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Theorem tgblthelfgott 46782
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 46780, ax-hgprmladder 46781 and bgoldbtbnd 46776. (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 46781 . 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 12493 . . . . . . . . . 10 1 ∈ β„•0
3 1nn 12228 . . . . . . . . . 10 1 ∈ β„•
42, 3decnncl 12702 . . . . . . . . 9 11 ∈ β„•
54nnzi 12591 . . . . . . . 8 11 ∈ β„€
6 8nn0 12500 . . . . . . . . . . 11 8 ∈ β„•0
7 8nn 12312 . . . . . . . . . . 11 8 ∈ β„•
86, 7decnncl 12702 . . . . . . . . . 10 88 ∈ β„•
9 10nn 12698 . . . . . . . . . . 11 10 ∈ β„•
10 2nn0 12494 . . . . . . . . . . . . 13 2 ∈ β„•0
11 9nn 12315 . . . . . . . . . . . . 13 9 ∈ β„•
1210, 11decnncl 12702 . . . . . . . . . . . 12 29 ∈ β„•
1312nnnn0i 12485 . . . . . . . . . . 11 29 ∈ β„•0
14 nnexpcl 14045 . . . . . . . . . . 11 ((10 ∈ β„• ∧ 29 ∈ β„•0) β†’ (10↑29) ∈ β„•)
159, 13, 14mp2an 689 . . . . . . . . . 10 (10↑29) ∈ β„•
168, 15nnmulcli 12242 . . . . . . . . 9 (88 Β· (10↑29)) ∈ β„•
1716nnzi 12591 . . . . . . . 8 (88 Β· (10↑29)) ∈ β„€
18 1re 11219 . . . . . . . . . . 11 1 ∈ ℝ
198nnrei 12226 . . . . . . . . . . 11 88 ∈ ℝ
2018, 19pm3.2i 470 . . . . . . . . . 10 (1 ∈ ℝ ∧ 88 ∈ ℝ)
21 0le1 11742 . . . . . . . . . . 11 0 ≀ 1
22 1lt10 12821 . . . . . . . . . . . 12 1 < 10
237, 6, 2, 22declti 12720 . . . . . . . . . . 11 1 < 88
2421, 23pm3.2i 470 . . . . . . . . . 10 (0 ≀ 1 ∧ 1 < 88)
25 nnexpcl 14045 . . . . . . . . . . . . 13 ((10 ∈ β„• ∧ 1 ∈ β„•0) β†’ (10↑1) ∈ β„•)
269, 2, 25mp2an 689 . . . . . . . . . . . 12 (10↑1) ∈ β„•
2726nnrei 12226 . . . . . . . . . . 11 (10↑1) ∈ ℝ
2815nnrei 12226 . . . . . . . . . . 11 (10↑29) ∈ ℝ
2927, 28pm3.2i 470 . . . . . . . . . 10 ((10↑1) ∈ ℝ ∧ (10↑29) ∈ ℝ)
30 0re 11221 . . . . . . . . . . . . 13 0 ∈ ℝ
31 10re 12701 . . . . . . . . . . . . 13 10 ∈ ℝ
32 10pos 12699 . . . . . . . . . . . . 13 0 < 10
3330, 31, 32ltleii 11342 . . . . . . . . . . . 12 0 ≀ 10
349nncni 12227 . . . . . . . . . . . . 13 10 ∈ β„‚
35 exp1 14038 . . . . . . . . . . . . 13 (10 ∈ β„‚ β†’ (10↑1) = 10)
3634, 35ax-mp 5 . . . . . . . . . . . 12 (10↑1) = 10
3733, 36breqtrri 5175 . . . . . . . . . . 11 0 ≀ (10↑1)
38 1z 12597 . . . . . . . . . . . . 13 1 ∈ β„€
3912nnzi 12591 . . . . . . . . . . . . 13 29 ∈ β„€
4031, 38, 393pm3.2i 1338 . . . . . . . . . . . 12 (10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 29 ∈ β„€)
41 2nn 12290 . . . . . . . . . . . . . 14 2 ∈ β„•
42 9nn0 12501 . . . . . . . . . . . . . 14 9 ∈ β„•0
4341, 42, 2, 22declti 12720 . . . . . . . . . . . . 13 1 < 29
4422, 43pm3.2i 470 . . . . . . . . . . . 12 (1 < 10 ∧ 1 < 29)
45 ltexp2a 14136 . . . . . . . . . . . 12 (((10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 29 ∈ β„€) ∧ (1 < 10 ∧ 1 < 29)) β†’ (10↑1) < (10↑29))
4640, 44, 45mp2an 689 . . . . . . . . . . 11 (10↑1) < (10↑29)
4737, 46pm3.2i 470 . . . . . . . . . 10 (0 ≀ (10↑1) ∧ (10↑1) < (10↑29))
48 ltmul12a 12075 . . . . . . . . . 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 690 . . . . . . . . 9 (1 Β· (10↑1)) < (88 Β· (10↑29))
5026nnzi 12591 . . . . . . . . . . . 12 (10↑1) ∈ β„€
51 zmulcl 12616 . . . . . . . . . . . 12 ((1 ∈ β„€ ∧ (10↑1) ∈ β„€) β†’ (1 Β· (10↑1)) ∈ β„€)
5238, 50, 51mp2an 689 . . . . . . . . . . 11 (1 Β· (10↑1)) ∈ β„€
53 zltp1le 12617 . . . . . . . . . . 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 689 . . . . . . . . . 10 ((1 Β· (10↑1)) < (88 Β· (10↑29)) ↔ ((1 Β· (10↑1)) + 1) ≀ (88 Β· (10↑29)))
55 1t10e1p1e11 46317 . . . . . . . . . . . 12 11 = ((1 Β· (10↑1)) + 1)
5655eqcomi 2740 . . . . . . . . . . 11 ((1 Β· (10↑1)) + 1) = 11
5756breq1i 5155 . . . . . . . . . 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 12833 . . . . . . . 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 1340 . . . . . . 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 12300 . . . . . . . . . 10 4 ∈ β„•
642, 7decnncl 12702 . . . . . . . . . . . 12 18 ∈ β„•
6564nnnn0i 12485 . . . . . . . . . . 11 18 ∈ β„•0
66 nnexpcl 14045 . . . . . . . . . . 11 ((10 ∈ β„• ∧ 18 ∈ β„•0) β†’ (10↑18) ∈ β„•)
679, 65, 66mp2an 689 . . . . . . . . . 10 (10↑18) ∈ β„•
6863, 67nnmulcli 12242 . . . . . . . . 9 (4 Β· (10↑18)) ∈ β„•
6968nnzi 12591 . . . . . . . 8 (4 Β· (10↑18)) ∈ β„€
70 4re 12301 . . . . . . . . . . 11 4 ∈ ℝ
7118, 70pm3.2i 470 . . . . . . . . . 10 (1 ∈ ℝ ∧ 4 ∈ ℝ)
72 1lt4 12393 . . . . . . . . . . 11 1 < 4
7321, 72pm3.2i 470 . . . . . . . . . 10 (0 ≀ 1 ∧ 1 < 4)
7467nnrei 12226 . . . . . . . . . . 11 (10↑18) ∈ ℝ
7527, 74pm3.2i 470 . . . . . . . . . 10 ((10↑1) ∈ ℝ ∧ (10↑18) ∈ ℝ)
7664nnzi 12591 . . . . . . . . . . . . 13 18 ∈ β„€
7731, 38, 763pm3.2i 1338 . . . . . . . . . . . 12 (10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 18 ∈ β„€)
783, 6, 2, 22declti 12720 . . . . . . . . . . . . 13 1 < 18
7922, 78pm3.2i 470 . . . . . . . . . . . 12 (1 < 10 ∧ 1 < 18)
80 ltexp2a 14136 . . . . . . . . . . . 12 (((10 ∈ ℝ ∧ 1 ∈ β„€ ∧ 18 ∈ β„€) ∧ (1 < 10 ∧ 1 < 18)) β†’ (10↑1) < (10↑18))
8177, 79, 80mp2an 689 . . . . . . . . . . 11 (10↑1) < (10↑18)
8237, 81pm3.2i 470 . . . . . . . . . 10 (0 ≀ (10↑1) ∧ (10↑1) < (10↑18))
83 ltmul12a 12075 . . . . . . . . . 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 690 . . . . . . . . 9 (1 Β· (10↑1)) < (4 Β· (10↑18))
85 4z 12601 . . . . . . . . . . . 12 4 ∈ β„€
8667nnzi 12591 . . . . . . . . . . . 12 (10↑18) ∈ β„€
87 zmulcl 12616 . . . . . . . . . . . 12 ((4 ∈ β„€ ∧ (10↑18) ∈ β„€) β†’ (4 Β· (10↑18)) ∈ β„€)
8885, 86, 87mp2an 689 . . . . . . . . . . 11 (4 Β· (10↑18)) ∈ β„€
89 zltp1le 12617 . . . . . . . . . . 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 689 . . . . . . . . . 10 ((1 Β· (10↑1)) < (4 Β· (10↑18)) ↔ ((1 Β· (10↑1)) + 1) ≀ (4 Β· (10↑18)))
9156breq1i 5155 . . . . . . . . . 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 12833 . . . . . . . 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 1340 . . . . . . 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 482 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 𝑛 ∈ Even )
98 simprl 768 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 4 < 𝑛)
99 evenz 46597 . . . . . . . . . . . . . 14 (𝑛 ∈ Even β†’ 𝑛 ∈ β„€)
10099zred 12671 . . . . . . . . . . . . 13 (𝑛 ∈ Even β†’ 𝑛 ∈ ℝ)
10168nnrei 12226 . . . . . . . . . . . . 13 (4 Β· (10↑18)) ∈ ℝ
102 ltle 11307 . . . . . . . . . . . . 13 ((𝑛 ∈ ℝ ∧ (4 Β· (10↑18)) ∈ ℝ) β†’ (𝑛 < (4 Β· (10↑18)) β†’ 𝑛 ≀ (4 Β· (10↑18))))
103100, 101, 102sylancl 585 . . . . . . . . . . . 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 418 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 𝑛 ≀ (4 Β· (10↑18)))
106 ax-bgbltosilva 46777 . . . . . . . . . 10 ((𝑛 ∈ Even ∧ 4 < 𝑛 ∧ 𝑛 ≀ (4 Β· (10↑18))) β†’ 𝑛 ∈ GoldbachEven )
10797, 98, 105, 106syl3anc 1370 . . . . . . . . 9 ((𝑛 ∈ Even ∧ (4 < 𝑛 ∧ 𝑛 < (4 Β· (10↑18)))) β†’ 𝑛 ∈ GoldbachEven )
108107ex 412 . . . . . . . 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 3144 . . . . . 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 482 . . . . . . 7 ((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) β†’ 𝑑 ∈ (β„€β‰₯β€˜3))
112111ad2antrr 723 . . . . . 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 484 . . . . . . 7 ((𝑑 ∈ (β„€β‰₯β€˜3) ∧ 𝑓 ∈ (RePartβ€˜π‘‘)) β†’ 𝑓 ∈ (RePartβ€˜π‘‘))
114113ad2antrr 723 . . . . . 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 484 . . . . . . 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 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†’ 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 1190 . . . . . . 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 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†’ (π‘“β€˜0) = 7)
119 simpl2 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€˜1) = 13)
120119ad2antlr 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†’ (π‘“β€˜1) = 13)
1216, 11decnncl 12702 . . . . . . . . . . . . 13 89 ∈ β„•
122121nnrei 12226 . . . . . . . . . . . 12 89 ∈ ℝ
12315nngt0i 12256 . . . . . . . . . . . . 13 0 < (10↑29)
12428, 123pm3.2i 470 . . . . . . . . . . . 12 ((10↑29) ∈ ℝ ∧ 0 < (10↑29))
12519, 122, 1243pm3.2i 1338 . . . . . . . . . . 11 (88 ∈ ℝ ∧ 89 ∈ ℝ ∧ ((10↑29) ∈ ℝ ∧ 0 < (10↑29)))
126 8lt9 12416 . . . . . . . . . . . 12 8 < 9
1276, 6, 11, 126declt 12710 . . . . . . . . . . 11 88 < 89
128 ltmul1a 12068 . . . . . . . . . . 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 689 . . . . . . . . . 10 (88 Β· (10↑29)) < (89 Β· (10↑29))
130 breq2 5152 . . . . . . . . . 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 1134 . . . . . . . 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 480 . . . . . . 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 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†’ (88 Β· (10↑29)) < (π‘“β€˜π‘‘))
135121, 15nnmulcli 12242 . . . . . . . . . . 11 (89 Β· (10↑29)) ∈ β„•
136135nnrei 12226 . . . . . . . . . 10 (89 Β· (10↑29)) ∈ ℝ
137 eleq1 2820 . . . . . . . . . 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 1134 . . . . . . . 8 (((π‘“β€˜0) = 7 ∧ (π‘“β€˜1) = 13 ∧ (π‘“β€˜π‘‘) = (89 Β· (10↑29))) β†’ (π‘“β€˜π‘‘) ∈ ℝ)
140139adantr 480 . . . . . . 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 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†’ (π‘“β€˜π‘‘) ∈ ℝ)
14262, 96, 110, 112, 114, 116, 118, 120, 134, 141bgoldbtbnd 46776 . . . . 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 419 . . . 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 3198 . . 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 5152 . . . . . . . 8 (𝑛 = 𝑁 β†’ (7 < 𝑛 ↔ 7 < 𝑁))
146 breq1 5151 . . . . . . . 8 (𝑛 = 𝑁 β†’ (𝑛 < (88 Β· (10↑29)) ↔ 𝑁 < (88 Β· (10↑29))))
147145, 146anbi12d 630 . . . . . . 7 (𝑛 = 𝑁 β†’ ((7 < 𝑛 ∧ 𝑛 < (88 Β· (10↑29))) ↔ (7 < 𝑁 ∧ 𝑁 < (88 Β· (10↑29)))))
148 eleq1 2820 . . . . . . 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 1115 . . 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069   βˆ– cdif 3945  {csn 4628   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7412  β„‚cc 11112  β„cr 11113  0cc0 11114  1c1 11115   + caddc 11117   Β· cmul 11119   < clt 11253   ≀ cle 11254   βˆ’ cmin 11449  β„•cn 12217  2c2 12272  3c3 12273  4c4 12274  7c7 12277  8c8 12278  9c9 12279  β„•0cn0 12477  β„€cz 12563  cdc 12682  β„€β‰₯cuz 12827  ..^cfzo 13632  β†‘cexp 14032  β„™cprime 16613  RePartciccp 46380   Even ceven 46591   Odd codd 46592   GoldbachEven cgbe 46712   GoldbachOdd cgbo 46714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192  ax-bgbltosilva 46777  ax-hgprmladder 46781
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9441  df-inf 9442  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-rp 12980  df-ico 13335  df-fz 13490  df-fzo 13633  df-seq 13972  df-exp 14033  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  df-dvds 16203  df-prm 16614  df-iccp 46381  df-even 46593  df-odd 46594  df-gbe 46715  df-gbo 46717
This theorem is referenced by:  tgoldbachlt  46783
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