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Mirrors > Home > MPE Home > Th. List > Mathboxes > gbowpos | Structured version Visualization version GIF version |
Description: Any weak odd Goldbach number is positive. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
gbowpos | ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isgbow 46973 | . 2 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) | |
2 | prmnn 16616 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ) | |
3 | prmnn 16616 | . . . . . . . . . . 11 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ) | |
4 | 2, 3 | anim12i 612 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → (𝑝 ∈ ℕ ∧ 𝑞 ∈ ℕ)) |
5 | 4 | adantr 480 | . . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ 𝑟 ∈ ℙ) → (𝑝 ∈ ℕ ∧ 𝑞 ∈ ℕ)) |
6 | nnaddcl 12236 | . . . . . . . . 9 ⊢ ((𝑝 ∈ ℕ ∧ 𝑞 ∈ ℕ) → (𝑝 + 𝑞) ∈ ℕ) | |
7 | 5, 6 | syl 17 | . . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ 𝑟 ∈ ℙ) → (𝑝 + 𝑞) ∈ ℕ) |
8 | prmnn 16616 | . . . . . . . . 9 ⊢ (𝑟 ∈ ℙ → 𝑟 ∈ ℕ) | |
9 | 8 | adantl 481 | . . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ 𝑟 ∈ ℙ) → 𝑟 ∈ ℕ) |
10 | 7, 9 | nnaddcld 12265 | . . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ 𝑟 ∈ ℙ) → ((𝑝 + 𝑞) + 𝑟) ∈ ℕ) |
11 | eleq1 2815 | . . . . . . 7 ⊢ (𝑍 = ((𝑝 + 𝑞) + 𝑟) → (𝑍 ∈ ℕ ↔ ((𝑝 + 𝑞) + 𝑟) ∈ ℕ)) | |
12 | 10, 11 | syl5ibrcom 246 | . . . . . 6 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ 𝑟 ∈ ℙ) → (𝑍 = ((𝑝 + 𝑞) + 𝑟) → 𝑍 ∈ ℕ)) |
13 | 12 | rexlimdva 3149 | . . . . 5 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → (∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟) → 𝑍 ∈ ℕ)) |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝑍 ∈ Odd → ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → (∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟) → 𝑍 ∈ ℕ))) |
15 | 14 | rexlimdvv 3204 | . . 3 ⊢ (𝑍 ∈ Odd → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟) → 𝑍 ∈ ℕ)) |
16 | 15 | imp 406 | . 2 ⊢ ((𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → 𝑍 ∈ ℕ) |
17 | 1, 16 | sylbi 216 | 1 ⊢ (𝑍 ∈ GoldbachOddW → 𝑍 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∃wrex 3064 (class class class)co 7404 + caddc 11112 ℕcn 12213 ℙcprime 16613 Odd codd 46846 GoldbachOddW cgbow 46967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 ax-1cn 11167 ax-addcl 11169 ax-addass 11174 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-nn 12214 df-prm 16614 df-gbow 46970 |
This theorem is referenced by: gbopos 46981 gbowge7 46984 |
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