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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 2763 | . . 3 ⊢ (deg1‘(ℂfld ↾s ℚ)) = (deg1‘(ℂfld ↾s ℚ)) | |
| 2 | eqid 2763 | . . 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 11163 | . . . . . . . . . . . 12 ⊢ i ∈ ℂ | |
| 7 | 6 | a1i 11 | . . . . . . . . . . 11 ⊢ (⊤ → i ∈ ℂ) |
| 8 | 2cnd 12323 | . . . . . . . . . . . 12 ⊢ (⊤ → 2 ∈ ℂ) | |
| 9 | picn 26630 | . . . . . . . . . . . . 13 ⊢ π ∈ ℂ | |
| 10 | 9 | a1i 11 | . . . . . . . . . . . 12 ⊢ (⊤ → π ∈ ℂ) |
| 11 | 8, 10 | mulcld 11233 | . . . . . . . . . . 11 ⊢ (⊤ → (2 · π) ∈ ℂ) |
| 12 | 7, 11 | mulcld 11233 | . . . . . . . . . 10 ⊢ (⊤ → (i · (2 · π)) ∈ ℂ) |
| 13 | 3cn 12326 | . . . . . . . . . . 11 ⊢ 3 ∈ ℂ | |
| 14 | 13 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ∈ ℂ) |
| 15 | 3ne0 12354 | . . . . . . . . . . 11 ⊢ 3 ≠ 0 | |
| 16 | 15 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → 3 ≠ 0) |
| 17 | 12, 14, 16 | divcld 11995 | . . . . . . . . 9 ⊢ (⊤ → ((i · (2 · π)) / 3) ∈ ℂ) |
| 18 | 17 | efcld 16141 | . . . . . . . 8 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ∈ ℂ) |
| 19 | 5, 18 | eqeltrid 2867 | . . . . . . 7 ⊢ (⊤ → 𝑂 ∈ ℂ) |
| 20 | 13, 15 | reccli 11949 | . . . . . . . 8 ⊢ (1 / 3) ∈ ℂ |
| 21 | 20 | a1i 11 | . . . . . . 7 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 22 | 19, 21 | cxpcld 26882 | . . . . . 6 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ∈ ℂ) |
| 23 | 4, 22 | eqeltrid 2867 | . . . . 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 16155 | . . . . . . . . 9 ⊢ (⊤ → (exp‘((i · (2 · π)) / 3)) ≠ 0) |
| 27 | 25, 26 | eqnetrd 3025 | . . . . . . . 8 ⊢ (⊤ → 𝑂 ≠ 0) |
| 28 | 19, 27, 21 | cxpne0d 26887 | . . . . . . 7 ⊢ (⊤ → (𝑂↑𝑐(1 / 3)) ≠ 0) |
| 29 | 24, 28 | eqnetrd 3025 | . . . . . 6 ⊢ (⊤ → 𝑍 ≠ 0) |
| 30 | 23, 29 | reccld 11988 | . . . . 5 ⊢ (⊤ → (1 / 𝑍) ∈ ℂ) |
| 31 | 23, 30 | addcld 11232 | . . . 4 ⊢ (⊤ → (𝑍 + (1 / 𝑍)) ∈ ℂ) |
| 32 | 3, 31 | eqeltrid 2867 | . . 3 ⊢ (⊤ → 𝐴 ∈ ℂ) |
| 33 | eqidd 2764 | . . 3 ⊢ (⊤ → ((ℂfld minPoly ℚ)‘𝐴) = ((ℂfld minPoly ℚ)‘𝐴)) | |
| 34 | eqid 2763 | . . . . . . . . 9 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 35 | eqid 2763 | . . . . . . . . 9 ⊢ (+g‘(Poly1‘(ℂfld ↾s ℚ))) = (+g‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 36 | eqid 2763 | . . . . . . . . 9 ⊢ (.r‘(Poly1‘(ℂfld ↾s ℚ))) = (.r‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 37 | eqid 2763 | . . . . . . . . 9 ⊢ (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) = (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) | |
| 38 | eqid 2763 | . . . . . . . . 9 ⊢ (Poly1‘(ℂfld ↾s ℚ)) = (Poly1‘(ℂfld ↾s ℚ)) | |
| 39 | eqid 2763 | . . . . . . . . 9 ⊢ (algSc‘(Poly1‘(ℂfld ↾s ℚ))) = (algSc‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 40 | eqid 2763 | . . . . . . . . 9 ⊢ (var1‘(ℂfld ↾s ℚ)) = (var1‘(ℂfld ↾s ℚ)) | |
| 41 | eqid 2763 | . . . . . . . . 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 34187 | . . . . . . . 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 488 | . . . . . . 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 6884 | . . . . . 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 490 | . . . . . 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 2788 | . . . . 5 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) = 3 |
| 47 | 3nn0 12526 | . . . . 5 ⊢ 3 ∈ ℕ0 | |
| 48 | 46, 47 | eqeltri 2859 | . . . 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 12631 | . . . . . . . . . 10 ⊢ 3 ∈ ℤ | |
| 52 | iddvds 16331 | . . . . . . . . . 10 ⊢ (3 ∈ ℤ → 3 ∥ 3) | |
| 53 | 51, 52 | ax-mp 5 | . . . . . . . . 9 ⊢ 3 ∥ 3 |
| 54 | simpr 489 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = (2↑𝑛)) | |
| 55 | 53, 54 | breqtrid 5148 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 ∥ (2↑𝑛)) |
| 56 | 3prm 16756 | . . . . . . . . . 10 ⊢ 3 ∈ ℙ | |
| 57 | 2prm 16754 | . . . . . . . . . 10 ⊢ 2 ∈ ℙ | |
| 58 | prmdvdsexpr 16780 | . . . . . . . . . 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 411 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 ∥ (2↑𝑛)) → 3 = 2) |
| 61 | 55, 60 | syldan 602 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = 2) |
| 62 | 2re 12319 | . . . . . . . . . 10 ⊢ 2 ∈ ℝ | |
| 63 | 2lt3 12418 | . . . . . . . . . 10 ⊢ 2 < 3 | |
| 64 | 62, 63 | gtneii 11326 | . . . . . . . . 9 ⊢ 3 ≠ 2 |
| 65 | 64 | neii 2960 | . . . . . . . 8 ⊢ ¬ 3 = 2 |
| 66 | 65 | a1i 11 | . . . . . . 7 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → ¬ 3 = 2) |
| 67 | 61, 66 | pm2.65da 828 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ¬ 3 = (2↑𝑛)) |
| 68 | 67 | neqned 2965 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → 3 ≠ (2↑𝑛)) |
| 69 | 50, 68 | eqnetrd 3025 | . . . 4 ⊢ (𝑛 ∈ ℕ0 → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) ≠ (2↑𝑛)) |
| 70 | 69 | adantl 486 | . . 3 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ0) → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘𝐴)) ≠ (2↑𝑛)) |
| 71 | 1, 2, 32, 33, 49, 70 | constrcon 34173 | . 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 400 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 ≠ wne 2958 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 ℂcc 11102 0cc0 11104 1c1 11105 ici 11106 + caddc 11107 · cmul 11109 -cneg 11446 / cdiv 11875 2c2 12299 3c3 12300 ℕ0cn0 12508 ℤcz 12595 ℚcq 12976 ↑cexp 14102 expce 16119 πcpi 16124 ∥ cdvds 16314 ℙcprime 16733 ↾s cress 17294 +gcplusg 17314 .rcmulr 17315 .gcmg 19137 mulGrpcmgp 20220 ℂfldccnfld 21531 algSccascl 22011 var1cv1 22345 Poly1cpl1 22346 deg1cdg1 26220 ↑𝑐ccxp 26729 minPoly cminply 34098 Constrcconstr 34128 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-reg 9550 ax-inf2 9606 ax-ac2 10451 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183 ax-mulf 11184 |
| This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-ofr 7675 df-rpss 7720 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-oadd 8453 df-er 8690 df-ec 8692 df-qs 8696 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-r1 9732 df-rank 9733 df-scott 9854 df-dju 9892 df-card 9930 df-acn 9933 df-ac 10105 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-xnn0 12582 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13380 df-ioc 13381 df-ico 13382 df-icc 13383 df-fz 13540 df-fzo 13688 df-fl 13830 df-mod 13908 df-seq 14043 df-exp 14103 df-fac 14315 df-bc 14344 df-hash 14372 df-word 14556 df-lsw 14605 df-concat 14613 df-s1 14639 df-substr 14684 df-pfx 14714 df-shft 15109 df-sgn 15129 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-limsup 15527 df-clim 15544 df-rlim 15545 df-sum 15743 df-ef 16125 df-sin 16127 df-cos 16128 df-pi 16130 df-dvds 16315 df-gcd 16557 df-prm 16734 df-pc 16901 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ocomp 17335 df-ds 17336 df-unif 17337 df-hom 17338 df-cco 17339 df-rest 17479 df-topn 17480 df-0g 17498 df-gsum 17499 df-topgen 17500 df-pt 17501 df-prds 17504 df-pws 17506 df-xrs 17560 df-qtop 17565 df-imas 17566 df-qus 17567 df-xps 17568 df-mre 17642 df-mrc 17643 df-mri 17644 df-acs 17645 df-proset 18354 df-drs 18355 df-poset 18373 df-ipo 18588 df-chn 18666 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-mhm 18845 df-submnd 18846 df-grp 19007 df-minusg 19008 df-sbg 19009 df-mulg 19138 df-subg 19193 df-nsg 19194 df-eqg 19195 df-ghm 19288 df-gim 19333 df-cntz 19391 df-oppg 19420 df-lsm 19710 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-srg 20273 df-ring 20321 df-cring 20322 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-irred 20446 df-invr 20475 df-dvr 20488 df-rhm 20559 df-nzr 20619 df-subrng 20654 df-subrg 20678 df-rlreg 20802 df-domn 20803 df-idom 20804 df-drng 20838 df-field 20839 df-sdrg 20899 df-lmod 20992 df-lss 21062 df-lsp 21102 df-lmhm 21152 df-lmim 21153 df-lmic 21154 df-lbs 21205 df-lvec 21233 df-sra 21303 df-rgmod 21304 df-lidl 21341 df-rsp 21342 df-2idl 21398 df-lpidl 21499 df-lpir 21500 df-pid 21514 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-fbas 21528 df-fg 21529 df-cnfld 21532 df-dsmm 21891 df-frlm 21906 df-uvc 21942 df-lindf 21965 df-linds 21966 df-assa 22012 df-asp 22013 df-ascl 22014 df-psr 22068 df-mvr 22069 df-mpl 22070 df-opsr 22072 df-evls 22234 df-evl 22235 df-psr1 22349 df-vr1 22350 df-ply1 22351 df-coe1 22352 df-evls1 22484 df-evl1 22485 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-cld 23185 df-ntr 23186 df-cls 23187 df-nei 23264 df-lp 23302 df-perf 23303 df-cn 23393 df-cnp 23394 df-haus 23481 df-tx 23728 df-hmeo 23921 df-fil 24012 df-fm 24104 df-flim 24105 df-flf 24106 df-xms 24486 df-ms 24487 df-tms 24488 df-cncf 25046 df-limc 26034 df-dv 26035 df-mdeg 26221 df-deg1 26222 df-mon1 26297 df-uc1p 26298 df-q1p 26299 df-r1p 26300 df-ig1p 26301 df-log 26730 df-cxp 26731 df-fldgen 33641 df-mxidl 33752 df-dim 33999 df-fldext 34040 df-extdg 34041 df-irng 34083 df-minply 34099 df-constr 34129 |
| This theorem is used by: cos9thpinconstr 34190 |
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