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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cos9thpinconstrlem2 | Structured version Visualization version GIF version | ||
| Description: The complex number 𝐴 is not constructible. (Contributed by Thierry Arnoux, 15-Nov-2025.) |
| Ref | Expression |
|---|---|
| cos9thpinconstr.1 | ⊢ 𝑂 = (exp‘((i · (2 · π)) / 3)) |
| cos9thpiminply.2 | ⊢ 𝑍 = (𝑂↑𝑐(1 / 3)) |
| cos9thpiminply.3 | ⊢ 𝐴 = (𝑍 + (1 / 𝑍)) |
| Ref | Expression |
|---|---|
| cos9thpinconstrlem2 | ⊢ ¬ 𝐴 ∈ Constr |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2762 | . . 3 ⊢ (deg1‘(ℂfld ↾s ℚ)) = (deg1‘(ℂfld ↾s ℚ)) | |
| 2 | eqid 2762 | . . 3 ⊢ (ℂfld minPoly ℚ) = (ℂfld minPoly ℚ) | |
| 3 | cos9thpiminply.3 | . . . 4 ⊢ 𝐴 = (𝑍 + (1 / 𝑍)) | |
| 4 | cos9thpiminply.2 | . . . . . 6 ⊢ 𝑍 = (𝑂↑𝑐(1 / 3)) | |
| 5 | cos9thpinconstr.1 | . . . . . . . 8 ⊢ 𝑂 = (exp‘((i · (2 · π)) / 3)) | |
| 6 | ax-icn 11187 | . . . . . . . . . . . 12 ⊢ i ∈ ℂ | |
| 7 | 6 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → i ∈ ℂ) |
| 8 | 2cnd 12347 | . . . . . . . . . . . 12 ⊢ (⊤ → 2 ∈ ℂ) | |
| 9 | picn 26701 | . . . . . . . . . . . . 13 ⊢ π ∈ ℂ | |
| 10 | 9 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → π ∈ ℂ) |
| 11 | 8, 10 | mulcld 11257 | . . . . . . . . . . 11 ⊢ (⊤ → (2 · π) ∈ ℂ) |
| 12 | 7, 11 | mulcld 11257 | . . . . . . . . . 10 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ) |
| 13 | 3cn 12350 | . . . . . . . . . . 11 ⊢ 3 ∈ ℂ | |
| 14 | 13 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℂ) |
| 15 | 3ne0 12378 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
| 16 | 15 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ≠ 0) |
| 17 | 12, 14, 16 | divcld 12019 | . . . . . . . . 9 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ) |
| 18 | 17 | efcld 16175 | . . . . . . . 8 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ) |
| 19 | 5, 18 | eqeltrid 2866 | . . . . . . 7 ⊢ (⊤ → 𝑂 ∈ ℂ) |
| 20 | 13, 15 | reccli 11973 | . . . . . . . 8 ⊢ (1 / 3) ∈ ℂ |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 22 | 19, 21 | cxpcld 26953 | . . . . . 6 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ) |
| 23 | 4, 22 | eqeltrid 2866 | . . . . 5 ⊢ (⊤ → 𝑍 ∈ ℂ) |
| 24 | 4 | a1i 11 | . . . . . . 7 ⊢ (⊤ → 𝑍 = (𝑂↑𝑐(1 / 3))) |
| 25 | 5 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → 𝑂 = (exp‘((i · (2 · π)) / 3))) |
| 26 | 17 | efne0d 16189 | . . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0) |
| 27 | 25, 26 | eqnetrd 3024 | . . . . . . . 8 ⊢ (⊤ → 𝑂 ≠ 0) |
| 28 | 19, 27, 21 | cxpne0d 26958 | . . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0) |
| 29 | 24, 28 | eqnetrd 3024 | . . . . . 6 ⊢ (⊤ → 𝑍 ≠ 0) |
| 30 | 23, 29 | reccld 12012 | . . . . 5 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ) |
| 31 | 23, 30 | addcld 11256 | . . . 4 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ) |
| 32 | 3, 31 | eqeltrid 2866 | . . 3 ⊢ (⊤ → 𝐴 ∈ ℂ) |
| 33 | eqidd 2763 | . . 3 ⊢ (⊤ → ((ℂfld minPoly ℚ)‘𝐴) = ((ℂfld minPoly ℚ)‘𝐴)) | |
| 34 | eqid 2762 | . . . . . . . . 9 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 35 | eqid 2762 | . . . . . . . . 9 ⊢ (+g‘(Poly1‘(ℂfld ↾s ℚ))) = (+g‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 36 | eqid 2762 | . . . . . . . . 9 ⊢ (.r‘(Poly1‘(ℂfld ↾s ℚ))) = (.r‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 37 | eqid 2762 | . . . . . . . . 9 ⊢ (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) = (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) | |
| 38 | eqid 2762 | . . . . . . . . 9 ⊢ (Poly1‘(ℂfld ↾s ℚ)) = (Poly1‘(ℂfld ↾s ℚ)) | |
| 39 | eqid 2762 | . . . . . . . . 9 ⊢ (algSc‘(Poly1‘(ℂfld ↾s ℚ))) = (algSc‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 40 | eqid 2762 | . . . . . . . . 9 ⊢ (var1‘(ℂfld ↾s ℚ)) = (var1‘(ℂfld ↾s ℚ)) | |
| 41 | eqid 2762 | . . . . . . . . 9 ⊢ ((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘-3)(.r‘(Poly1‘(ℂfld ↾s ℚ)))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘1))) = ((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘-3)(.r‘(Poly1‘(ℂfld ↾s ℚ)))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘1))) | |
| 42 | 5, 4, 3, 34, 35, 36, 37, 38, 39, 40, 1, 41, 2 | cos9thpiminply 34306 | . . . . . . . 8 ⊢ (((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘-3)(.r‘(Poly1‘(ℂfld ↾s ℚ)))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘1))) = ((ℂfld minPoly ℚ)‘𝐴) ∧ ((deg1‘(ℂfld ↾s ℚ))‘((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘-3)(.r‘(Poly1‘(ℂfld ↾s ℚ)))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘1)))) = 3) |
| 43 | 42 | simpli 489 | . . . . . . 7 ⊢ ((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘-3)(.r‘(Poly1‘(ℂfld ↾s ℚ)))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘1))) = ((ℂfld minPoly ℚ)‘𝐴) |
| 44 | 43 | fveq2i 6885 | . . . . . 6 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘-3)(.r‘(Poly1‘(ℂfld ↾s ℚ)))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘1)))) = ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) |
| 45 | 42 | simpri 491 | . . . . . 6 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘-3)(.r‘(Poly1‘(ℂfld ↾s ℚ)))(var1‘(ℂfld ↾s ℚ)))(+g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘1)))) = 3 |
| 46 | 44, 45 | eqtr3i 2787 | . . . . 5 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) = 3 |
| 47 | 3nn0 12550 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
| 48 | 46, 47 | eqeltri 2858 | . . . 4 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) ∈ ℕ0 |
| 49 | 48 | a1i 11 | . . 3 ⊢ (⊤ → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) ∈ ℕ0) |
| 50 | 46 | a1i 11 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) = 3) |
| 51 | 3z 12655 | . . . . . . . . . 10 ⊢ 3 ∈ ℤ | |
| 52 | iddvds 16365 | . . . . . . . . . 10 ⊢ (3 ∈ ℤ → 3 ∥ 3) | |
| 53 | 51, 52 | ax-mp 5 | . . . . . . . . 9 ⊢ 3 ∥ 3 |
| 54 | simpr 490 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = (2↑𝑛)) | |
| 55 | 53, 54 | breqtrid 5146 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 ∥ (2↑𝑛)) |
| 56 | 3prm 16790 | . . . . . . . . . 10 ⊢ 3 ∈ ℙ | |
| 57 | 2prm 16788 | . . . . . . . . . 10 ⊢ 2 ∈ ℙ | |
| 58 | prmdvdsexpr 16814 | . . . . . . . . . 10 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (3 ∥ (2↑𝑛) → 3 = 2)) | |
| 59 | 56, 57, 58 | mp3an12 1480 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (3 ∥ (2↑𝑛) → 3 = 2)) |
| 60 | 59 | imp 412 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 ∥ (2↑𝑛)) → 3 = 2) |
| 61 | 55, 60 | syldan 603 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = 2) |
| 62 | 2re 12343 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
| 63 | 2lt3 12442 | . . . . . . . . . 10 ⊢ 2 < 3 | |
| 64 | 62, 63 | gtneii 11350 | . . . . . . . . 9 ⊢ 3 ≠ 2 |
| 65 | 64 | neii 2959 | . . . . . . . 8 ⊢ ¬ 3 = 2 |
| 66 | 65 | a1i 11 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → ¬ 3 = 2) |
| 67 | 61, 66 | pm2.65da 829 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ¬ 3 = (2↑𝑛)) |
| 68 | 67 | neqned 2964 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → 3 ≠ (2↑𝑛)) |
| 69 | 50, 68 | eqnetrd 3024 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) ≠ (2↑𝑛)) |
| 70 | 69 | adantl 487 | . . 3 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ0) → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) ≠ (2↑𝑛)) |
| 71 | 1, 2, 32, 33, 49, 70 | constrcon 34292 | . 2 ⊢ (⊤ → ¬ 𝐴 ∈ Constr) |
| 72 | 71 | mptru 1577 | 1 ⊢ ¬ 𝐴 ∈ Constr |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 = wceq 1570 ⊤wtru 1571 ∈ wcel 2145 ≠ wne 2957 class class class wbr 5107 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 0cc0 11128 1c1 11129 ici 11130 + caddc 11131 · cmul 11133 -cneg 11470 / cdiv 11899 2c2 12323 3c3 12324 ℕ0cn0 12532 ℤcz 12619 ℚcq 13001 ↑cexp 14129 expce 16153 πcpi 16158 ∥ cdvds 16348 ℙcprime 16767 ↾s cress 17328 +gcplusg 17348 .rcmulr 17349 .gcmg 19196 mulGrpcmgp 20279 ℂfldccnfld 21591 algSccascl 22073 var1cv1 22407 Poly1cpl1 22408 deg1cdg1 26286 ↑𝑐ccxp 26800 minPoly cminply 34217 Constrcconstr 34247 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-reg 9568 ax-inf2 9624 ax-ac2 10469 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206 ax-addf 11207 ax-mulf 11208 |
| This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-of 7682 df-ofr 7683 df-rpss 7728 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-ec 8702 df-qs 8706 df-map 8832 df-pm 8833 df-ixp 8909 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fsupp 9336 df-fi 9385 df-sup 9416 df-inf 9417 df-oi 9486 df-r1 9750 df-rank 9751 df-scott 9872 df-dju 9910 df-card 9948 df-acn 9951 df-ac 10123 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-xnn0 12606 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-ioo 13406 df-ioc 13407 df-ico 13408 df-icc 13409 df-fz 13566 df-fzo 13714 df-fl 13857 df-mod 13935 df-seq 14070 df-exp 14130 df-fac 14342 df-bc 14371 df-hash 14399 df-word 14583 df-lsw 14632 df-concat 14640 df-s1 14667 df-substr 14713 df-pfx 14745 df-shft 15144 df-sgn 15164 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-limsup 15562 df-clim 15579 df-rlim 15580 df-sum 15778 df-ef 16159 df-sin 16161 df-cos 16162 df-pi 16164 df-dvds 16349 df-gcd 16591 df-prm 16768 df-pc 16935 df-struct 17245 df-sets 17262 df-slot 17280 df-ndx 17292 df-base 17308 df-ress 17329 df-plusg 17361 df-mulr 17362 df-starv 17363 df-sca 17364 df-vsca 17365 df-ip 17366 df-tset 17367 df-ple 17368 df-ocomp 17369 df-ds 17370 df-unif 17371 df-hom 17372 df-cco 17373 df-rest 17513 df-topn 17514 df-0g 17532 df-gsum 17533 df-topgen 17534 df-pt 17535 df-prds 17538 df-pws 17540 df-xrs 17594 df-qtop 17599 df-imas 17600 df-qus 17601 df-xps 17602 df-mre 17676 df-mrc 17677 df-mri 17678 df-acs 17679 df-proset 18388 df-drs 18389 df-poset 18407 df-ipo 18622 df-chn 18700 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-mhm 18897 df-submnd 18898 df-grp 19066 df-minusg 19067 df-sbg 19068 df-mulg 19197 df-subg 19252 df-nsg 19253 df-eqg 19254 df-ghm 19347 df-gim 19392 df-cntz 19450 df-oppg 19479 df-lsm 19769 df-cmn 19915 df-abl 19916 df-mgp 20280 df-rng 20294 df-ur 20327 df-srg 20332 df-ring 20380 df-cring 20381 df-oppr 20484 df-dvdsr 20504 df-unit 20505 df-irred 20506 df-invr 20535 df-dvr 20548 df-rhm 20619 df-nzr 20679 df-subrng 20714 df-subrg 20738 df-rlreg 20862 df-domn 20863 df-idom 20864 df-drng 20898 df-field 20899 df-sdrg 20959 df-lmod 21052 df-lss 21122 df-lsp 21162 df-lmhm 21212 df-lmim 21213 df-lmic 21214 df-lbs 21265 df-lvec 21293 df-sra 21363 df-rgmod 21364 df-lidl 21401 df-rsp 21402 df-2idl 21458 df-lpidl 21559 df-lpir 21560 df-pid 21574 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-fbas 21588 df-fg 21589 df-cnfld 21592 df-dsmm 21951 df-frlm 21966 df-uvc 22002 df-lindf 22025 df-linds 22026 df-assa 22074 df-asp 22075 df-ascl 22076 df-psr 22130 df-mvr 22131 df-mpl 22132 df-opsr 22134 df-evls 22296 df-evl 22297 df-psr1 22411 df-vr1 22412 df-ply1 22413 df-coe1 22414 df-evls1 22546 df-evl1 22547 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-cld 23250 df-ntr 23251 df-cls 23252 df-nei 23329 df-lp 23367 df-perf 23368 df-cn 23458 df-cnp 23459 df-haus 23546 df-tx 23794 df-hmeo 23987 df-fil 24078 df-fm 24170 df-flim 24171 df-flf 24172 df-xms 24552 df-ms 24553 df-tms 24554 df-cncf 25112 df-limc 26100 df-dv 26101 df-mdeg 26287 df-deg1 26288 df-mon1 26363 df-uc1p 26364 df-q1p 26365 df-r1p 26366 df-ig1p 26367 df-log 26801 df-cxp 26802 df-fldgen 33760 df-mxidl 33871 df-dim 34118 df-fldext 34159 df-extdg 34160 df-irng 34202 df-minply 34218 df-constr 34248 |
| This theorem is used by: cos9thpinconstr 34309 |
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