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Mirrors > Home > MPE Home > Th. List > Mathboxes > gbogbow | Structured version Visualization version GIF version |
Description: A (strong) odd Goldbach number is a weak Goldbach number. (Contributed by AV, 26-Jul-2020.) |
Ref | Expression |
---|---|
gbogbow | ⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ GoldbachOddW ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 487 | . . . . . 6 ⊢ (((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → 𝑍 = ((𝑝 + 𝑞) + 𝑟)) | |
2 | 1 | reximi 3243 | . . . . 5 ⊢ (∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) |
3 | 2 | reximi 3243 | . . . 4 ⊢ (∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) |
4 | 3 | reximi 3243 | . . 3 ⊢ (∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) |
5 | 4 | anim2i 618 | . 2 ⊢ ((𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) → (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) |
6 | isgbo 43912 | . 2 ⊢ (𝑍 ∈ GoldbachOdd ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))) | |
7 | isgbow 43911 | . 2 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) | |
8 | 5, 6, 7 | 3imtr4i 294 | 1 ⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ GoldbachOddW ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 (class class class)co 7150 + caddc 10534 ℙcprime 16009 Odd codd 43784 GoldbachOddW cgbow 43905 GoldbachOdd cgbo 43906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-gbow 43908 df-gbo 43909 |
This theorem is referenced by: gboodd 43916 gbopos 43919 stgoldbwt 43935 |
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