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Mirrors > Home > MPE Home > Th. List > Mathboxes > 7gbow | Structured version Visualization version GIF version |
Description: 7 is a weak odd Goldbach number. (Contributed by AV, 20-Jul-2020.) |
Ref | Expression |
---|---|
7gbow | ⊢ 7 ∈ GoldbachOddW |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7odd 46890 | . 2 ⊢ 7 ∈ Odd | |
2 | 2prm 16628 | . . 3 ⊢ 2 ∈ ℙ | |
3 | 3prm 16630 | . . . 4 ⊢ 3 ∈ ℙ | |
4 | gbpart7 46945 | . . . 4 ⊢ 7 = ((2 + 2) + 3) | |
5 | oveq2 7410 | . . . . 5 ⊢ (𝑟 = 3 → ((2 + 2) + 𝑟) = ((2 + 2) + 3)) | |
6 | 5 | rspceeqv 3626 | . . . 4 ⊢ ((3 ∈ ℙ ∧ 7 = ((2 + 2) + 3)) → ∃𝑟 ∈ ℙ 7 = ((2 + 2) + 𝑟)) |
7 | 3, 4, 6 | mp2an 689 | . . 3 ⊢ ∃𝑟 ∈ ℙ 7 = ((2 + 2) + 𝑟) |
8 | oveq1 7409 | . . . . . . 7 ⊢ (𝑝 = 2 → (𝑝 + 𝑞) = (2 + 𝑞)) | |
9 | 8 | oveq1d 7417 | . . . . . 6 ⊢ (𝑝 = 2 → ((𝑝 + 𝑞) + 𝑟) = ((2 + 𝑞) + 𝑟)) |
10 | 9 | eqeq2d 2735 | . . . . 5 ⊢ (𝑝 = 2 → (7 = ((𝑝 + 𝑞) + 𝑟) ↔ 7 = ((2 + 𝑞) + 𝑟))) |
11 | 10 | rexbidv 3170 | . . . 4 ⊢ (𝑝 = 2 → (∃𝑟 ∈ ℙ 7 = ((𝑝 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ ℙ 7 = ((2 + 𝑞) + 𝑟))) |
12 | oveq2 7410 | . . . . . . 7 ⊢ (𝑞 = 2 → (2 + 𝑞) = (2 + 2)) | |
13 | 12 | oveq1d 7417 | . . . . . 6 ⊢ (𝑞 = 2 → ((2 + 𝑞) + 𝑟) = ((2 + 2) + 𝑟)) |
14 | 13 | eqeq2d 2735 | . . . . 5 ⊢ (𝑞 = 2 → (7 = ((2 + 𝑞) + 𝑟) ↔ 7 = ((2 + 2) + 𝑟))) |
15 | 14 | rexbidv 3170 | . . . 4 ⊢ (𝑞 = 2 → (∃𝑟 ∈ ℙ 7 = ((2 + 𝑞) + 𝑟) ↔ ∃𝑟 ∈ ℙ 7 = ((2 + 2) + 𝑟))) |
16 | 11, 15 | rspc2ev 3617 | . . 3 ⊢ ((2 ∈ ℙ ∧ 2 ∈ ℙ ∧ ∃𝑟 ∈ ℙ 7 = ((2 + 2) + 𝑟)) → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 7 = ((𝑝 + 𝑞) + 𝑟)) |
17 | 2, 2, 7, 16 | mp3an 1457 | . 2 ⊢ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 7 = ((𝑝 + 𝑞) + 𝑟) |
18 | isgbow 46930 | . 2 ⊢ (7 ∈ GoldbachOddW ↔ (7 ∈ Odd ∧ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ ∃𝑟 ∈ ℙ 7 = ((𝑝 + 𝑞) + 𝑟))) | |
19 | 1, 17, 18 | mpbir2an 708 | 1 ⊢ 7 ∈ GoldbachOddW |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 ∃wrex 3062 (class class class)co 7402 + caddc 11110 2c2 12265 3c3 12266 7c7 12270 ℙcprime 16607 Odd codd 46803 GoldbachOddW cgbow 46924 |
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 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-1st 7969 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-sub 11444 df-neg 11445 df-div 11870 df-nn 12211 df-2 12273 df-3 12274 df-4 12275 df-5 12276 df-6 12277 df-7 12278 df-n0 12471 df-z 12557 df-uz 12821 df-rp 12973 df-fz 13483 df-seq 13965 df-exp 14026 df-cj 15044 df-re 15045 df-im 15046 df-sqrt 15180 df-abs 15181 df-dvds 16197 df-prm 16608 df-even 46804 df-odd 46805 df-gbow 46927 |
This theorem is referenced by: stgoldbwt 46954 sbgoldbwt 46955 |
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