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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 2737 | . . . 4 ⊢ (deg1‘(ℂfld ↾s ℚ)) = (deg1‘(ℂfld ↾s ℚ)) | |
| 2 | eqid 2737 | . . . 4 ⊢ (ℂfld minPoly ℚ) = (ℂfld minPoly ℚ) | |
| 3 | 2cnd 12235 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
| 4 | 3cn 12238 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
| 5 | 3ne0 12263 | . . . . . . 7 ⊢ 3 ≠ 0 | |
| 6 | 4, 5 | reccli 11883 | . . . . . 6 ⊢ (1 / 3) ∈ ℂ |
| 7 | 6 | a1i 11 | . . . . 5 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 8 | 3, 7 | cxpcld 26685 | . . . 4 ⊢ (⊤ → (2↑𝑐(1 / 3)) ∈ ℂ) |
| 9 | eqidd 2738 | . . . 4 ⊢ (⊤ → ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3))) = ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) | |
| 10 | eqid 2737 | . . . . . . . . . 10 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 11 | eqid 2737 | . . . . . . . . . 10 ⊢ (-g‘(Poly1‘(ℂfld ↾s ℚ))) = (-g‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 12 | eqid 2737 | . . . . . . . . . 10 ⊢ (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) = (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) | |
| 13 | eqid 2737 | . . . . . . . . . 10 ⊢ (Poly1‘(ℂfld ↾s ℚ)) = (Poly1‘(ℂfld ↾s ℚ)) | |
| 14 | eqid 2737 | . . . . . . . . . 10 ⊢ (algSc‘(Poly1‘(ℂfld ↾s ℚ))) = (algSc‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 15 | eqid 2737 | . . . . . . . . . 10 ⊢ (var1‘(ℂfld ↾s ℚ)) = (var1‘(ℂfld ↾s ℚ)) | |
| 16 | eqid 2737 | . . . . . . . . . 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 2737 | . . . . . . . . . 10 ⊢ (2↑𝑐(1 / 3)) = (2↑𝑐(1 / 3)) | |
| 18 | 10, 11, 12, 13, 14, 15, 1, 16, 17, 2 | 2sqr3minply 33957 | . . . . . . . . 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 6845 | . . . . . . 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 2762 | . . . . . 6 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) = 3 |
| 23 | 3nn0 12431 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
| 24 | 22, 23 | eqeltri 2833 | . . . . 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 12536 | . . . . . . . . . . 11 ⊢ 3 ∈ ℤ | |
| 28 | iddvds 16208 | . . . . . . . . . . 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 5137 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 ∥ (2↑𝑛)) |
| 32 | 3prm 16633 | . . . . . . . . . . 11 ⊢ 3 ∈ ℙ | |
| 33 | 2prm 16631 | . . . . . . . . . . 11 ⊢ 2 ∈ ℙ | |
| 34 | prmdvdsexpr 16656 | . . . . . . . . . . 11 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (3 ∥ (2↑𝑛) → 3 = 2)) | |
| 35 | 32, 33, 34 | mp3an12 1454 | . . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 → (3 ∥ (2↑𝑛) → 3 = 2)) |
| 36 | 35 | imp 406 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 ∥ (2↑𝑛)) → 3 = 2) |
| 37 | 31, 36 | syldan 592 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 = 2) |
| 38 | 2re 12231 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
| 39 | 2lt3 12324 | . . . . . . . . . . 11 ⊢ 2 < 3 | |
| 40 | 38, 39 | gtneii 11257 | . . . . . . . . . 10 ⊢ 3 ≠ 2 |
| 41 | 40 | neii 2935 | . . . . . . . . 9 ⊢ ¬ 3 = 2 |
| 42 | 41 | a1i 11 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → ¬ 3 = 2) |
| 43 | 37, 42 | pm2.65da 817 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ0 → ¬ 3 = (2↑𝑛)) |
| 44 | 43 | neqned 2940 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 3 ≠ (2↑𝑛)) |
| 45 | 26, 44 | eqnetrd 3000 | . . . . 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 33951 | . . 3 ⊢ (⊤ → ¬ (2↑𝑐(1 / 3)) ∈ Constr) |
| 48 | 47 | mptru 1549 | . 2 ⊢ ¬ (2↑𝑐(1 / 3)) ∈ Constr |
| 49 | 48 | nelir 3040 | 1 ⊢ (2↑𝑐(1 / 3)) ∉ Constr |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 ≠ wne 2933 ∉ wnel 3037 class class class wbr 5100 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 1c1 11039 / cdiv 11806 2c2 12212 3c3 12213 ℕ0cn0 12413 ℤcz 12500 ℚcq 12873 ↑cexp 13996 ∥ cdvds 16191 ℙcprime 16610 ↾s cress 17169 -gcsg 18877 .gcmg 19009 mulGrpcmgp 20087 ℂfldccnfld 21321 algSccascl 21819 var1cv1 22128 Poly1cpl1 22129 deg1cdg1 26027 ↑𝑐ccxp 26532 minPoly cminply 33876 Constrcconstr 33906 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-reg 9509 ax-inf2 9562 ax-ac2 10385 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-ofr 7633 df-rpss 7678 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-tpos 8178 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-ec 8647 df-qs 8651 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-r1 9688 df-rank 9689 df-dju 9825 df-card 9863 df-acn 9866 df-ac 10038 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-word 14449 df-lsw 14498 df-concat 14506 df-s1 14532 df-substr 14577 df-pfx 14607 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-pi 16007 df-dvds 16192 df-gcd 16434 df-prm 16611 df-numer 16674 df-denom 16675 df-pc 16777 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ocomp 17210 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-pws 17381 df-xrs 17435 df-qtop 17440 df-imas 17441 df-qus 17442 df-xps 17443 df-mre 17517 df-mrc 17518 df-mri 17519 df-acs 17520 df-proset 18229 df-drs 18230 df-poset 18248 df-ipo 18463 df-chn 18541 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-nsg 19066 df-eqg 19067 df-ghm 19154 df-gim 19200 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-srg 20134 df-ring 20182 df-cring 20183 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-irred 20307 df-invr 20336 df-dvr 20349 df-rhm 20420 df-nzr 20458 df-subrng 20491 df-subrg 20515 df-rlreg 20639 df-domn 20640 df-idom 20641 df-drng 20676 df-field 20677 df-sdrg 20732 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lmhm 20986 df-lmim 20987 df-lmic 20988 df-lbs 21039 df-lvec 21067 df-sra 21137 df-rgmod 21138 df-lidl 21175 df-rsp 21176 df-2idl 21217 df-lpidl 21289 df-lpir 21290 df-pid 21304 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-dsmm 21699 df-frlm 21714 df-uvc 21750 df-lindf 21773 df-linds 21774 df-assa 21820 df-asp 21821 df-ascl 21822 df-psr 21877 df-mvr 21878 df-mpl 21879 df-opsr 21881 df-evls 22041 df-evl 22042 df-psr1 22132 df-vr1 22133 df-ply1 22134 df-coe1 22135 df-evls1 22271 df-evl1 22272 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836 df-mdeg 26028 df-deg1 26029 df-mon1 26104 df-uc1p 26105 df-q1p 26106 df-r1p 26107 df-ig1p 26108 df-log 26533 df-cxp 26534 df-fldgen 33404 df-mxidl 33552 df-dim 33776 df-fldext 33818 df-extdg 33819 df-irng 33861 df-minply 33877 df-constr 33907 |
| This theorem is referenced by: (None) |
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