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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 45165 | . 2 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) | |
2 | prmnn 16369 | . . . . . . . . . . 11 ⊢ (𝑝 ∈ ℙ → 𝑝 ∈ ℕ) | |
3 | prmnn 16369 | . . . . . . . . . . 11 ⊢ (𝑞 ∈ ℙ → 𝑞 ∈ ℕ) | |
4 | 2, 3 | anim12i 613 | . . . . . . . . . 10 ⊢ ((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) → (𝑝 ∈ ℕ ∧ 𝑞 ∈ ℕ)) |
5 | 4 | adantr 481 | . . . . . . . . 9 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ 𝑟 ∈ ℙ) → (𝑝 ∈ ℕ ∧ 𝑞 ∈ ℕ)) |
6 | nnaddcl 11988 | . . . . . . . . 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 16369 | . . . . . . . . 9 ⊢ (𝑟 ∈ ℙ → 𝑟 ∈ ℕ) | |
9 | 8 | adantl 482 | . . . . . . . 8 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ 𝑟 ∈ ℙ) → 𝑟 ∈ ℕ) |
10 | 7, 9 | nnaddcld 12017 | . . . . . . 7 ⊢ (((𝑝 ∈ ℙ ∧ 𝑞 ∈ ℙ) ∧ 𝑟 ∈ ℙ) → ((𝑝 + 𝑞) + 𝑟) ∈ ℕ) |
11 | eleq1 2828 | . . . . . . 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 3215 | . . . . 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 3224 | . . 3 ⊢ (𝑍 ∈ Odd → (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟) → 𝑍 ∈ ℕ)) |
16 | 15 | imp 407 | . 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 396 = wceq 1542 ∈ wcel 2110 ∃wrex 3067 (class class class)co 7269 + caddc 10867 ℕcn 11965 ℙcprime 16366 Odd codd 45038 GoldbachOddW cgbow 45159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 ax-un 7580 ax-1cn 10922 ax-addcl 10924 ax-addass 10929 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-ov 7272 df-om 7702 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-nn 11966 df-prm 16367 df-gbow 45162 |
This theorem is referenced by: gbopos 45173 gbowge7 45176 |
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