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Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > constrsdrg | Structured version Visualization version GIF version | ||
| Description: Constructible numbers form a subfield of the complex numbers. (Contributed by Thierry Arnoux, 5-Nov-2025.) |
| Ref | Expression |
|---|---|
| constrsdrg | ⊢ Constr ∈ (SubDRing‘ℂfld) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnfldfld 33287 | . . . . 5 ⊢ ℂfld ∈ Field | |
| 2 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → ℂfld ∈ Field) |
| 3 | 2 | flddrngd 20626 | . . 3 ⊢ (⊤ → ℂfld ∈ DivRing) |
| 4 | 3 | drngringd 20622 | . . . 4 ⊢ (⊤ → ℂfld ∈ Ring) |
| 5 | 3 | drnggrpd 20623 | . . . . 5 ⊢ (⊤ → ℂfld ∈ Grp) |
| 6 | simpr 484 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ Constr) | |
| 7 | 6 | constrcn 33723 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ ℂ) |
| 8 | 7 | ex 412 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ Constr → 𝑥 ∈ ℂ)) |
| 9 | 8 | ssrdv 3949 | . . . . 5 ⊢ (⊤ → Constr ⊆ ℂ) |
| 10 | 1zzd 12540 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ) | |
| 11 | 10 | zconstr 33727 | . . . . . 6 ⊢ (⊤ → 1 ∈ Constr) |
| 12 | 11 | ne0d 4301 | . . . . 5 ⊢ (⊤ → Constr ≠ ∅) |
| 13 | simplr 768 | . . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → 𝑥 ∈ Constr) | |
| 14 | simpr 484 | . . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → 𝑦 ∈ Constr) | |
| 15 | 13, 14 | constraddcl 33725 | . . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 + 𝑦) ∈ Constr) |
| 16 | 15 | ralrimiva 3125 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr) |
| 17 | cnfldneg 21283 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((invg‘ℂfld)‘𝑥) = -𝑥) | |
| 18 | 7, 17 | syl 17 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ((invg‘ℂfld)‘𝑥) = -𝑥) |
| 19 | 6 | constrnegcl 33726 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → -𝑥 ∈ Constr) |
| 20 | 18, 19 | eqeltrd 2828 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ((invg‘ℂfld)‘𝑥) ∈ Constr) |
| 21 | 16, 20 | jca 511 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → (∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr ∧ ((invg‘ℂfld)‘𝑥) ∈ Constr)) |
| 22 | 21 | ralrimiva 3125 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ Constr (∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr ∧ ((invg‘ℂfld)‘𝑥) ∈ Constr)) |
| 23 | cnfldbas 21244 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
| 24 | cnfldadd 21246 | . . . . . . 7 ⊢ + = (+g‘ℂfld) | |
| 25 | eqid 2729 | . . . . . . 7 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
| 26 | 23, 24, 25 | issubg2 19049 | . . . . . 6 ⊢ (ℂfld ∈ Grp → (Constr ∈ (SubGrp‘ℂfld) ↔ (Constr ⊆ ℂ ∧ Constr ≠ ∅ ∧ ∀𝑥 ∈ Constr (∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr ∧ ((invg‘ℂfld)‘𝑥) ∈ Constr)))) |
| 27 | 26 | biimpar 477 | . . . . 5 ⊢ ((ℂfld ∈ Grp ∧ (Constr ⊆ ℂ ∧ Constr ≠ ∅ ∧ ∀𝑥 ∈ Constr (∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr ∧ ((invg‘ℂfld)‘𝑥) ∈ Constr))) → Constr ∈ (SubGrp‘ℂfld)) |
| 28 | 5, 9, 12, 22, 27 | syl13anc 1374 | . . . 4 ⊢ (⊤ → Constr ∈ (SubGrp‘ℂfld)) |
| 29 | 13, 14 | constrmulcl 33734 | . . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 · 𝑦) ∈ Constr) |
| 30 | 29 | anasss 466 | . . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ Constr ∧ 𝑦 ∈ Constr)) → (𝑥 · 𝑦) ∈ Constr) |
| 31 | 30 | ralrimivva 3178 | . . . 4 ⊢ (⊤ → ∀𝑥 ∈ Constr ∀𝑦 ∈ Constr (𝑥 · 𝑦) ∈ Constr) |
| 32 | cnfld1 21281 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
| 33 | cnfldmul 21248 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 34 | 23, 32, 33 | issubrg2 20477 | . . . . 5 ⊢ (ℂfld ∈ Ring → (Constr ∈ (SubRing‘ℂfld) ↔ (Constr ∈ (SubGrp‘ℂfld) ∧ 1 ∈ Constr ∧ ∀𝑥 ∈ Constr ∀𝑦 ∈ Constr (𝑥 · 𝑦) ∈ Constr))) |
| 35 | 34 | biimpar 477 | . . . 4 ⊢ ((ℂfld ∈ Ring ∧ (Constr ∈ (SubGrp‘ℂfld) ∧ 1 ∈ Constr ∧ ∀𝑥 ∈ Constr ∀𝑦 ∈ Constr (𝑥 · 𝑦) ∈ Constr)) → Constr ∈ (SubRing‘ℂfld)) |
| 36 | 4, 28, 11, 31, 35 | syl13anc 1374 | . . 3 ⊢ (⊤ → Constr ∈ (SubRing‘ℂfld)) |
| 37 | simpr 484 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ (Constr ∖ {0})) | |
| 38 | 37 | eldifad 3923 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ Constr) |
| 39 | 38 | constrcn 33723 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ ℂ) |
| 40 | eldifsni 4750 | . . . . . . 7 ⊢ (𝑥 ∈ (Constr ∖ {0}) → 𝑥 ≠ 0) | |
| 41 | 40 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ≠ 0) |
| 42 | cnfldinv 21290 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) | |
| 43 | 39, 41, 42 | syl2anc 584 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) |
| 44 | 38, 41 | constrinvcl 33736 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → (1 / 𝑥) ∈ Constr) |
| 45 | 43, 44 | eqeltrd 2828 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → ((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 46 | 45 | ralrimiva 3125 | . . 3 ⊢ (⊤ → ∀𝑥 ∈ (Constr ∖ {0})((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 47 | eqid 2729 | . . . 4 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
| 48 | cnfld0 21280 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
| 49 | 47, 48 | issdrg2 20680 | . . 3 ⊢ (Constr ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ Constr ∈ (SubRing‘ℂfld) ∧ ∀𝑥 ∈ (Constr ∖ {0})((invr‘ℂfld)‘𝑥) ∈ Constr)) |
| 50 | 3, 36, 46, 49 | syl3anbrc 1344 | . 2 ⊢ (⊤ → Constr ∈ (SubDRing‘ℂfld)) |
| 51 | 50 | mptru 1547 | 1 ⊢ Constr ∈ (SubDRing‘ℂfld) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4292 {csn 4585 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 0cc0 11044 1c1 11045 + caddc 11047 · cmul 11049 -cneg 11382 / cdiv 11811 Grpcgrp 18841 invgcminusg 18842 SubGrpcsubg 19028 Ringcrg 20118 invrcinvr 20272 SubRingcsubrg 20454 DivRingcdr 20614 Fieldcfield 20615 SubDRingcsdrg 20671 ℂfldccnfld 21240 Constrcconstr 33692 |
| 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 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124 |
| 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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-ioo 13286 df-ioc 13287 df-ico 13288 df-icc 13289 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-fac 14215 df-bc 14244 df-hash 14272 df-shft 15009 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-limsup 15413 df-clim 15430 df-rlim 15431 df-sum 15629 df-ef 16009 df-sin 16011 df-cos 16012 df-pi 16014 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-mulg 18976 df-subg 19031 df-cntz 19225 df-cmn 19688 df-abl 19689 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-cring 20121 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-subrng 20431 df-subrg 20455 df-drng 20616 df-field 20617 df-sdrg 20672 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744 df-log 26441 df-constr 33693 |
| This theorem is referenced by: constrfld 33739 |
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