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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. (Contributed by Thierry Arnoux and Saveliy Skresanov, 26-Oct-2025.) |
| Ref | Expression |
|---|---|
| 2sqr3nconstr | ⊢ (2↑𝑐(1 / 3)) ∉ Constr |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2734 | . . . 4 ⊢ (deg1‘(ℂfld ↾s ℚ)) = (deg1‘(ℂfld ↾s ℚ)) | |
| 2 | eqid 2734 | . . . 4 ⊢ (ℂfld minPoly ℚ) = (ℂfld minPoly ℚ) | |
| 3 | 2cnd 12311 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
| 4 | 3cn 12314 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
| 5 | 3ne0 12339 | . . . . . . 7 ⊢ 3 ≠ 0 | |
| 6 | 4, 5 | reccli 11964 | . . . . . 6 ⊢ (1 / 3) ∈ ℂ |
| 7 | 6 | a1i 11 | . . . . 5 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 8 | 3, 7 | cxpcld 26655 | . . . 4 ⊢ (⊤ → (2↑𝑐(1 / 3)) ∈ ℂ) |
| 9 | eqidd 2735 | . . . 4 ⊢ (⊤ → ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3))) = ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) | |
| 10 | eqid 2734 | . . . . . . . . . 10 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 11 | eqid 2734 | . . . . . . . . . 10 ⊢ (-g‘(Poly1‘(ℂfld ↾s ℚ))) = (-g‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 12 | eqid 2734 | . . . . . . . . . 10 ⊢ (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) = (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) | |
| 13 | eqid 2734 | . . . . . . . . . 10 ⊢ (Poly1‘(ℂfld ↾s ℚ)) = (Poly1‘(ℂfld ↾s ℚ)) | |
| 14 | eqid 2734 | . . . . . . . . . 10 ⊢ (algSc‘(Poly1‘(ℂfld ↾s ℚ))) = (algSc‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 15 | eqid 2734 | . . . . . . . . . 10 ⊢ (var1‘(ℂfld ↾s ℚ)) = (var1‘(ℂfld ↾s ℚ)) | |
| 16 | eqid 2734 | . . . . . . . . . 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 2734 | . . . . . . . . . 10 ⊢ (2↑𝑐(1 / 3)) = (2↑𝑐(1 / 3)) | |
| 18 | 10, 11, 12, 13, 14, 15, 1, 16, 17, 2 | 2sqr3minply 33749 | . . . . . . . . 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 483 | . . . . . . . 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 6876 | . . . . . . 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 485 | . . . . . . 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 2759 | . . . . . 6 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) = 3 |
| 23 | 3nn0 12512 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
| 24 | 22, 23 | eqeltri 2829 | . . . . 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 12618 | . . . . . . . . . . 11 ⊢ 3 ∈ ℤ | |
| 28 | iddvds 16276 | . . . . . . . . . . 11 ⊢ (3 ∈ ℤ → 3 ∥ 3) | |
| 29 | 27, 28 | ax-mp 5 | . . . . . . . . . 10 ⊢ 3 ∥ 3 |
| 30 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = (2↑𝑛)) | |
| 31 | 29, 30 | breqtrid 5154 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 ∥ (2↑𝑛)) |
| 32 | 3prm 16700 | . . . . . . . . . . 11 ⊢ 3 ∈ ℙ | |
| 33 | 2prm 16698 | . . . . . . . . . . 11 ⊢ 2 ∈ ℙ | |
| 34 | prmdvdsexpr 16723 | . . . . . . . . . . 11 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (3 ∥ (2↑𝑛) → 3 = 2)) | |
| 35 | 32, 33, 34 | mp3an12 1452 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → (3 ∥ (2↑𝑛) → 3 = 2)) |
| 36 | 35 | imp 406 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 ∥ (2↑𝑛)) → 3 = 2) |
| 37 | 31, 36 | syldan 591 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = 2) |
| 38 | 2re 12307 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
| 39 | 2lt3 12405 | . . . . . . . . . . 11 ⊢ 2 < 3 | |
| 40 | 38, 39 | gtneii 11340 | . . . . . . . . . 10 ⊢ 3 ≠ 2 |
| 41 | 40 | neii 2933 | . . . . . . . . 9 ⊢ ¬ 3 = 2 |
| 42 | 41 | a1i 11 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → ¬ 3 = 2) |
| 43 | 37, 42 | pm2.65da 816 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → ¬ 3 = (2↑𝑛)) |
| 44 | 43 | neqned 2938 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 3 ≠ (2↑𝑛)) |
| 45 | 26, 44 | eqnetrd 2998 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) ≠ (2↑𝑛)) |
| 46 | 45 | adantl 481 | . . . 4 ⊢ ((⊤ ∧ 𝑛 ∈ ℕ0) → ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) ≠ (2↑𝑛)) |
| 47 | 1, 2, 8, 9, 25, 46 | constrcon 33743 | . . 3 ⊢ (⊤ → ¬ (2↑𝑐(1 / 3)) ∈ Constr) |
| 48 | 47 | mptru 1546 | . 2 ⊢ ¬ (2↑𝑐(1 / 3)) ∈ Constr |
| 49 | 48 | nelir 3038 | 1 ⊢ (2↑𝑐(1 / 3)) ∉ Constr |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ⊤wtru 1540 ∈ wcel 2107 ≠ wne 2931 ∉ wnel 3035 class class class wbr 5117 ‘cfv 6528 (class class class)co 7400 ℂcc 11120 1c1 11123 / cdiv 11887 2c2 12288 3c3 12289 ℕ0cn0 12494 ℤcz 12581 ℚcq 12957 ↑cexp 14069 ∥ cdvds 16259 ℙcprime 16677 ↾s cress 17238 -gcsg 18905 .gcmg 19037 mulGrpcmgp 20087 ℂfldccnfld 21302 algSccascl 21799 var1cv1 22098 Poly1cpl1 22099 deg1cdg1 25998 ↑𝑐ccxp 26502 minPoly cminply 33668 Constrcconstr 33698 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5247 ax-sep 5264 ax-nul 5274 ax-pow 5333 ax-pr 5400 ax-un 7724 ax-reg 9599 ax-inf2 9648 ax-ac2 10470 ax-cnex 11178 ax-resscn 11179 ax-1cn 11180 ax-icn 11181 ax-addcl 11182 ax-addrcl 11183 ax-mulcl 11184 ax-mulrcl 11185 ax-mulcom 11186 ax-addass 11187 ax-mulass 11188 ax-distr 11189 ax-i2m1 11190 ax-1ne0 11191 ax-1rid 11192 ax-rnegex 11193 ax-rrecex 11194 ax-cnre 11195 ax-pre-lttri 11196 ax-pre-lttrn 11197 ax-pre-ltadd 11198 ax-pre-mulgt0 11199 ax-pre-sup 11200 ax-addf 11201 ax-mulf 11202 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3357 df-reu 3358 df-rab 3414 df-v 3459 df-sbc 3764 df-csb 3873 df-dif 3927 df-un 3929 df-in 3931 df-ss 3941 df-pss 3944 df-nul 4307 df-if 4499 df-pw 4575 df-sn 4600 df-pr 4602 df-tp 4604 df-op 4606 df-uni 4882 df-int 4921 df-iun 4967 df-iin 4968 df-br 5118 df-opab 5180 df-mpt 5200 df-tr 5228 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6288 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6530 df-fn 6531 df-f 6532 df-f1 6533 df-fo 6534 df-f1o 6535 df-fv 6536 df-isom 6537 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7666 df-ofr 7667 df-rpss 7712 df-om 7857 df-1st 7983 df-2nd 7984 df-supp 8155 df-tpos 8220 df-frecs 8275 df-wrecs 8306 df-recs 8380 df-rdg 8419 df-1o 8475 df-2o 8476 df-oadd 8479 df-er 8714 df-ec 8716 df-qs 8720 df-map 8837 df-pm 8838 df-ixp 8907 df-en 8955 df-dom 8956 df-sdom 8957 df-fin 8958 df-fsupp 9369 df-fi 9418 df-sup 9449 df-inf 9450 df-oi 9517 df-r1 9771 df-rank 9772 df-dju 9908 df-card 9946 df-acn 9949 df-ac 10123 df-pnf 11264 df-mnf 11265 df-xr 11266 df-ltxr 11267 df-le 11268 df-sub 11461 df-neg 11462 df-div 11888 df-nn 12234 df-2 12296 df-3 12297 df-4 12298 df-5 12299 df-6 12300 df-7 12301 df-8 12302 df-9 12303 df-n0 12495 df-xnn0 12568 df-z 12582 df-dec 12702 df-uz 12846 df-q 12958 df-rp 13002 df-xneg 13121 df-xadd 13122 df-xmul 13123 df-ioo 13358 df-ioc 13359 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13662 df-fl 13799 df-mod 13877 df-seq 14010 df-exp 14070 df-fac 14282 df-bc 14311 df-hash 14339 df-word 14522 df-lsw 14570 df-concat 14578 df-s1 14603 df-substr 14648 df-pfx 14678 df-shft 15075 df-cj 15107 df-re 15108 df-im 15109 df-sqrt 15243 df-abs 15244 df-limsup 15476 df-clim 15493 df-rlim 15494 df-sum 15692 df-ef 16072 df-sin 16074 df-cos 16075 df-pi 16077 df-dvds 16260 df-gcd 16501 df-prm 16678 df-numer 16741 df-denom 16742 df-pc 16844 df-struct 17153 df-sets 17170 df-slot 17188 df-ndx 17200 df-base 17216 df-ress 17239 df-plusg 17271 df-mulr 17272 df-starv 17273 df-sca 17274 df-vsca 17275 df-ip 17276 df-tset 17277 df-ple 17278 df-ocomp 17279 df-ds 17280 df-unif 17281 df-hom 17282 df-cco 17283 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-pws 17450 df-xrs 17503 df-qtop 17508 df-imas 17509 df-qus 17510 df-xps 17511 df-mre 17585 df-mrc 17586 df-mri 17587 df-acs 17588 df-proset 18293 df-drs 18294 df-poset 18312 df-ipo 18525 df-mgm 18605 df-sgrp 18684 df-mnd 18700 df-mhm 18748 df-submnd 18749 df-grp 18906 df-minusg 18907 df-sbg 18908 df-mulg 19038 df-subg 19093 df-nsg 19094 df-eqg 19095 df-ghm 19183 df-gim 19229 df-cntz 19287 df-oppg 19316 df-lsm 19604 df-cmn 19750 df-abl 19751 df-mgp 20088 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-oppr 20284 df-dvdsr 20304 df-unit 20305 df-irred 20306 df-invr 20335 df-dvr 20348 df-rhm 20419 df-nzr 20460 df-subrng 20493 df-subrg 20517 df-rlreg 20641 df-domn 20642 df-idom 20643 df-drng 20678 df-field 20679 df-sdrg 20734 df-lmod 20806 df-lss 20876 df-lsp 20916 df-lmhm 20967 df-lmim 20968 df-lmic 20969 df-lbs 21020 df-lvec 21048 df-sra 21118 df-rgmod 21119 df-lidl 21156 df-rsp 21157 df-2idl 21198 df-lpidl 21270 df-lpir 21271 df-pid 21285 df-psmet 21294 df-xmet 21295 df-met 21296 df-bl 21297 df-mopn 21298 df-fbas 21299 df-fg 21300 df-cnfld 21303 df-dsmm 21679 df-frlm 21694 df-uvc 21730 df-lindf 21753 df-linds 21754 df-assa 21800 df-asp 21801 df-ascl 21802 df-psr 21856 df-mvr 21857 df-mpl 21858 df-opsr 21860 df-evls 22019 df-evl 22020 df-psr1 22102 df-vr1 22103 df-ply1 22104 df-coe1 22105 df-evls1 22240 df-evl1 22241 df-top 22819 df-topon 22836 df-topsp 22858 df-bases 22871 df-cld 22944 df-ntr 22945 df-cls 22946 df-nei 23023 df-lp 23061 df-perf 23062 df-cn 23152 df-cnp 23153 df-haus 23240 df-tx 23487 df-hmeo 23680 df-fil 23771 df-fm 23863 df-flim 23864 df-flf 23865 df-xms 24246 df-ms 24247 df-tms 24248 df-cncf 24809 df-limc 25806 df-dv 25807 df-mdeg 25999 df-deg1 26000 df-mon1 26075 df-uc1p 26076 df-q1p 26077 df-r1p 26078 df-ig1p 26079 df-log 26503 df-cxp 26504 df-chn 32923 df-fldgen 33242 df-mxidl 33412 df-dim 33574 df-fldext 33617 df-extdg 33618 df-irng 33660 df-minply 33669 df-constr 33699 |
| This theorem is referenced by: (None) |
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