Nearby theorems
|
|
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
| 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 2761 | . . 3 ⊢ (deg1‘(ℂfld ↾s ℚ)) = (deg1‘(ℂfld ↾s ℚ)) | |
| 2 | eqid 2761 | . . 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 11240 | . . . . . . . . . . . 12 ⊢ i ∈ ℂ | |
| 7 | 6 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → i ∈ ℂ) |
| 8 | 2cnd 12402 | . . . . . . . . . . . 12 ⊢ (⊤ → 2 ∈ ℂ) | |
| 9 | picn 26767 | . . . . . . . . . . . . 13 ⊢ π ∈ ℂ | |
| 10 | 9 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → π ∈ ℂ) |
| 11 | 8, 10 | mulcld 11310 | . . . . . . . . . . 11 ⊢ (⊤ → (2 · π) ∈ ℂ) |
| 12 | 7, 11 | mulcld 11310 | . . . . . . . . . 10 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ) |
| 13 | 3cn 12405 | . . . . . . . . . . 11 ⊢ 3 ∈ ℂ | |
| 14 | 13 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℂ) |
| 15 | 3ne0 12433 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
| 16 | 15 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ≠ 0) |
| 17 | 12, 14, 16 | divcld 12074 | . . . . . . . . 9 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ) |
| 18 | 17 | efcld 16229 | . . . . . . . 8 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ) |
| 19 | 5, 18 | eqeltrid 2865 | . . . . . . 7 ⊢ (⊤ → 𝑂 ∈ ℂ) |
| 20 | 13, 15 | reccli 12028 | . . . . . . . 8 ⊢ (1 / 3) ∈ ℂ |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 22 | 19, 21 | cxpcld 27018 | . . . . . 6 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ) |
| 23 | 4, 22 | eqeltrid 2865 | . . . . 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 16243 | . . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0) |
| 27 | 25, 26 | eqnetrd 3023 | . . . . . . . 8 ⊢ (⊤ → 𝑂 ≠ 0) |
| 28 | 19, 27, 21 | cxpne0d 27023 | . . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0) |
| 29 | 24, 28 | eqnetrd 3023 | . . . . . 6 ⊢ (⊤ → 𝑍 ≠ 0) |
| 30 | 23, 29 | reccld 12067 | . . . . 5 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ) |
| 31 | 23, 30 | addcld 11309 | . . . 4 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ) |
| 32 | 3, 31 | eqeltrid 2865 | . . 3 ⊢ (⊤ → 𝐴 ∈ ℂ) |
| 33 | eqidd 2762 | . . 3 ⊢ (⊤ → ((ℂfld minPoly ℚ)‘𝐴) = ((ℂfld minPoly ℚ)‘𝐴)) | |
| 34 | eqid 2761 | . . . . . . . . 9 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 35 | eqid 2761 | . . . . . . . . 9 ⊢ (+g‘(Poly1‘(ℂfld ↾s ℚ))) = (+g‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 36 | eqid 2761 | . . . . . . . . 9 ⊢ (.r‘(Poly1‘(ℂfld ↾s ℚ))) = (.r‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 37 | eqid 2761 | . . . . . . . . 9 ⊢ (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) = (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) | |
| 38 | eqid 2761 | . . . . . . . . 9 ⊢ (Poly1‘(ℂfld ↾s ℚ)) = (Poly1‘(ℂfld ↾s ℚ)) | |
| 39 | eqid 2761 | . . . . . . . . 9 ⊢ (algSc‘(Poly1‘(ℂfld ↾s ℚ))) = (algSc‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 40 | eqid 2761 | . . . . . . . . 9 ⊢ (var1‘(ℂfld ↾s ℚ)) = (var1‘(ℂfld ↾s ℚ)) | |
| 41 | eqid 2761 | . . . . . . . . 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 34402 | . . . . . . . 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 6880 | . . . . . 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 2786 | . . . . 5 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) = 3 |
| 47 | 3nn0 12605 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
| 48 | 46, 47 | eqeltri 2857 | . . . 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 12710 | . . . . . . . . . 10 ⊢ 3 ∈ ℤ | |
| 52 | iddvds 16419 | . . . . . . . . . 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 5142 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 ∥ (2↑𝑛)) |
| 56 | 3prm 16849 | . . . . . . . . . 10 ⊢ 3 ∈ ℙ | |
| 57 | 2prm 16847 | . . . . . . . . . 10 ⊢ 2 ∈ ℙ | |
| 58 | prmdvdsexpr 16873 | . . . . . . . . . 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 12398 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
| 63 | 2lt3 12497 | . . . . . . . . . 10 ⊢ 2 < 3 | |
| 64 | 62, 63 | gtneii 11403 | . . . . . . . . 9 ⊢ 3 ≠ 2 |
| 65 | 64 | neii 2958 | . . . . . . . 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 2963 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → 3 ≠ (2↑𝑛)) |
| 69 | 50, 68 | eqnetrd 3023 | . . . 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 34388 | . 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 2956 class class class wbr 5103 ‘cfv 6531 (class class class)co 7412 ℂcc 11179 0cc0 11181 1c1 11182 ici 11183 + caddc 11184 · cmul 11186 -cneg 11523 / cdiv 11954 2c2 12378 3c3 12379 ℕ0cn0 12587 ℤcz 12674 ℚcq 13056 ↑cexp 14184 expce 16207 πcpi 16212 ∥ cdvds 16402 ℙcprime 16826 ↾s cress 17388 +gcplusg 17408 .rcmulr 17409 .gcmg 19257 mulGrpcmgp 20340 ℂfldccnfld 21658 algSccascl 22140 var1cv1 22474 Poly1cpl1 22475 deg1cdg1 26352 ↑𝑐ccxp 26865 minPoly cminply 34313 Constrcconstr 34343 |
| 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 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-reg 9570 ax-inf2 9626 ax-ac2 10522 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-pre-sup 11259 ax-addf 11260 ax-mulf 11261 |
| 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 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-ofr 7683 df-rpss 7728 df-om 7867 df-1st 7990 df-2nd 7991 df-supp 8162 df-tpos 8227 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-oadd 8464 df-er 8701 df-ec 8703 df-qs 8707 df-map 8833 df-pm 8834 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9338 df-fi 9387 df-sup 9418 df-inf 9419 df-oi 9488 df-r1 9752 df-rank 9753 df-scott 9910 df-dju 9963 df-card 10001 df-acn 10004 df-ac 10176 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-7 12391 df-8 12392 df-9 12393 df-n0 12588 df-xnn0 12661 df-z 12675 df-dec 12796 df-uz 12947 df-q 13057 df-rp 13102 df-xneg 13222 df-xadd 13223 df-xmul 13224 df-ioo 13461 df-ioc 13462 df-ico 13463 df-icc 13464 df-fz 13621 df-fzo 13769 df-fl 13912 df-mod 13990 df-seq 14125 df-exp 14185 df-fac 14398 df-bc 14427 df-hash 14455 df-word 14639 df-lsw 14688 df-concat 14696 df-s1 14723 df-substr 14769 df-pfx 14801 df-shft 15200 df-sgn 15220 df-cj 15246 df-re 15247 df-im 15248 df-sqrt 15382 df-abs 15383 df-limsup 15618 df-clim 15635 df-rlim 15636 df-sum 15834 df-ef 16213 df-sin 16215 df-cos 16216 df-pi 16218 df-dvds 16403 df-gcd 16645 df-prm 16827 df-pc 16995 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-starv 17423 df-sca 17424 df-vsca 17425 df-ip 17426 df-tset 17427 df-ple 17428 df-ocomp 17429 df-ds 17430 df-unif 17431 df-hom 17432 df-cco 17433 df-rest 17573 df-topn 17574 df-0g 17592 df-gsum 17593 df-topgen 17594 df-pt 17595 df-prds 17598 df-pws 17600 df-xrs 17654 df-qtop 17659 df-imas 17660 df-qus 17661 df-xps 17662 df-mre 17736 df-mrc 17737 df-mri 17738 df-acs 17739 df-proset 18448 df-drs 18449 df-poset 18467 df-ipo 18682 df-chn 18760 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-mhm 18958 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-mulg 19258 df-subg 19313 df-nsg 19314 df-eqg 19315 df-ghm 19408 df-gim 19453 df-cntz 19511 df-oppg 19540 df-lsm 19830 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-srg 20393 df-ring 20441 df-cring 20442 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-irred 20569 df-invr 20598 df-dvr 20611 df-rhm 20682 df-nzr 20743 df-subrng 20778 df-subrg 20802 df-rlreg 20926 df-domn 20927 df-idom 20928 df-drng 20962 df-field 20963 df-sdrg 21024 df-lmod 21117 df-lss 21187 df-lsp 21227 df-lmhm 21277 df-lmim 21278 df-lmic 21279 df-lbs 21330 df-lvec 21358 df-sra 21428 df-rgmod 21429 df-lidl 21466 df-rsp 21467 df-2idl 21523 df-lpidl 21626 df-lpir 21627 df-pid 21641 df-psmet 21650 df-xmet 21651 df-met 21652 df-bl 21653 df-mopn 21654 df-fbas 21655 df-fg 21656 df-cnfld 21659 df-dsmm 22018 df-frlm 22033 df-uvc 22069 df-lindf 22092 df-linds 22093 df-assa 22141 df-asp 22142 df-ascl 22143 df-psr 22197 df-mvr 22198 df-mpl 22199 df-opsr 22201 df-evls 22363 df-evl 22364 df-psr1 22478 df-vr1 22479 df-ply1 22480 df-coe1 22481 df-evls1 22613 df-evl1 22614 df-top 23192 df-topon 23209 df-topsp 23231 df-bases 23244 df-cld 23317 df-ntr 23318 df-cls 23319 df-nei 23396 df-lp 23434 df-perf 23435 df-cn 23525 df-cnp 23526 df-haus 23613 df-tx 23861 df-hmeo 24054 df-fil 24145 df-fm 24237 df-flim 24238 df-flf 24239 df-xms 24619 df-ms 24620 df-tms 24621 df-cncf 25179 df-limc 26166 df-dv 26167 df-mdeg 26353 df-deg1 26354 df-mon1 26429 df-uc1p 26430 df-q1p 26431 df-r1p 26432 df-ig1p 26433 df-log 26866 df-cxp 26867 df-fldgen 33855 df-mxidl 33967 df-dim 34214 df-fldext 34255 df-extdg 34256 df-irng 34298 df-minply 34314 df-constr 34344 |
| This theorem is used by: cos9thpinconstr 34405 |
| Copyright terms: Public domain | W3C validator |