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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 2756 | . . 3 ⊢ (deg1‘(ℂfld ↾s ℚ)) = (deg1‘(ℂfld ↾s ℚ)) | |
| 2 | eqid 2756 | . . 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 11122 | . . . . . . . . . . . 12 ⊢ i ∈ ℂ | |
| 7 | 6 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → i ∈ ℂ) |
| 8 | 2cnd 12286 | . . . . . . . . . . . 12 ⊢ (⊤ → 2 ∈ ℂ) | |
| 9 | picn 26491 | . . . . . . . . . . . . 13 ⊢ π ∈ ℂ | |
| 10 | 9 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → π ∈ ℂ) |
| 11 | 8, 10 | mulcld 11192 | . . . . . . . . . . 11 ⊢ (⊤ → (2 · π) ∈ ℂ) |
| 12 | 7, 11 | mulcld 11192 | . . . . . . . . . 10 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ) |
| 13 | 3cn 12289 | . . . . . . . . . . 11 ⊢ 3 ∈ ℂ | |
| 14 | 13 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℂ) |
| 15 | 3ne0 12317 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
| 16 | 15 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ≠ 0) |
| 17 | 12, 14, 16 | divcld 11957 | . . . . . . . . 9 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ) |
| 18 | 17 | efcld 16089 | . . . . . . . 8 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ) |
| 19 | 5, 18 | eqeltrid 2860 | . . . . . . 7 ⊢ (⊤ → 𝑂 ∈ ℂ) |
| 20 | 13, 15 | reccli 11911 | . . . . . . . 8 ⊢ (1 / 3) ∈ ℂ |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 22 | 19, 21 | cxpcld 26743 | . . . . . 6 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ) |
| 23 | 4, 22 | eqeltrid 2860 | . . . . 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 16103 | . . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0) |
| 27 | 25, 26 | eqnetrd 3018 | . . . . . . . 8 ⊢ (⊤ → 𝑂 ≠ 0) |
| 28 | 19, 27, 21 | cxpne0d 26748 | . . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0) |
| 29 | 24, 28 | eqnetrd 3018 | . . . . . 6 ⊢ (⊤ → 𝑍 ≠ 0) |
| 30 | 23, 29 | reccld 11950 | . . . . 5 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ) |
| 31 | 23, 30 | addcld 11191 | . . . 4 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ) |
| 32 | 3, 31 | eqeltrid 2860 | . . 3 ⊢ (⊤ → 𝐴 ∈ ℂ) |
| 33 | eqidd 2757 | . . 3 ⊢ (⊤ → ((ℂfld minPoly ℚ)‘𝐴) = ((ℂfld minPoly ℚ)‘𝐴)) | |
| 34 | eqid 2756 | . . . . . . . . 9 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 35 | eqid 2756 | . . . . . . . . 9 ⊢ (+g‘(Poly1‘(ℂfld ↾s ℚ))) = (+g‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 36 | eqid 2756 | . . . . . . . . 9 ⊢ (.r‘(Poly1‘(ℂfld ↾s ℚ))) = (.r‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 37 | eqid 2756 | . . . . . . . . 9 ⊢ (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) = (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) | |
| 38 | eqid 2756 | . . . . . . . . 9 ⊢ (Poly1‘(ℂfld ↾s ℚ)) = (Poly1‘(ℂfld ↾s ℚ)) | |
| 39 | eqid 2756 | . . . . . . . . 9 ⊢ (algSc‘(Poly1‘(ℂfld ↾s ℚ))) = (algSc‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 40 | eqid 2756 | . . . . . . . . 9 ⊢ (var1‘(ℂfld ↾s ℚ)) = (var1‘(ℂfld ↾s ℚ)) | |
| 41 | eqid 2756 | . . . . . . . . 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 34039 | . . . . . . . 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 486 | . . . . . . 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 6859 | . . . . . 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 488 | . . . . . 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 2781 | . . . . 5 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) = 3 |
| 47 | 3nn0 12489 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
| 48 | 46, 47 | eqeltri 2852 | . . . 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 12594 | . . . . . . . . . 10 ⊢ 3 ∈ ℤ | |
| 52 | iddvds 16279 | . . . . . . . . . 10 ⊢ (3 ∈ ℤ → 3 ∥ 3) | |
| 53 | 51, 52 | ax-mp 5 | . . . . . . . . 9 ⊢ 3 ∥ 3 |
| 54 | simpr 487 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = (2↑𝑛)) | |
| 55 | 53, 54 | breqtrid 5131 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 ∥ (2↑𝑛)) |
| 56 | 3prm 16704 | . . . . . . . . . 10 ⊢ 3 ∈ ℙ | |
| 57 | 2prm 16702 | . . . . . . . . . 10 ⊢ 2 ∈ ℙ | |
| 58 | prmdvdsexpr 16728 | . . . . . . . . . 10 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (3 ∥ (2↑𝑛) → 3 = 2)) | |
| 59 | 56, 57, 58 | mp3an12 1466 | . . . . . . . . 9 ⊢ (𝑛 ∈ ℕ0 → (3 ∥ (2↑𝑛) → 3 = 2)) |
| 60 | 59 | imp 409 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 ∥ (2↑𝑛)) → 3 = 2) |
| 61 | 55, 60 | syldan 599 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = 2) |
| 62 | 2re 12282 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
| 63 | 2lt3 12381 | . . . . . . . . . 10 ⊢ 2 < 3 | |
| 64 | 62, 63 | gtneii 11285 | . . . . . . . . 9 ⊢ 3 ≠ 2 |
| 65 | 64 | neii 2953 | . . . . . . . 8 ⊢ ¬ 3 = 2 |
| 66 | 65 | a1i 11 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → ¬ 3 = 2) |
| 67 | 61, 66 | pm2.65da 824 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ¬ 3 = (2↑𝑛)) |
| 68 | 67 | neqned 2958 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → 3 ≠ (2↑𝑛)) |
| 69 | 50, 68 | eqnetrd 3018 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) ≠ (2↑𝑛)) |
| 70 | 69 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ0) → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) ≠ (2↑𝑛)) |
| 71 | 1, 2, 32, 33, 49, 70 | constrcon 34025 | . 2 ⊢ (⊤ → ¬ 𝐴 ∈ Constr) |
| 72 | 71 | mptru 1561 | 1 ⊢ ¬ 𝐴 ∈ Constr |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1554 ⊤wtru 1555 ∈ wcel 2136 ≠ wne 2951 class class class wbr 5094 ‘cfv 6510 (class class class)co 7385 ℂcc 11061 0cc0 11063 1c1 11064 ici 11065 + caddc 11066 · cmul 11068 -cneg 11405 / cdiv 11834 2c2 12262 3c3 12263 ℕ0cn0 12471 ℤcz 12558 ℚcq 12939 ↑cexp 14064 expce 16067 πcpi 16072 ∥ cdvds 16262 ℙcprime 16681 ↾s cress 17242 +gcplusg 17262 .rcmulr 17263 .gcmg 19085 mulGrpcmgp 20162 ℂfldccnfld 21397 algSccascl 21877 var1cv1 22211 Poly1cpl1 22212 deg1cdg1 26087 ↑𝑐ccxp 26590 minPoly cminply 33950 Constrcconstr 33980 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 ax-reg 9530 ax-inf2 9586 ax-ac2 10410 ax-cnex 11119 ax-resscn 11120 ax-1cn 11121 ax-icn 11122 ax-addcl 11123 ax-addrcl 11124 ax-mulcl 11125 ax-mulrcl 11126 ax-mulcom 11127 ax-addass 11128 ax-mulass 11129 ax-distr 11130 ax-i2m1 11131 ax-1ne0 11132 ax-1rid 11133 ax-rnegex 11134 ax-rrecex 11135 ax-cnre 11136 ax-pre-lttri 11137 ax-pre-lttrn 11138 ax-pre-ltadd 11139 ax-pre-mulgt0 11140 ax-pre-sup 11141 ax-addf 11142 ax-mulf 11143 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-nel 3056 df-ral 3071 df-rex 3081 df-rmo 3361 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4900 df-iun 4945 df-iin 4946 df-br 5095 df-opab 5157 df-mpt 5176 df-tr 5202 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-isom 6519 df-riota 7342 df-ov 7388 df-oprab 7389 df-mpo 7390 df-of 7649 df-ofr 7650 df-rpss 7695 df-om 7836 df-1st 7959 df-2nd 7960 df-supp 8129 df-tpos 8194 df-frecs 8250 df-wrecs 8281 df-recs 8330 df-rdg 8369 df-1o 8425 df-2o 8426 df-oadd 8429 df-er 8666 df-ec 8668 df-qs 8672 df-map 8798 df-pm 8799 df-ixp 8869 df-en 8917 df-dom 8918 df-sdom 8919 df-fin 8920 df-fsupp 9298 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9448 df-r1 9712 df-rank 9713 df-dju 9849 df-card 9887 df-acn 9890 df-ac 10062 df-pnf 11208 df-mnf 11209 df-xr 11210 df-ltxr 11211 df-le 11212 df-sub 11406 df-neg 11407 df-div 11835 df-nn 12201 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-7 12275 df-8 12276 df-9 12277 df-n0 12472 df-xnn0 12545 df-z 12559 df-dec 12679 df-uz 12830 df-q 12940 df-rp 12984 df-xneg 13104 df-xadd 13105 df-xmul 13106 df-ioo 13343 df-ioc 13344 df-ico 13345 df-icc 13346 df-fz 13503 df-fzo 13650 df-fl 13792 df-mod 13870 df-seq 14005 df-exp 14065 df-fac 14277 df-bc 14306 df-hash 14334 df-word 14517 df-lsw 14566 df-concat 14574 df-s1 14600 df-substr 14645 df-pfx 14675 df-shft 15070 df-sgn 15090 df-cj 15102 df-re 15103 df-im 15104 df-sqrt 15238 df-abs 15239 df-limsup 15474 df-clim 15491 df-rlim 15492 df-sum 15690 df-ef 16073 df-sin 16075 df-cos 16076 df-pi 16078 df-dvds 16263 df-gcd 16505 df-prm 16682 df-pc 16849 df-struct 17159 df-sets 17176 df-slot 17194 df-ndx 17206 df-base 17222 df-ress 17243 df-plusg 17275 df-mulr 17276 df-starv 17277 df-sca 17278 df-vsca 17279 df-ip 17280 df-tset 17281 df-ple 17282 df-ocomp 17283 df-ds 17284 df-unif 17285 df-hom 17286 df-cco 17287 df-rest 17427 df-topn 17428 df-0g 17446 df-gsum 17447 df-topgen 17448 df-pt 17449 df-prds 17452 df-pws 17454 df-xrs 17508 df-qtop 17513 df-imas 17514 df-qus 17515 df-xps 17516 df-mre 17590 df-mrc 17591 df-mri 17592 df-acs 17593 df-proset 18302 df-drs 18303 df-poset 18321 df-ipo 18536 df-chn 18614 df-mgm 18650 df-sgrp 18729 df-mnd 18745 df-mhm 18793 df-submnd 18794 df-grp 18954 df-minusg 18955 df-sbg 18956 df-mulg 19086 df-subg 19141 df-nsg 19142 df-eqg 19143 df-ghm 19230 df-gim 19275 df-cntz 19333 df-oppg 19362 df-lsm 19652 df-cmn 19798 df-abl 19799 df-mgp 20163 df-rng 20175 df-ur 20204 df-srg 20209 df-ring 20257 df-cring 20258 df-oppr 20358 df-dvdsr 20378 df-unit 20379 df-irred 20380 df-invr 20409 df-dvr 20422 df-rhm 20493 df-nzr 20535 df-subrng 20568 df-subrg 20592 df-rlreg 20716 df-domn 20717 df-idom 20718 df-drng 20753 df-field 20754 df-sdrg 20809 df-lmod 20902 df-lss 20972 df-lsp 21012 df-lmhm 21062 df-lmim 21063 df-lmic 21064 df-lbs 21115 df-lvec 21143 df-sra 21213 df-rgmod 21214 df-lidl 21251 df-rsp 21252 df-2idl 21293 df-lpidl 21365 df-lpir 21366 df-pid 21380 df-psmet 21389 df-xmet 21390 df-met 21391 df-bl 21392 df-mopn 21393 df-fbas 21394 df-fg 21395 df-cnfld 21398 df-dsmm 21757 df-frlm 21772 df-uvc 21808 df-lindf 21831 df-linds 21832 df-assa 21878 df-asp 21879 df-ascl 21880 df-psr 21934 df-mvr 21935 df-mpl 21936 df-opsr 21938 df-evls 22100 df-evl 22101 df-psr1 22215 df-vr1 22216 df-ply1 22217 df-coe1 22218 df-evls1 22351 df-evl1 22352 df-top 22927 df-topon 22944 df-topsp 22966 df-bases 22979 df-cld 23052 df-ntr 23053 df-cls 23054 df-nei 23131 df-lp 23169 df-perf 23170 df-cn 23260 df-cnp 23261 df-haus 23348 df-tx 23595 df-hmeo 23788 df-fil 23879 df-fm 23971 df-flim 23972 df-flf 23973 df-xms 24353 df-ms 24354 df-tms 24355 df-cncf 24913 df-limc 25901 df-dv 25902 df-mdeg 26088 df-deg1 26089 df-mon1 26164 df-uc1p 26165 df-q1p 26166 df-r1p 26167 df-ig1p 26168 df-log 26591 df-cxp 26592 df-fldgen 33452 df-mxidl 33602 df-dim 33851 df-fldext 33892 df-extdg 33893 df-irng 33935 df-minply 33951 df-constr 33981 |
| This theorem is referenced by: cos9thpinconstr 34042 |
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