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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 486 | . . . . . 6 ⊢ (((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → 𝑍 = ((𝑝 + 𝑞) + 𝑟)) | |
2 | 1 | reximi 3084 | . . . . 5 ⊢ (∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) → ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟)) |
3 | 2 | reximi 3084 | . . . 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 3084 | . . 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 46035 | . 2 ⊢ (𝑍 ∈ GoldbachOdd ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ ((𝑝 ∈ Odd ∧ 𝑞 ∈ Odd ∧ 𝑟 ∈ Odd ) ∧ 𝑍 = ((𝑝 + 𝑞) + 𝑟)))) | |
7 | isgbow 46034 | . 2 ⊢ (𝑍 ∈ GoldbachOddW ↔ (𝑍 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 𝑍 = ((𝑝 + 𝑞) + 𝑟))) | |
8 | 5, 6, 7 | 3imtr4i 292 | 1 ⊢ (𝑍 ∈ GoldbachOdd → 𝑍 ∈ GoldbachOddW ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3070 (class class class)co 7361 + caddc 11062 ℙcprime 16555 Odd codd 45907 GoldbachOddW cgbow 46028 GoldbachOdd cgbo 46029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rex 3071 df-rab 3407 df-v 3449 df-gbow 46031 df-gbo 46032 |
This theorem is referenced by: gboodd 46039 gbopos 46042 stgoldbwt 46058 |
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