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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 2766 | . . 3 ⊢ (deg1‘(ℂfld ↾s ℚ)) = (deg1‘(ℂfld ↾s ℚ)) | |
| 2 | eqid 2766 | . . 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 11177 | . . . . . . . . . . . 12 ⊢ i ∈ ℂ | |
| 7 | 6 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → i ∈ ℂ) |
| 8 | 2cnd 12337 | . . . . . . . . . . . 12 ⊢ (⊤ → 2 ∈ ℂ) | |
| 9 | picn 26658 | . . . . . . . . . . . . 13 ⊢ π ∈ ℂ | |
| 10 | 9 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → π ∈ ℂ) |
| 11 | 8, 10 | mulcld 11247 | . . . . . . . . . . 11 ⊢ (⊤ → (2 · π) ∈ ℂ) |
| 12 | 7, 11 | mulcld 11247 | . . . . . . . . . 10 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ) |
| 13 | 3cn 12340 | . . . . . . . . . . 11 ⊢ 3 ∈ ℂ | |
| 14 | 13 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℂ) |
| 15 | 3ne0 12368 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
| 16 | 15 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ≠ 0) |
| 17 | 12, 14, 16 | divcld 12009 | . . . . . . . . 9 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ) |
| 18 | 17 | efcld 16162 | . . . . . . . 8 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ) |
| 19 | 5, 18 | eqeltrid 2870 | . . . . . . 7 ⊢ (⊤ → 𝑂 ∈ ℂ) |
| 20 | 13, 15 | reccli 11963 | . . . . . . . 8 ⊢ (1 / 3) ∈ ℂ |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 22 | 19, 21 | cxpcld 26910 | . . . . . 6 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ) |
| 23 | 4, 22 | eqeltrid 2870 | . . . . 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 16176 | . . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0) |
| 27 | 25, 26 | eqnetrd 3028 | . . . . . . . 8 ⊢ (⊤ → 𝑂 ≠ 0) |
| 28 | 19, 27, 21 | cxpne0d 26915 | . . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0) |
| 29 | 24, 28 | eqnetrd 3028 | . . . . . 6 ⊢ (⊤ → 𝑍 ≠ 0) |
| 30 | 23, 29 | reccld 12002 | . . . . 5 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ) |
| 31 | 23, 30 | addcld 11246 | . . . 4 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ) |
| 32 | 3, 31 | eqeltrid 2870 | . . 3 ⊢ (⊤ → 𝐴 ∈ ℂ) |
| 33 | eqidd 2767 | . . 3 ⊢ (⊤ → ((ℂfld minPoly ℚ)‘𝐴) = ((ℂfld minPoly ℚ)‘𝐴)) | |
| 34 | eqid 2766 | . . . . . . . . 9 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 35 | eqid 2766 | . . . . . . . . 9 ⊢ (+g‘(Poly1‘(ℂfld ↾s ℚ))) = (+g‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 36 | eqid 2766 | . . . . . . . . 9 ⊢ (.r‘(Poly1‘(ℂfld ↾s ℚ))) = (.r‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 37 | eqid 2766 | . . . . . . . . 9 ⊢ (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) = (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) | |
| 38 | eqid 2766 | . . . . . . . . 9 ⊢ (Poly1‘(ℂfld ↾s ℚ)) = (Poly1‘(ℂfld ↾s ℚ)) | |
| 39 | eqid 2766 | . . . . . . . . 9 ⊢ (algSc‘(Poly1‘(ℂfld ↾s ℚ))) = (algSc‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 40 | eqid 2766 | . . . . . . . . 9 ⊢ (var1‘(ℂfld ↾s ℚ)) = (var1‘(ℂfld ↾s ℚ)) | |
| 41 | eqid 2766 | . . . . . . . . 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 34209 | . . . . . . . 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 6891 | . . . . . 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 2791 | . . . . 5 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) = 3 |
| 47 | 3nn0 12540 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
| 48 | 46, 47 | eqeltri 2862 | . . . 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 12645 | . . . . . . . . . 10 ⊢ 3 ∈ ℤ | |
| 52 | iddvds 16352 | . . . . . . . . . 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 5153 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 ∥ (2↑𝑛)) |
| 56 | 3prm 16777 | . . . . . . . . . 10 ⊢ 3 ∈ ℙ | |
| 57 | 2prm 16775 | . . . . . . . . . 10 ⊢ 2 ∈ ℙ | |
| 58 | prmdvdsexpr 16801 | . . . . . . . . . 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 12333 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
| 63 | 2lt3 12432 | . . . . . . . . . 10 ⊢ 2 < 3 | |
| 64 | 62, 63 | gtneii 11340 | . . . . . . . . 9 ⊢ 3 ≠ 2 |
| 65 | 64 | neii 2963 | . . . . . . . 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 2968 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → 3 ≠ (2↑𝑛)) |
| 69 | 50, 68 | eqnetrd 3028 | . . . 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 34195 | . 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 2146 ≠ wne 2961 class class class wbr 5114 ‘cfv 6543 (class class class)co 7423 ℂcc 11116 0cc0 11118 1c1 11119 ici 11120 + caddc 11121 · cmul 11123 -cneg 11460 / cdiv 11889 2c2 12313 3c3 12314 ℕ0cn0 12522 ℤcz 12609 ℚcq 12990 ↑cexp 14117 expce 16140 πcpi 16145 ∥ cdvds 16335 ℙcprime 16754 ↾s cress 17315 +gcplusg 17335 .rcmulr 17336 .gcmg 19164 mulGrpcmgp 20247 ℂfldccnfld 21559 algSccascl 22039 var1cv1 22373 Poly1cpl1 22374 deg1cdg1 26248 ↑𝑐ccxp 26757 minPoly cminply 34120 Constrcconstr 34150 |
| 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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-reg 9564 ax-inf2 9620 ax-ac2 10465 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196 ax-addf 11197 ax-mulf 11198 |
| 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 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4878 df-int 4918 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-se 5620 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-of 7687 df-ofr 7688 df-rpss 7733 df-om 7872 df-1st 7995 df-2nd 7996 df-supp 8166 df-tpos 8231 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8703 df-ec 8705 df-qs 8709 df-map 8835 df-pm 8836 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-r1 9746 df-rank 9747 df-scott 9868 df-dju 9906 df-card 9944 df-acn 9947 df-ac 10119 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12523 df-xnn0 12596 df-z 12610 df-dec 12730 df-uz 12881 df-q 12991 df-rp 13035 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13394 df-ioc 13395 df-ico 13396 df-icc 13397 df-fz 13554 df-fzo 13702 df-fl 13845 df-mod 13923 df-seq 14058 df-exp 14118 df-fac 14330 df-bc 14359 df-hash 14387 df-word 14571 df-lsw 14620 df-concat 14628 df-s1 14655 df-substr 14701 df-pfx 14733 df-shft 15130 df-sgn 15150 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-limsup 15548 df-clim 15565 df-rlim 15566 df-sum 15764 df-ef 16146 df-sin 16148 df-cos 16149 df-pi 16151 df-dvds 16336 df-gcd 16578 df-prm 16755 df-pc 16922 df-struct 17232 df-sets 17249 df-slot 17267 df-ndx 17279 df-base 17295 df-ress 17316 df-plusg 17348 df-mulr 17349 df-starv 17350 df-sca 17351 df-vsca 17352 df-ip 17353 df-tset 17354 df-ple 17355 df-ocomp 17356 df-ds 17357 df-unif 17358 df-hom 17359 df-cco 17360 df-rest 17500 df-topn 17501 df-0g 17519 df-gsum 17520 df-topgen 17521 df-pt 17522 df-prds 17525 df-pws 17527 df-xrs 17581 df-qtop 17586 df-imas 17587 df-qus 17588 df-xps 17589 df-mre 17663 df-mrc 17664 df-mri 17665 df-acs 17666 df-proset 18375 df-drs 18376 df-poset 18394 df-ipo 18609 df-chn 18687 df-mgm 18723 df-sgrp 18806 df-mnd 18822 df-mhm 18872 df-submnd 18873 df-grp 19034 df-minusg 19035 df-sbg 19036 df-mulg 19165 df-subg 19220 df-nsg 19221 df-eqg 19222 df-ghm 19315 df-gim 19360 df-cntz 19418 df-oppg 19447 df-lsm 19737 df-cmn 19883 df-abl 19884 df-mgp 20248 df-rng 20262 df-ur 20295 df-srg 20300 df-ring 20348 df-cring 20349 df-oppr 20452 df-dvdsr 20472 df-unit 20473 df-irred 20474 df-invr 20503 df-dvr 20516 df-rhm 20587 df-nzr 20647 df-subrng 20682 df-subrg 20706 df-rlreg 20830 df-domn 20831 df-idom 20832 df-drng 20866 df-field 20867 df-sdrg 20927 df-lmod 21020 df-lss 21090 df-lsp 21130 df-lmhm 21180 df-lmim 21181 df-lmic 21182 df-lbs 21233 df-lvec 21261 df-sra 21331 df-rgmod 21332 df-lidl 21369 df-rsp 21370 df-2idl 21426 df-lpidl 21527 df-lpir 21528 df-pid 21542 df-psmet 21551 df-xmet 21552 df-met 21553 df-bl 21554 df-mopn 21555 df-fbas 21556 df-fg 21557 df-cnfld 21560 df-dsmm 21919 df-frlm 21934 df-uvc 21970 df-lindf 21993 df-linds 21994 df-assa 22040 df-asp 22041 df-ascl 22042 df-psr 22096 df-mvr 22097 df-mpl 22098 df-opsr 22100 df-evls 22262 df-evl 22263 df-psr1 22377 df-vr1 22378 df-ply1 22379 df-coe1 22380 df-evls1 22512 df-evl1 22513 df-top 23088 df-topon 23105 df-topsp 23127 df-bases 23140 df-cld 23213 df-ntr 23214 df-cls 23215 df-nei 23292 df-lp 23330 df-perf 23331 df-cn 23421 df-cnp 23422 df-haus 23509 df-tx 23756 df-hmeo 23949 df-fil 24040 df-fm 24132 df-flim 24133 df-flf 24134 df-xms 24514 df-ms 24515 df-tms 24516 df-cncf 25074 df-limc 26062 df-dv 26063 df-mdeg 26249 df-deg1 26250 df-mon1 26325 df-uc1p 26326 df-q1p 26327 df-r1p 26328 df-ig1p 26329 df-log 26758 df-cxp 26759 df-fldgen 33663 df-mxidl 33774 df-dim 34021 df-fldext 34062 df-extdg 34063 df-irng 34105 df-minply 34121 df-constr 34151 |
| This theorem is used by: cos9thpinconstr 34212 |
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