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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 2730 | . . . 4 ⊢ (deg1‘(ℂfld ↾s ℚ)) = (deg1‘(ℂfld ↾s ℚ)) | |
| 2 | eqid 2730 | . . . 4 ⊢ (ℂfld minPoly ℚ) = (ℂfld minPoly ℚ) | |
| 3 | 2cnd 12275 | . . . . 5 ⊢ (⊤ → 2 ∈ ℂ) | |
| 4 | 3cn 12278 | . . . . . . 7 ⊢ 3 ∈ ℂ | |
| 5 | 3ne0 12303 | . . . . . . 7 ⊢ 3 ≠ 0 | |
| 6 | 4, 5 | reccli 11928 | . . . . . 6 ⊢ (1 / 3) ∈ ℂ |
| 7 | 6 | a1i 11 | . . . . 5 ⊢ (⊤ → (1 / 3) ∈ ℂ) |
| 8 | 3, 7 | cxpcld 26624 | . . . 4 ⊢ (⊤ → (2↑𝑐(1 / 3)) ∈ ℂ) |
| 9 | eqidd 2731 | . . . 4 ⊢ (⊤ → ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3))) = ((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) | |
| 10 | eqid 2730 | . . . . . . . . . 10 ⊢ (ℂfld ↾s ℚ) = (ℂfld ↾s ℚ) | |
| 11 | eqid 2730 | . . . . . . . . . 10 ⊢ (-g‘(Poly1‘(ℂfld ↾s ℚ))) = (-g‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 12 | eqid 2730 | . . . . . . . . . 10 ⊢ (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) = (.g‘(mulGrp‘(Poly1‘(ℂfld ↾s ℚ)))) | |
| 13 | eqid 2730 | . . . . . . . . . 10 ⊢ (Poly1‘(ℂfld ↾s ℚ)) = (Poly1‘(ℂfld ↾s ℚ)) | |
| 14 | eqid 2730 | . . . . . . . . . 10 ⊢ (algSc‘(Poly1‘(ℂfld ↾s ℚ))) = (algSc‘(Poly1‘(ℂfld ↾s ℚ))) | |
| 15 | eqid 2730 | . . . . . . . . . 10 ⊢ (var1‘(ℂfld ↾s ℚ)) = (var1‘(ℂfld ↾s ℚ)) | |
| 16 | eqid 2730 | . . . . . . . . . 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 2730 | . . . . . . . . . 10 ⊢ (2↑𝑐(1 / 3)) = (2↑𝑐(1 / 3)) | |
| 18 | 10, 11, 12, 13, 14, 15, 1, 16, 17, 2 | 2sqr3minply 33778 | . . . . . . . . 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 6868 | . . . . . . 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 2755 | . . . . . 6 ⊢ ((deg1‘(ℂfld ↾s ℚ))‘((ℂfld minPoly ℚ)‘(2↑𝑐(1 / 3)))) = 3 |
| 23 | 3nn0 12476 | . . . . . 6 ⊢ 3 ∈ ℕ0 | |
| 24 | 22, 23 | eqeltri 2825 | . . . . 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 12582 | . . . . . . . . . . 11 ⊢ 3 ∈ ℤ | |
| 28 | iddvds 16246 | . . . . . . . . . . 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 5152 | . . . . . . . . 9 ⊢ ((𝑛 ∈ ℕ0 ∧ 3 = (2↑𝑛)) → 3 ∥ (2↑𝑛)) |
| 32 | 3prm 16670 | . . . . . . . . . . 11 ⊢ 3 ∈ ℙ | |
| 33 | 2prm 16668 | . . . . . . . . . . 11 ⊢ 2 ∈ ℙ | |
| 34 | prmdvdsexpr 16693 | . . . . . . . . . . 11 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ ∧ 𝑛 ∈ ℕ0) → (3 ∥ (2↑𝑛) → 3 = 2)) | |
| 35 | 32, 33, 34 | mp3an12 1453 | . . . . . . . . . 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 12271 | . . . . . . . . . . 11 ⊢ 2 ∈ ℝ | |
| 39 | 2lt3 12369 | . . . . . . . . . . 11 ⊢ 2 < 3 | |
| 40 | 38, 39 | gtneii 11304 | . . . . . . . . . 10 ⊢ 3 ≠ 2 |
| 41 | 40 | neii 2929 | . . . . . . . . 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 2934 | . . . . . 6 ⊢ (𝑛 ∈ ℕ0 → 3 ≠ (2↑𝑛)) |
| 45 | 26, 44 | eqnetrd 2994 | . . . . 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 33772 | . . 3 ⊢ (⊤ → ¬ (2↑𝑐(1 / 3)) ∈ Constr) |
| 48 | 47 | mptru 1547 | . 2 ⊢ ¬ (2↑𝑐(1 / 3)) ∈ Constr |
| 49 | 48 | nelir 3034 | 1 ⊢ (2↑𝑐(1 / 3)) ∉ Constr |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2927 ∉ wnel 3031 class class class wbr 5115 ‘cfv 6519 (class class class)co 7394 ℂcc 11084 1c1 11087 / cdiv 11851 2c2 12252 3c3 12253 ℕ0cn0 12458 ℤcz 12545 ℚcq 12921 ↑cexp 14036 ∥ cdvds 16229 ℙcprime 16647 ↾s cress 17206 -gcsg 18873 .gcmg 19005 mulGrpcmgp 20055 ℂfldccnfld 21270 algSccascl 21767 var1cv1 22066 Poly1cpl1 22067 deg1cdg1 25966 ↑𝑐ccxp 26471 minPoly cminply 33697 Constrcconstr 33727 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5242 ax-sep 5259 ax-nul 5269 ax-pow 5328 ax-pr 5395 ax-un 7718 ax-reg 9563 ax-inf2 9612 ax-ac2 10434 ax-cnex 11142 ax-resscn 11143 ax-1cn 11144 ax-icn 11145 ax-addcl 11146 ax-addrcl 11147 ax-mulcl 11148 ax-mulrcl 11149 ax-mulcom 11150 ax-addass 11151 ax-mulass 11152 ax-distr 11153 ax-i2m1 11154 ax-1ne0 11155 ax-1rid 11156 ax-rnegex 11157 ax-rrecex 11158 ax-cnre 11159 ax-pre-lttri 11160 ax-pre-lttrn 11161 ax-pre-ltadd 11162 ax-pre-mulgt0 11163 ax-pre-sup 11164 ax-addf 11165 ax-mulf 11166 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2880 df-ne 2928 df-nel 3032 df-ral 3047 df-rex 3056 df-rmo 3357 df-reu 3358 df-rab 3412 df-v 3457 df-sbc 3762 df-csb 3871 df-dif 3925 df-un 3927 df-in 3929 df-ss 3939 df-pss 3942 df-nul 4305 df-if 4497 df-pw 4573 df-sn 4598 df-pr 4600 df-tp 4602 df-op 4604 df-uni 4880 df-int 4919 df-iun 4965 df-iin 4966 df-br 5116 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5541 df-eprel 5546 df-po 5554 df-so 5555 df-fr 5599 df-se 5600 df-we 5601 df-xp 5652 df-rel 5653 df-cnv 5654 df-co 5655 df-dm 5656 df-rn 5657 df-res 5658 df-ima 5659 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6472 df-fun 6521 df-fn 6522 df-f 6523 df-f1 6524 df-fo 6525 df-f1o 6526 df-fv 6527 df-isom 6528 df-riota 7351 df-ov 7397 df-oprab 7398 df-mpo 7399 df-of 7660 df-ofr 7661 df-rpss 7706 df-om 7851 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8349 df-rdg 8387 df-1o 8443 df-2o 8444 df-oadd 8447 df-er 8682 df-ec 8684 df-qs 8688 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9331 df-fi 9380 df-sup 9411 df-inf 9412 df-oi 9481 df-r1 9735 df-rank 9736 df-dju 9872 df-card 9910 df-acn 9913 df-ac 10087 df-pnf 11228 df-mnf 11229 df-xr 11230 df-ltxr 11231 df-le 11232 df-sub 11425 df-neg 11426 df-div 11852 df-nn 12198 df-2 12260 df-3 12261 df-4 12262 df-5 12263 df-6 12264 df-7 12265 df-8 12266 df-9 12267 df-n0 12459 df-xnn0 12532 df-z 12546 df-dec 12666 df-uz 12810 df-q 12922 df-rp 12966 df-xneg 13085 df-xadd 13086 df-xmul 13087 df-ioo 13323 df-ioc 13324 df-ico 13325 df-icc 13326 df-fz 13482 df-fzo 13629 df-fl 13766 df-mod 13844 df-seq 13977 df-exp 14037 df-fac 14249 df-bc 14278 df-hash 14306 df-word 14489 df-lsw 14538 df-concat 14546 df-s1 14571 df-substr 14616 df-pfx 14646 df-shft 15043 df-cj 15075 df-re 15076 df-im 15077 df-sqrt 15211 df-abs 15212 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-pi 16045 df-dvds 16230 df-gcd 16471 df-prm 16648 df-numer 16711 df-denom 16712 df-pc 16814 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ocomp 17247 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17391 df-topn 17392 df-0g 17410 df-gsum 17411 df-topgen 17412 df-pt 17413 df-prds 17416 df-pws 17418 df-xrs 17471 df-qtop 17476 df-imas 17477 df-qus 17478 df-xps 17479 df-mre 17553 df-mrc 17554 df-mri 17555 df-acs 17556 df-proset 18261 df-drs 18262 df-poset 18280 df-ipo 18493 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18716 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-mulg 19006 df-subg 19061 df-nsg 19062 df-eqg 19063 df-ghm 19151 df-gim 19197 df-cntz 19255 df-oppg 19284 df-lsm 19572 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-srg 20102 df-ring 20150 df-cring 20151 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-irred 20274 df-invr 20303 df-dvr 20316 df-rhm 20387 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rlreg 20609 df-domn 20610 df-idom 20611 df-drng 20646 df-field 20647 df-sdrg 20702 df-lmod 20774 df-lss 20844 df-lsp 20884 df-lmhm 20935 df-lmim 20936 df-lmic 20937 df-lbs 20988 df-lvec 21016 df-sra 21086 df-rgmod 21087 df-lidl 21124 df-rsp 21125 df-2idl 21166 df-lpidl 21238 df-lpir 21239 df-pid 21253 df-psmet 21262 df-xmet 21263 df-met 21264 df-bl 21265 df-mopn 21266 df-fbas 21267 df-fg 21268 df-cnfld 21271 df-dsmm 21647 df-frlm 21662 df-uvc 21698 df-lindf 21721 df-linds 21722 df-assa 21768 df-asp 21769 df-ascl 21770 df-psr 21824 df-mvr 21825 df-mpl 21826 df-opsr 21828 df-evls 21987 df-evl 21988 df-psr1 22070 df-vr1 22071 df-ply1 22072 df-coe1 22073 df-evls1 22208 df-evl1 22209 df-top 22787 df-topon 22804 df-topsp 22826 df-bases 22839 df-cld 22912 df-ntr 22913 df-cls 22914 df-nei 22991 df-lp 23029 df-perf 23030 df-cn 23120 df-cnp 23121 df-haus 23208 df-tx 23455 df-hmeo 23648 df-fil 23739 df-fm 23831 df-flim 23832 df-flf 23833 df-xms 24214 df-ms 24215 df-tms 24216 df-cncf 24777 df-limc 25774 df-dv 25775 df-mdeg 25967 df-deg1 25968 df-mon1 26043 df-uc1p 26044 df-q1p 26045 df-r1p 26046 df-ig1p 26047 df-log 26472 df-cxp 26473 df-chn 32939 df-fldgen 33269 df-mxidl 33439 df-dim 33603 df-fldext 33645 df-extdg 33646 df-irng 33687 df-minply 33698 df-constr 33728 |
| This theorem is referenced by: (None) |
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