . . 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 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†’ (π‘“β€˜0) = 7)
119 simpl2 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€˜1) = 13)
120119ad2antlr 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†’ (π‘“β€˜1) = 13)
1216, 11decnncl 12693 . . . . . . . . . . . . 13 89 ∈ β„•
122121nnrei 12217 . . . . . . . . . . . 12 89 ∈ ℝ
12315nngt0i 12247 . . . . . . . . . . . . 13 0 < (10↑29)
12428, 123pm3.2i 471 . . . . . . . . . . . 12 ((10↑29) ∈ ℝ ∧ 0 < (10↑29))
12519, 122, 1243pm3.2i 1339 . . . . . . . . . . 11 (88 ∈ ℝ ∧ 89 ∈ ℝ ∧ ((10↑29) ∈ ℝ ∧ 0 < (10↑29)))
126 8lt9 12407 . . . . . . . . . . . 12 8 < 9
1276, 6, 11, 126declt 12701 . . . . . . . . . . 11 88 < 89
128 ltmul1a 12059 . . . . . . . . . . 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 690 . . . . . . . . . 10 (88 Β· (10↑29)) < (89 Β· (10↑29))
130 breq2 5151 . . . . . . . . . 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 257 . . . . . . . . 9 ((π‘“β€˜π‘‘) = (89 Β· (10↑29)) β†’ (88 Β· (10↑29)) < (π‘“β€˜π‘‘))
1321313ad2ant3 1135 . . . . . . . 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 481 . . . . . . 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 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†’ (88 Β· (10↑29)) < (π‘“β€˜π‘‘))
135121, 15nnmulcli 12233 . . . . . . . . . . 11 (89 Β· (10↑29)) ∈ β„•
136135nnrei 12217 . . . . . . . . . 10 (89 Β· (10↑29)) ∈ ℝ
137 eleq1 2821 . . . . . . . . . 10 ((π‘“β€˜π‘‘) = (89 Β· (10↑29)) β†’ ((π‘“β€˜π‘‘) ∈ ℝ ↔ (89 Β· (10↑29)) ∈ ℝ))
138136, 137mpbiri 257 . . . . . . . . 9 ((π‘“β€˜π‘‘) = (89 Β· (10↑29)) β†’ (π‘“β€˜π‘‘) ∈ ℝ)
1391383ad2ant3 1135 . . . . . . . 8 (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) β†’ (π‘“β€˜π‘‘) ∈ ℝ)
140139adantr 481 . . . . . . 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 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†’ (π‘“β€˜π‘‘) ∈ ℝ)
14262, 96, 110, 112, 114, 116, 118, 120, 134, 141bgoldbtbnd 46463 . . . . 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 420 . . . 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 3199 . . 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 5151 . . . . . . . 8 (𝑛 = 𝑁 β†’ (7 < 𝑛 ↔ 7 < 𝑁))
146 breq1 5150 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝑛 < (88 Β· (10↑29)) ↔ 𝑁 < (88 Β· (10↑29))))
147145, 146anbi12d 631 . . . . . . 7 (𝑛 = 𝑁 β†’ ((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) ↔ (7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))))
148 eleq1 2821 . . . . . . 7 (𝑛 = 𝑁 β†’ (𝑛 ∈ GoldbachOdd ↔ 𝑁 ∈ GoldbachOdd ))
149147, 148imbi12d 344 . . . . . 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 3608 . . . . 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 1116 . . 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βˆ– cdif 3944  {csn 4627   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   Β· cmul 11111   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440  β„•cn 12208  2c2 12263  3c3 12264  4c4 12265  7c7 12268  8c8 12269  9c9 12270  β„•0cn0 12468  β„€cz 12554  cdc 12673  β„€β‰₯cuz 12818  ..^cfzo 13623  β†‘cexp 14023  β„™cprime 16604  RePartciccp 46067   Even ceven 46278   Odd codd 46279   GoldbachEven cgbe 46399   GoldbachOdd cgbo 46401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184  ax-bgbltosilva 46464  ax-hgprmladder 46468
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-2o 8463  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-z 12555  df-dec 12674  df-uz 12819  df-rp 12971  df-ico 13326  df-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-dvds 16194  df-prm 16605  df-iccp 46068  df-even 46280  df-odd 46281  df-gbe 46402  df-gbo 46404
This theorem is referenced by:  tgoldbachlt  46470
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