Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nnsum4primesevenALTV Structured version   Visualization version   GIF version
Theorem nnsum4primesevenALTV 46455
Description: If the (strong) ternary Goldbach conjecture is valid, then every even integer greater than 10 is the sum of 4 primes. (Contributed by AV, 27-Jul-2020.)
Assertion
Ref Expression
nnsum4primesevenALTV (βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) β†’ ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even ) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜)))
Distinct variable group:   𝑓,𝑁,π‘˜,π‘š
Proof of Theorem nnsum4primesevenALTV
Dummy variables π‘œ 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplll 773 . . . . 5 ((((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) ∧ π‘œ ∈ GoldbachOdd ) ∧ 𝑁 = (π‘œ + 3)) β†’ βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ))
2 8nn 12303 . . . . . . . . . 10 8 ∈ β„•
32nnzi 12582 . . . . . . . . 9 8 ∈ β„€
43a1i 11 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ 8 ∈ β„€)
5 3z 12591 . . . . . . . . 9 3 ∈ β„€
65a1i 11 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ 3 ∈ β„€)
74, 6zaddcld 12666 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ (8 + 3) ∈ β„€)
8 eluzelz 12828 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ 𝑁 ∈ β„€)
9 eluz2 12824 . . . . . . . . . 10 (𝑁 ∈ (β„€β‰₯β€˜12) ↔ (12 ∈ β„€ ∧ 𝑁 ∈ β„€ ∧ 12 ≀ 𝑁))
10 8p4e12 12755 . . . . . . . . . . . . . 14 (8 + 4) = 12
1110breq1i 5154 . . . . . . . . . . . . 13 ((8 + 4) ≀ 𝑁 ↔ 12 ≀ 𝑁)
12 1nn0 12484 . . . . . . . . . . . . . . . 16 1 ∈ β„•0
13 2nn 12281 . . . . . . . . . . . . . . . 16 2 ∈ β„•
14 1lt2 12379 . . . . . . . . . . . . . . . 16 1 < 2
1512, 12, 13, 14declt 12701 . . . . . . . . . . . . . . 15 11 < 12
16 8p3e11 12754 . . . . . . . . . . . . . . 15 (8 + 3) = 11
1715, 16, 103brtr4i 5177 . . . . . . . . . . . . . 14 (8 + 3) < (8 + 4)
18 8re 12304 . . . . . . . . . . . . . . . . 17 8 ∈ ℝ
1918a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„€ β†’ 8 ∈ ℝ)
20 3re 12288 . . . . . . . . . . . . . . . . 17 3 ∈ ℝ
2120a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„€ β†’ 3 ∈ ℝ)
2219, 21readdcld 11239 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„€ β†’ (8 + 3) ∈ ℝ)
23 4re 12292 . . . . . . . . . . . . . . . . 17 4 ∈ ℝ
2423a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„€ β†’ 4 ∈ ℝ)
2519, 24readdcld 11239 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„€ β†’ (8 + 4) ∈ ℝ)
26 zre 12558 . . . . . . . . . . . . . . 15 (𝑁 ∈ β„€ β†’ 𝑁 ∈ ℝ)
27 ltleletr 11303 . . . . . . . . . . . . . . 15 (((8 + 3) ∈ ℝ ∧ (8 + 4) ∈ ℝ ∧ 𝑁 ∈ ℝ) β†’ (((8 + 3) < (8 + 4) ∧ (8 + 4) ≀ 𝑁) β†’ (8 + 3) ≀ 𝑁))
2822, 25, 26, 27syl3anc 1371 . . . . . . . . . . . . . 14 (𝑁 ∈ β„€ β†’ (((8 + 3) < (8 + 4) ∧ (8 + 4) ≀ 𝑁) β†’ (8 + 3) ≀ 𝑁))
2917, 28mpani 694 . . . . . . . . . . . . 13 (𝑁 ∈ β„€ β†’ ((8 + 4) ≀ 𝑁 β†’ (8 + 3) ≀ 𝑁))
3011, 29biimtrrid 242 . . . . . . . . . . . 12 (𝑁 ∈ β„€ β†’ (12 ≀ 𝑁 β†’ (8 + 3) ≀ 𝑁))
3130imp 407 . . . . . . . . . . 11 ((𝑁 ∈ β„€ ∧ 12 ≀ 𝑁) β†’ (8 + 3) ≀ 𝑁)
32313adant1 1130 . . . . . . . . . 10 ((12 ∈ β„€ ∧ 𝑁 ∈ β„€ ∧ 12 ≀ 𝑁) β†’ (8 + 3) ≀ 𝑁)
339, 32sylbi 216 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ (8 + 3) ≀ 𝑁)
34 eluz2 12824 . . . . . . . . 9 (𝑁 ∈ (β„€β‰₯β€˜(8 + 3)) ↔ ((8 + 3) ∈ β„€ ∧ 𝑁 ∈ β„€ ∧ (8 + 3) ≀ 𝑁))
357, 8, 33, 34syl3anbrc 1343 . . . . . . . 8 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ 𝑁 ∈ (β„€β‰₯β€˜(8 + 3)))
36 eluzsub 12848 . . . . . . . 8 ((8 ∈ β„€ ∧ 3 ∈ β„€ ∧ 𝑁 ∈ (β„€β‰₯β€˜(8 + 3))) β†’ (𝑁 βˆ’ 3) ∈ (β„€β‰₯β€˜8))
374, 6, 35, 36syl3anc 1371 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ (𝑁 βˆ’ 3) ∈ (β„€β‰₯β€˜8))
3837adantr 481 . . . . . 6 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even ) β†’ (𝑁 βˆ’ 3) ∈ (β„€β‰₯β€˜8))
3938ad3antlr 729 . . . . 5 ((((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) ∧ π‘œ ∈ GoldbachOdd ) ∧ 𝑁 = (π‘œ + 3)) β†’ (𝑁 βˆ’ 3) ∈ (β„€β‰₯β€˜8))
40 3odd 46362 . . . . . . . . . . . 12 3 ∈ Odd
4140a1i 11 . . . . . . . . . . 11 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ 3 ∈ Odd )
4241anim1i 615 . . . . . . . . . 10 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even ) β†’ (3 ∈ Odd ∧ 𝑁 ∈ Even ))
4342adantl 482 . . . . . . . . 9 ((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) β†’ (3 ∈ Odd ∧ 𝑁 ∈ Even ))
4443ancomd 462 . . . . . . . 8 ((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) β†’ (𝑁 ∈ Even ∧ 3 ∈ Odd ))
4544adantr 481 . . . . . . 7 (((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) ∧ π‘œ ∈ GoldbachOdd ) β†’ (𝑁 ∈ Even ∧ 3 ∈ Odd ))
4645adantr 481 . . . . . 6 ((((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) ∧ π‘œ ∈ GoldbachOdd ) ∧ 𝑁 = (π‘œ + 3)) β†’ (𝑁 ∈ Even ∧ 3 ∈ Odd ))
47 emoo 46358 . . . . . 6 ((𝑁 ∈ Even ∧ 3 ∈ Odd ) β†’ (𝑁 βˆ’ 3) ∈ Odd )
4846, 47syl 17 . . . . 5 ((((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) ∧ π‘œ ∈ GoldbachOdd ) ∧ 𝑁 = (π‘œ + 3)) β†’ (𝑁 βˆ’ 3) ∈ Odd )
49 nnsum4primesoddALTV 46451 . . . . . 6 (βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) β†’ (((𝑁 βˆ’ 3) ∈ (β„€β‰₯β€˜8) ∧ (𝑁 βˆ’ 3) ∈ Odd ) β†’ βˆƒπ‘” ∈ (β„™ ↑m (1...3))(𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)))
5049imp 407 . . . . 5 ((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ ((𝑁 βˆ’ 3) ∈ (β„€β‰₯β€˜8) ∧ (𝑁 βˆ’ 3) ∈ Odd )) β†’ βˆƒπ‘” ∈ (β„™ ↑m (1...3))(𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜))
511, 39, 48, 50syl12anc 835 . . . 4 ((((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) ∧ π‘œ ∈ GoldbachOdd ) ∧ 𝑁 = (π‘œ + 3)) β†’ βˆƒπ‘” ∈ (β„™ ↑m (1...3))(𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜))
52 simpr 485 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ 𝑔:(1...3)βŸΆβ„™)
53 4z 12592 . . . . . . . . . . . . . . . . 17 4 ∈ β„€
54 fzonel 13642 . . . . . . . . . . . . . . . . . 18 Β¬ 4 ∈ (1..^4)
55 fzoval 13629 . . . . . . . . . . . . . . . . . . . . . 22 (4 ∈ β„€ β†’ (1..^4) = (1...(4 βˆ’ 1)))
5653, 55ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (1..^4) = (1...(4 βˆ’ 1))
57 4cn 12293 . . . . . . . . . . . . . . . . . . . . . . 23 4 ∈ β„‚
58 ax-1cn 11164 . . . . . . . . . . . . . . . . . . . . . . 23 1 ∈ β„‚
59 3cn 12289 . . . . . . . . . . . . . . . . . . . . . . 23 3 ∈ β„‚
60 3p1e4 12353 . . . . . . . . . . . . . . . . . . . . . . . 24 (3 + 1) = 4
61 subadd2 11460 . . . . . . . . . . . . . . . . . . . . . . . 24 ((4 ∈ β„‚ ∧ 1 ∈ β„‚ ∧ 3 ∈ β„‚) β†’ ((4 βˆ’ 1) = 3 ↔ (3 + 1) = 4))
6260, 61mpbiri 257 . . . . . . . . . . . . . . . . . . . . . . 23 ((4 ∈ β„‚ ∧ 1 ∈ β„‚ ∧ 3 ∈ β„‚) β†’ (4 βˆ’ 1) = 3)
6357, 58, 59, 62mp3an 1461 . . . . . . . . . . . . . . . . . . . . . 22 (4 βˆ’ 1) = 3
6463oveq2i 7416 . . . . . . . . . . . . . . . . . . . . 21 (1...(4 βˆ’ 1)) = (1...3)
6556, 64eqtri 2760 . . . . . . . . . . . . . . . . . . . 20 (1..^4) = (1...3)
6665eqcomi 2741 . . . . . . . . . . . . . . . . . . 19 (1...3) = (1..^4)
6766eleq2i 2825 . . . . . . . . . . . . . . . . . 18 (4 ∈ (1...3) ↔ 4 ∈ (1..^4))
6854, 67mtbir 322 . . . . . . . . . . . . . . . . 17 Β¬ 4 ∈ (1...3)
6953, 68pm3.2i 471 . . . . . . . . . . . . . . . 16 (4 ∈ β„€ ∧ Β¬ 4 ∈ (1...3))
7069a1i 11 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ (4 ∈ β„€ ∧ Β¬ 4 ∈ (1...3)))
71 3prm 16627 . . . . . . . . . . . . . . . 16 3 ∈ β„™
7271a1i 11 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ 3 ∈ β„™)
73 fsnunf 7179 . . . . . . . . . . . . . . 15 ((𝑔:(1...3)βŸΆβ„™ ∧ (4 ∈ β„€ ∧ Β¬ 4 ∈ (1...3)) ∧ 3 ∈ β„™) β†’ (𝑔 βˆͺ {⟨4, 3⟩}):((1...3) βˆͺ {4})βŸΆβ„™)
7452, 70, 72, 73syl3anc 1371 . . . . . . . . . . . . . 14 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ (𝑔 βˆͺ {⟨4, 3⟩}):((1...3) βˆͺ {4})βŸΆβ„™)
75 fzval3 13697 . . . . . . . . . . . . . . . . 17 (4 ∈ β„€ β†’ (1...4) = (1..^(4 + 1)))
7653, 75ax-mp 5 . . . . . . . . . . . . . . . 16 (1...4) = (1..^(4 + 1))
77 1z 12588 . . . . . . . . . . . . . . . . . 18 1 ∈ β„€
78 1re 11210 . . . . . . . . . . . . . . . . . . 19 1 ∈ ℝ
79 1lt4 12384 . . . . . . . . . . . . . . . . . . 19 1 < 4
8078, 23, 79ltleii 11333 . . . . . . . . . . . . . . . . . 18 1 ≀ 4
81 eluz2 12824 . . . . . . . . . . . . . . . . . 18 (4 ∈ (β„€β‰₯β€˜1) ↔ (1 ∈ β„€ ∧ 4 ∈ β„€ ∧ 1 ≀ 4))
8277, 53, 80, 81mpbir3an 1341 . . . . . . . . . . . . . . . . 17 4 ∈ (β„€β‰₯β€˜1)
83 fzosplitsn 13736 . . . . . . . . . . . . . . . . 17 (4 ∈ (β„€β‰₯β€˜1) β†’ (1..^(4 + 1)) = ((1..^4) βˆͺ {4}))
8482, 83ax-mp 5 . . . . . . . . . . . . . . . 16 (1..^(4 + 1)) = ((1..^4) βˆͺ {4})
8565uneq1i 4158 . . . . . . . . . . . . . . . 16 ((1..^4) βˆͺ {4}) = ((1...3) βˆͺ {4})
8676, 84, 853eqtri 2764 . . . . . . . . . . . . . . 15 (1...4) = ((1...3) βˆͺ {4})
8786feq2i 6706 . . . . . . . . . . . . . 14 ((𝑔 βˆͺ {⟨4, 3⟩}):(1...4)βŸΆβ„™ ↔ (𝑔 βˆͺ {⟨4, 3⟩}):((1...3) βˆͺ {4})βŸΆβ„™)
8874, 87sylibr 233 . . . . . . . . . . . . 13 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ (𝑔 βˆͺ {⟨4, 3⟩}):(1...4)βŸΆβ„™)
89 prmex 16610 . . . . . . . . . . . . . . 15 β„™ ∈ V
90 ovex 7438 . . . . . . . . . . . . . . 15 (1...4) ∈ V
9189, 90pm3.2i 471 . . . . . . . . . . . . . 14 (β„™ ∈ V ∧ (1...4) ∈ V)
92 elmapg 8829 . . . . . . . . . . . . . 14 ((β„™ ∈ V ∧ (1...4) ∈ V) β†’ ((𝑔 βˆͺ {⟨4, 3⟩}) ∈ (β„™ ↑m (1...4)) ↔ (𝑔 βˆͺ {⟨4, 3⟩}):(1...4)βŸΆβ„™))
9391, 92mp1i 13 . . . . . . . . . . . . 13 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑔 βˆͺ {⟨4, 3⟩}) ∈ (β„™ ↑m (1...4)) ↔ (𝑔 βˆͺ {⟨4, 3⟩}):(1...4)βŸΆβ„™))
9488, 93mpbird 256 . . . . . . . . . . . 12 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ (𝑔 βˆͺ {⟨4, 3⟩}) ∈ (β„™ ↑m (1...4)))
9594adantr 481 . . . . . . . . . . 11 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)) β†’ (𝑔 βˆͺ {⟨4, 3⟩}) ∈ (β„™ ↑m (1...4)))
96 fveq1 6887 . . . . . . . . . . . . . 14 (𝑓 = (𝑔 βˆͺ {⟨4, 3⟩}) β†’ (π‘“β€˜π‘˜) = ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜))
9796sumeq2sdv 15646 . . . . . . . . . . . . 13 (𝑓 = (𝑔 βˆͺ {⟨4, 3⟩}) β†’ Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜) = Ξ£π‘˜ ∈ (1...4)((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜))
9897eqeq2d 2743 . . . . . . . . . . . 12 (𝑓 = (𝑔 βˆͺ {⟨4, 3⟩}) β†’ (𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜) ↔ 𝑁 = Ξ£π‘˜ ∈ (1...4)((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜)))
9998adantl 482 . . . . . . . . . . 11 ((((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)) ∧ 𝑓 = (𝑔 βˆͺ {⟨4, 3⟩})) β†’ (𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜) ↔ 𝑁 = Ξ£π‘˜ ∈ (1...4)((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜)))
10082a1i 11 . . . . . . . . . . . . . 14 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ 4 ∈ (β„€β‰₯β€˜1))
10186eleq2i 2825 . . . . . . . . . . . . . . . . 17 (π‘˜ ∈ (1...4) ↔ π‘˜ ∈ ((1...3) βˆͺ {4}))
102 elun 4147 . . . . . . . . . . . . . . . . 17 (π‘˜ ∈ ((1...3) βˆͺ {4}) ↔ (π‘˜ ∈ (1...3) ∨ π‘˜ ∈ {4}))
103 velsn 4643 . . . . . . . . . . . . . . . . . 18 (π‘˜ ∈ {4} ↔ π‘˜ = 4)
104103orbi2i 911 . . . . . . . . . . . . . . . . 17 ((π‘˜ ∈ (1...3) ∨ π‘˜ ∈ {4}) ↔ (π‘˜ ∈ (1...3) ∨ π‘˜ = 4))
105101, 102, 1043bitri 296 . . . . . . . . . . . . . . . 16 (π‘˜ ∈ (1...4) ↔ (π‘˜ ∈ (1...3) ∨ π‘˜ = 4))
106 elfz2 13487 . . . . . . . . . . . . . . . . . . . . . . . 24 (π‘˜ ∈ (1...3) ↔ ((1 ∈ β„€ ∧ 3 ∈ β„€ ∧ π‘˜ ∈ β„€) ∧ (1 ≀ π‘˜ ∧ π‘˜ ≀ 3)))
10720, 23pm3.2i 471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (3 ∈ ℝ ∧ 4 ∈ ℝ)
108 3lt4 12382 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 < 4
109 ltnle 11289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((3 ∈ ℝ ∧ 4 ∈ ℝ) β†’ (3 < 4 ↔ Β¬ 4 ≀ 3))
110108, 109mpbii 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((3 ∈ ℝ ∧ 4 ∈ ℝ) β†’ Β¬ 4 ≀ 3)
111107, 110ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Β¬ 4 ≀ 3
112 breq1 5150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (π‘˜ = 4 β†’ (π‘˜ ≀ 3 ↔ 4 ≀ 3))
113112eqcoms 2740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (4 = π‘˜ β†’ (π‘˜ ≀ 3 ↔ 4 ≀ 3))
114111, 113mtbiri 326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (4 = π‘˜ β†’ Β¬ π‘˜ ≀ 3)
115114a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (π‘˜ ∈ β„€ β†’ (4 = π‘˜ β†’ Β¬ π‘˜ ≀ 3))
116115necon2ad 2955 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (π‘˜ ∈ β„€ β†’ (π‘˜ ≀ 3 β†’ 4 β‰  π‘˜))
117116adantld 491 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (π‘˜ ∈ β„€ β†’ ((1 ≀ π‘˜ ∧ π‘˜ ≀ 3) β†’ 4 β‰  π‘˜))
1181173ad2ant3 1135 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((1 ∈ β„€ ∧ 3 ∈ β„€ ∧ π‘˜ ∈ β„€) β†’ ((1 ≀ π‘˜ ∧ π‘˜ ≀ 3) β†’ 4 β‰  π‘˜))
119118imp 407 . . . . . . . . . . . . . . . . . . . . . . . 24 (((1 ∈ β„€ ∧ 3 ∈ β„€ ∧ π‘˜ ∈ β„€) ∧ (1 ≀ π‘˜ ∧ π‘˜ ≀ 3)) β†’ 4 β‰  π‘˜)
120106, 119sylbi 216 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘˜ ∈ (1...3) β†’ 4 β‰  π‘˜)
121120adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘˜ ∈ (1...3) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ 4 β‰  π‘˜)
122 fvunsn 7173 . . . . . . . . . . . . . . . . . . . . . 22 (4 β‰  π‘˜ β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) = (π‘”β€˜π‘˜))
123121, 122syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((π‘˜ ∈ (1...3) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) = (π‘”β€˜π‘˜))
124 ffvelcdm 7080 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑔:(1...3)βŸΆβ„™ ∧ π‘˜ ∈ (1...3)) β†’ (π‘”β€˜π‘˜) ∈ β„™)
125124ancoms 459 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘˜ ∈ (1...3) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ (π‘”β€˜π‘˜) ∈ β„™)
126 prmz 16608 . . . . . . . . . . . . . . . . . . . . . . 23 ((π‘”β€˜π‘˜) ∈ β„™ β†’ (π‘”β€˜π‘˜) ∈ β„€)
127125, 126syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((π‘˜ ∈ (1...3) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ (π‘”β€˜π‘˜) ∈ β„€)
128127zcnd 12663 . . . . . . . . . . . . . . . . . . . . 21 ((π‘˜ ∈ (1...3) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ (π‘”β€˜π‘˜) ∈ β„‚)
129123, 128eqeltrd 2833 . . . . . . . . . . . . . . . . . . . 20 ((π‘˜ ∈ (1...3) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) ∈ β„‚)
130129ex 413 . . . . . . . . . . . . . . . . . . 19 (π‘˜ ∈ (1...3) β†’ (𝑔:(1...3)βŸΆβ„™ β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) ∈ β„‚))
131130adantld 491 . . . . . . . . . . . . . . . . . 18 (π‘˜ ∈ (1...3) β†’ ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) ∈ β„‚))
132 fveq2 6888 . . . . . . . . . . . . . . . . . . . . 21 (π‘˜ = 4 β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) = ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4))
13353a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔:(1...3)βŸΆβ„™ β†’ 4 ∈ β„€)
1345a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔:(1...3)βŸΆβ„™ β†’ 3 ∈ β„€)
135 fdm 6723 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔:(1...3)βŸΆβ„™ β†’ dom 𝑔 = (1...3))
136 eleq2 2822 . . . . . . . . . . . . . . . . . . . . . . . . 25 (dom 𝑔 = (1...3) β†’ (4 ∈ dom 𝑔 ↔ 4 ∈ (1...3)))
13768, 136mtbiri 326 . . . . . . . . . . . . . . . . . . . . . . . 24 (dom 𝑔 = (1...3) β†’ Β¬ 4 ∈ dom 𝑔)
138135, 137syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔:(1...3)βŸΆβ„™ β†’ Β¬ 4 ∈ dom 𝑔)
139 fsnunfv 7181 . . . . . . . . . . . . . . . . . . . . . . 23 ((4 ∈ β„€ ∧ 3 ∈ β„€ ∧ Β¬ 4 ∈ dom 𝑔) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4) = 3)
140133, 134, 138, 139syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔:(1...3)βŸΆβ„™ β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4) = 3)
141140adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4) = 3)
142132, 141sylan9eq 2792 . . . . . . . . . . . . . . . . . . . 20 ((π‘˜ = 4 ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™)) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) = 3)
143142, 59eqeltrdi 2841 . . . . . . . . . . . . . . . . . . 19 ((π‘˜ = 4 ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™)) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) ∈ β„‚)
144143ex 413 . . . . . . . . . . . . . . . . . 18 (π‘˜ = 4 β†’ ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) ∈ β„‚))
145131, 144jaoi 855 . . . . . . . . . . . . . . . . 17 ((π‘˜ ∈ (1...3) ∨ π‘˜ = 4) β†’ ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) ∈ β„‚))
146145com12 32 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((π‘˜ ∈ (1...3) ∨ π‘˜ = 4) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) ∈ β„‚))
147105, 146biimtrid 241 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ (π‘˜ ∈ (1...4) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) ∈ β„‚))
148147imp 407 . . . . . . . . . . . . . 14 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ π‘˜ ∈ (1...4)) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) ∈ β„‚)
149100, 148, 132fsumm1 15693 . . . . . . . . . . . . 13 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ Ξ£π‘˜ ∈ (1...4)((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) = (Ξ£π‘˜ ∈ (1...(4 βˆ’ 1))((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) + ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4)))
150149adantr 481 . . . . . . . . . . . 12 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)) β†’ Ξ£π‘˜ ∈ (1...4)((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) = (Ξ£π‘˜ ∈ (1...(4 βˆ’ 1))((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) + ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4)))
15163eqcomi 2741 . . . . . . . . . . . . . . . . . . 19 3 = (4 βˆ’ 1)
152151oveq2i 7416 . . . . . . . . . . . . . . . . . 18 (1...3) = (1...(4 βˆ’ 1))
153152a1i 11 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ (1...3) = (1...(4 βˆ’ 1)))
154120adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ π‘˜ ∈ (1...3)) β†’ 4 β‰  π‘˜)
155154, 122syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ π‘˜ ∈ (1...3)) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) = (π‘”β€˜π‘˜))
156155eqcomd 2738 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ π‘˜ ∈ (1...3)) β†’ (π‘”β€˜π‘˜) = ((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜))
157153, 156sumeq12dv 15648 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜) = Ξ£π‘˜ ∈ (1...(4 βˆ’ 1))((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜))
158157eqeq2d 2743 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜) ↔ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...(4 βˆ’ 1))((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜)))
159158biimpa 477 . . . . . . . . . . . . . 14 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)) β†’ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...(4 βˆ’ 1))((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜))
160159eqcomd 2738 . . . . . . . . . . . . 13 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)) β†’ Ξ£π‘˜ ∈ (1...(4 βˆ’ 1))((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) = (𝑁 βˆ’ 3))
161160oveq1d 7420 . . . . . . . . . . . 12 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)) β†’ (Ξ£π‘˜ ∈ (1...(4 βˆ’ 1))((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜) + ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4)) = ((𝑁 βˆ’ 3) + ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4)))
16253a1i 11 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ 4 ∈ β„€)
1635a1i 11 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ 3 ∈ β„€)
164138adantl 482 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ Β¬ 4 ∈ dom 𝑔)
165162, 163, 164, 139syl3anc 1371 . . . . . . . . . . . . . . 15 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4) = 3)
166165oveq2d 7421 . . . . . . . . . . . . . 14 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑁 βˆ’ 3) + ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4)) = ((𝑁 βˆ’ 3) + 3))
167 eluzelcn 12830 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ 𝑁 ∈ β„‚)
16859a1i 11 . . . . . . . . . . . . . . . 16 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ 3 ∈ β„‚)
169167, 168npcand 11571 . . . . . . . . . . . . . . 15 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ ((𝑁 βˆ’ 3) + 3) = 𝑁)
170169adantr 481 . . . . . . . . . . . . . 14 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑁 βˆ’ 3) + 3) = 𝑁)
171166, 170eqtrd 2772 . . . . . . . . . . . . 13 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑁 βˆ’ 3) + ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4)) = 𝑁)
172171adantr 481 . . . . . . . . . . . 12 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)) β†’ ((𝑁 βˆ’ 3) + ((𝑔 βˆͺ {⟨4, 3⟩})β€˜4)) = 𝑁)
173150, 161, 1723eqtrrd 2777 . . . . . . . . . . 11 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)) β†’ 𝑁 = Ξ£π‘˜ ∈ (1...4)((𝑔 βˆͺ {⟨4, 3⟩})β€˜π‘˜))
17495, 99, 173rspcedvd 3614 . . . . . . . . . 10 (((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) ∧ (𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜)) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜))
175174ex 413 . . . . . . . . 9 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑔:(1...3)βŸΆβ„™) β†’ ((𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜)))
176175expcom 414 . . . . . . . 8 (𝑔:(1...3)βŸΆβ„™ β†’ (𝑁 ∈ (β„€β‰₯β€˜12) β†’ ((𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜))))
177 elmapi 8839 . . . . . . . 8 (𝑔 ∈ (β„™ ↑m (1...3)) β†’ 𝑔:(1...3)βŸΆβ„™)
178176, 177syl11 33 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ (𝑔 ∈ (β„™ ↑m (1...3)) β†’ ((𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜))))
179178rexlimdv 3153 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜12) β†’ (βˆƒπ‘” ∈ (β„™ ↑m (1...3))(𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜)))
180179adantr 481 . . . . 5 ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even ) β†’ (βˆƒπ‘” ∈ (β„™ ↑m (1...3))(𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜)))
181180ad3antlr 729 . . . 4 ((((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) ∧ π‘œ ∈ GoldbachOdd ) ∧ 𝑁 = (π‘œ + 3)) β†’ (βˆƒπ‘” ∈ (β„™ ↑m (1...3))(𝑁 βˆ’ 3) = Ξ£π‘˜ ∈ (1...3)(π‘”β€˜π‘˜) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜)))
18251, 181mpd 15 . . 3 ((((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) ∧ π‘œ ∈ GoldbachOdd ) ∧ 𝑁 = (π‘œ + 3)) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜))
183 evengpoap3 46453 . . . 4 (βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) β†’ ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even ) β†’ βˆƒπ‘œ ∈ GoldbachOdd 𝑁 = (π‘œ + 3)))
184183imp 407 . . 3 ((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) β†’ βˆƒπ‘œ ∈ GoldbachOdd 𝑁 = (π‘œ + 3))
185182, 184r19.29a 3162 . 2 ((βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) ∧ (𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even )) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜))
186185ex 413 1 (βˆ€π‘š ∈ Odd (7 < π‘š β†’ π‘š ∈ GoldbachOdd ) β†’ ((𝑁 ∈ (β„€β‰₯β€˜12) ∧ 𝑁 ∈ Even ) β†’ βˆƒπ‘“ ∈ (β„™ ↑m (1...4))𝑁 = Ξ£π‘˜ ∈ (1...4)(π‘“β€˜π‘˜)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆͺ cun 3945  {csn 4627  βŸ¨cop 4633   class class class wbr 5147  dom cdm 5675  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ↑m cmap 8816  β„‚cc 11104  β„cr 11105  1c1 11107   + caddc 11109   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440  2c2 12263  3c3 12264  4c4 12265  7c7 12268  8c8 12269  β„€cz 12554  cdc 12673  β„€β‰₯cuz 12818  ...cfz 13480  ..^cfzo 13623  Ξ£csu 15628  β„™cprime 16604   Even ceven 46278   Odd codd 46279   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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  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
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-tp 4632  df-op 4634  df-uni 4908  df-int 4950  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-se 5631  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-isom 6549  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-oi 9501  df-card 9930  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-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-dvds 16194  df-prm 16605  df-even 46280  df-odd 46281  df-gbo 46404
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator