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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 2sqr3nconstr | Structured version Visualization version GIF version | ||
| Description: Doubling the cube is an impossible construction, i.e. the cube root of 2 is not constructible with straightedge and compass. Given a cube of edge of length one, a cube of double volume would have an edge of length (2↑𝑐(1 / 3)), however that number is not constructible. This is the first part of Metamath 100 proof #8. Theorem 7.13 of [Stewart] p. 99. (Contributed by Thierry Arnoux and Saveliy Skresanov, 26-Oct-2025.) |
| Ref | Expression |
|---|---|
| 2sqr3nconstr | ⊢ (2↑𝑐(1 / 3)) ∉ Constr |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2741 | . . . 4 ⊢ (deg1‘(ℂfld ↾s ℚ)) = (deg1‘(ℂfld ↾s ℚ)) | |
| 2 | eqid 2741 | . . . 4 ⊢ (ℂfld minPoly ℚ) = (ℂfld minPoly ℚ) | |
| 3 | 2cnd 12254 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
| 4 | 3cn 12257 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
| 5 | 3ne0 12282 | . . . . . . 7 ⊢ 3 ≠ 0 | |
| 6 | 4, 5 | reccli 11880 | . . . . . 6 ⊢ (1 / 3) ∈ ℂ |
| 7 | 6 | a1i 11 | . . . . 5 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 8 | 3, 7 | cxpcld 26693 | . . . 4 ⊢ (⊤ → (2↑𝑐(1 / 3)) ∈ ℂ) |
| 9 | eqidd 2742 | . . . 4 ⊢ (⊤ → ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3))) = ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) | |
| 10 | eqid 2741 | . . . . . . . . . 10 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 11 | eqid 2741 | . . . . . . . . . 10 ⊢ (-g‘(Poly1‘(ℂfld ↾s ℚ))) = (-g‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 12 | eqid 2741 | . . . . . . . . . 10 ⊢ (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) = (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) | |
| 13 | eqid 2741 | . . . . . . . . . 10 ⊢ (Poly1‘(ℂfld ↾s ℚ)) = (Poly1‘(ℂfld ↾s ℚ)) | |
| 14 | eqid 2741 | . . . . . . . . . 10 ⊢ (algSc‘(Poly1‘(ℂfld ↾s ℚ))) = (algSc‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 15 | eqid 2741 | . . . . . . . . . 10 ⊢ (var1‘(ℂfld ↾s ℚ)) = (var1‘(ℂfld ↾s ℚ)) | |
| 16 | eqid 2741 | . . . . . . . . . 10 ⊢ ((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(-g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘2)) = ((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(-g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘2)) | |
| 17 | eqid 2741 | . . . . . . . . . 10 ⊢ (2↑𝑐(1 / 3)) = (2↑𝑐(1 / 3)) | |
| 18 | 10, 11, 12, 13, 14, 15, 1, 16, 17, 2 | 2sqr3minply 33974 | . . . . . . . . 9 ⊢ (((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(-g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘2)) = ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3))) ∧ ((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)))‘2))) = 3) |
| 19 | 18 | simpli 485 | . . . . . . . 8 ⊢ ((3(.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ))))(var1‘(ℂfld ↾s ℚ)))(-g‘(Poly1‘(ℂfld ↾s ℚ)))((algSc‘(Poly1‘(ℂfld ↾s ℚ)))‘2)) = ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3))) |
| 20 | 19 | fveq2i 6833 | . . . . . . 7 ⊢ ((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)))‘2))) = ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) |
| 21 | 18 | simpri 487 | . . . . . . 7 ⊢ ((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)))‘2))) = 3 |
| 22 | 20, 21 | eqtr3i 2766 | . . . . . 6 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) = 3 |
| 23 | 3nn0 12450 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
| 24 | 22, 23 | eqeltri 2837 | . . . . 5 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) ∈ ℕ0 |
| 25 | 24 | a1i 11 | . . . 4 ⊢ (⊤ → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) ∈ ℕ0) |
| 26 | 22 | a1i 11 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) = 3) |
| 27 | 3z 12555 | . . . . . . . . . . 11 ⊢ 3 ∈ ℤ | |
| 28 | iddvds 16233 | . . . . . . . . . . 11 ⊢ (3 ∈ ℤ → 3 ∥ 3) | |
| 29 | 27, 28 | ax-mp 5 | . . . . . . . . . 10 ⊢ 3 ∥ 3 |
| 30 | simpr 486 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = (2↑𝑛)) | |
| 31 | 29, 30 | breqtrid 5111 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 ∥ (2↑𝑛)) |
| 32 | 3prm 16658 | . . . . . . . . . . 11 ⊢ 3 ∈ ℙ | |
| 33 | 2prm 16656 | . . . . . . . . . . 11 ⊢ 2 ∈ ℙ | |
| 34 | prmdvdsexpr 16682 | . . . . . . . . . . 11 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (3 ∥ (2↑𝑛) → 3 = 2)) | |
| 35 | 32, 33, 34 | mp3an12 1460 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → (3 ∥ (2↑𝑛) → 3 = 2)) |
| 36 | 35 | imp 408 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 ∥ (2↑𝑛)) → 3 = 2) |
| 37 | 31, 36 | syldan 598 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = 2) |
| 38 | 2re 12250 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
| 39 | 2lt3 12343 | . . . . . . . . . . 11 ⊢ 2 < 3 | |
| 40 | 38, 39 | gtneii 11254 | . . . . . . . . . 10 ⊢ 3 ≠ 2 |
| 41 | 40 | neii 2938 | . . . . . . . . 9 ⊢ ¬ 3 = 2 |
| 42 | 41 | a1i 11 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → ¬ 3 = 2) |
| 43 | 37, 42 | pm2.65da 823 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → ¬ 3 = (2↑𝑛)) |
| 44 | 43 | neqned 2943 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 3 ≠ (2↑𝑛)) |
| 45 | 26, 44 | eqnetrd 3003 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) ≠ (2↑𝑛)) |
| 46 | 45 | adantl 483 | . . . 4 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ0) → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) ≠ (2↑𝑛)) |
| 47 | 1, 2, 8, 9, 25, 46 | constrcon 33968 | . . 3 ⊢ (⊤ → ¬ (2↑𝑐(1 / 3)) ∈ Constr) |
| 48 | 47 | mptru 1555 | . 2 ⊢ ¬ (2↑𝑐(1 / 3)) ∈ Constr |
| 49 | 48 | nelir 3043 | 1 ⊢ (2↑𝑐(1 / 3)) ∉ Constr |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1548 ⊤wtru 1549 ∈ wcel 2121 ≠ wne 2936 ∉ wnel 3040 class class class wbr 5074 ‘cfv 6488 (class class class)co 7359 ℂcc 11032 1c1 11035 / cdiv 11803 2c2 12231 3c3 12232 ℕ0cn0 12432 ℤcz 12519 ℚcq 12893 ↑cexp 14018 ∥ cdvds 16216 ℙcprime 16635 ↾s cress 17195 -gcsg 18906 .gcmg 19038 mulGrpcmgp 20115 ℂfldccnfld 21350 algSccascl 21830 var1cv1 22164 Poly1cpl1 22165 deg1cdg1 26040 ↑𝑐ccxp 26540 minPoly cminply 33893 Constrcconstr 33923 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-reg 9501 ax-inf2 9557 ax-ac2 10381 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112 ax-addf 11113 ax-mulf 11114 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-ofr 7624 df-rpss 7669 df-om 7810 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-ec 8639 df-qs 8643 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-r1 9683 df-rank 9684 df-dju 9820 df-card 9858 df-acn 9861 df-ac 10033 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-word 14471 df-lsw 14520 df-concat 14528 df-s1 14554 df-substr 14599 df-pfx 14629 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-ef 16027 df-sin 16029 df-cos 16030 df-pi 16032 df-dvds 16217 df-gcd 16459 df-prm 16636 df-numer 16700 df-denom 16701 df-pc 16803 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ocomp 17236 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-pws 17407 df-xrs 17461 df-qtop 17466 df-imas 17467 df-qus 17468 df-xps 17469 df-mre 17543 df-mrc 17544 df-mri 17545 df-acs 17546 df-proset 18255 df-drs 18256 df-poset 18274 df-ipo 18489 df-chn 18567 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-nsg 19095 df-eqg 19096 df-ghm 19183 df-gim 19228 df-cntz 19286 df-oppg 19315 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-srg 20162 df-ring 20210 df-cring 20211 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-irred 20333 df-invr 20362 df-dvr 20375 df-rhm 20446 df-nzr 20488 df-subrng 20521 df-subrg 20545 df-rlreg 20669 df-domn 20670 df-idom 20671 df-drng 20706 df-field 20707 df-sdrg 20762 df-lmod 20855 df-lss 20925 df-lsp 20965 df-lmhm 21015 df-lmim 21016 df-lmic 21017 df-lbs 21068 df-lvec 21096 df-sra 21166 df-rgmod 21167 df-lidl 21204 df-rsp 21205 df-2idl 21246 df-lpidl 21318 df-lpir 21319 df-pid 21333 df-psmet 21342 df-xmet 21343 df-met 21344 df-bl 21345 df-mopn 21346 df-fbas 21347 df-fg 21348 df-cnfld 21351 df-dsmm 21710 df-frlm 21725 df-uvc 21761 df-lindf 21784 df-linds 21785 df-assa 21831 df-asp 21832 df-ascl 21833 df-psr 21887 df-mvr 21888 df-mpl 21889 df-opsr 21891 df-evls 22053 df-evl 22054 df-psr1 22168 df-vr1 22169 df-ply1 22170 df-coe1 22171 df-evls1 22304 df-evl1 22305 df-top 22880 df-topon 22897 df-topsp 22919 df-bases 22932 df-cld 23005 df-ntr 23006 df-cls 23007 df-nei 23084 df-lp 23122 df-perf 23123 df-cn 23213 df-cnp 23214 df-haus 23301 df-tx 23548 df-hmeo 23741 df-fil 23832 df-fm 23924 df-flim 23925 df-flf 23926 df-xms 24306 df-ms 24307 df-tms 24308 df-cncf 24866 df-limc 25854 df-dv 25855 df-mdeg 26041 df-deg1 26042 df-mon1 26117 df-uc1p 26118 df-q1p 26119 df-r1p 26120 df-ig1p 26121 df-log 26541 df-cxp 26542 df-fldgen 33397 df-mxidl 33545 df-dim 33794 df-fldext 33835 df-extdg 33836 df-irng 33878 df-minply 33894 df-constr 33924 |
| This theorem is referenced by: (None) |
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