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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 33299 | . . . . 5 ⊢ ℂfld ∈ Field | |
| 2 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → ℂfld ∈ Field) |
| 3 | 2 | flddrngd 20651 | . . 3 ⊢ (⊤ → ℂfld ∈ DivRing) |
| 4 | 3 | drngringd 20647 | . . . 4 ⊢ (⊤ → ℂfld ∈ Ring) |
| 5 | 3 | drnggrpd 20648 | . . . . 5 ⊢ (⊤ → ℂfld ∈ Grp) |
| 6 | simpr 484 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ Constr) | |
| 7 | 6 | constrcn 33765 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ ℂ) |
| 8 | 7 | ex 412 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ Constr → 𝑥 ∈ ℂ)) |
| 9 | 8 | ssrdv 3935 | . . . . 5 ⊢ (⊤ → Constr ⊆ ℂ) |
| 10 | 1zzd 12498 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ) | |
| 11 | 10 | zconstr 33769 | . . . . . 6 ⊢ (⊤ → 1 ∈ Constr) |
| 12 | 11 | ne0d 4287 | . . . . 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 33767 | . . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 + 𝑦) ∈ Constr) |
| 16 | 15 | ralrimiva 3124 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr) |
| 17 | cnfldneg 21327 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((invg‘ℂfld)‘𝑥) = -𝑥) | |
| 18 | 7, 17 | syl 17 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ((invg‘ℂfld)‘𝑥) = -𝑥) |
| 19 | 6 | constrnegcl 33768 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → -𝑥 ∈ Constr) |
| 20 | 18, 19 | eqeltrd 2831 | . . . . . . 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 3124 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ Constr (∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr ∧ ((invg‘ℂfld)‘𝑥) ∈ Constr)) |
| 23 | cnfldbas 21290 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
| 24 | cnfldadd 21292 | . . . . . . 7 ⊢ + = (+g‘ℂfld) | |
| 25 | eqid 2731 | . . . . . . 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 33776 | . . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 · 𝑦) ∈ Constr) |
| 30 | 29 | anasss 466 | . . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ Constr ∧ 𝑦 ∈ Constr)) → (𝑥 · 𝑦) ∈ Constr) |
| 31 | 30 | ralrimivva 3175 | . . . 4 ⊢ (⊤ → ∀𝑥 ∈ Constr ∀𝑦 ∈ Constr (𝑥 · 𝑦) ∈ Constr) |
| 32 | cnfld1 21325 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
| 33 | cnfldmul 21294 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 34 | 23, 32, 33 | issubrg2 20502 | . . . . 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 3909 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ Constr) |
| 39 | 38 | constrcn 33765 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ ℂ) |
| 40 | eldifsni 4737 | . . . . . . 7 ⊢ (𝑥 ∈ (Constr ∖ {0}) → 𝑥 ≠ 0) | |
| 41 | 40 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ≠ 0) |
| 42 | cnfldinv 21334 | . . . . . 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 33778 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → (1 / 𝑥) ∈ Constr) |
| 45 | 43, 44 | eqeltrd 2831 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → ((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 46 | 45 | ralrimiva 3124 | . . 3 ⊢ (⊤ → ∀𝑥 ∈ (Constr ∖ {0})((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 47 | eqid 2731 | . . . 4 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
| 48 | cnfld0 21324 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
| 49 | 47, 48 | issdrg2 20705 | . . 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 1548 | 1 ⊢ Constr ∈ (SubDRing‘ℂfld) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∧ w3a 1086 = wceq 1541 ⊤wtru 1542 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∖ cdif 3894 ⊆ wss 3897 ∅c0 4278 {csn 4571 ‘cfv 6476 (class class class)co 7341 ℂcc 10999 0cc0 11001 1c1 11002 + caddc 11004 · cmul 11006 -cneg 11340 / cdiv 11769 Grpcgrp 18841 invgcminusg 18842 SubGrpcsubg 19028 Ringcrg 20146 invrcinvr 20300 SubRingcsubrg 20479 DivRingcdr 20639 Fieldcfield 20640 SubDRingcsdrg 20696 ℂfldccnfld 21286 Constrcconstr 33734 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5212 ax-sep 5229 ax-nul 5239 ax-pow 5298 ax-pr 5365 ax-un 7663 ax-inf2 9526 ax-cnex 11057 ax-resscn 11058 ax-1cn 11059 ax-icn 11060 ax-addcl 11061 ax-addrcl 11062 ax-mulcl 11063 ax-mulrcl 11064 ax-mulcom 11065 ax-addass 11066 ax-mulass 11067 ax-distr 11068 ax-i2m1 11069 ax-1ne0 11070 ax-1rid 11071 ax-rnegex 11072 ax-rrecex 11073 ax-cnre 11074 ax-pre-lttri 11075 ax-pre-lttrn 11076 ax-pre-ltadd 11077 ax-pre-mulgt0 11078 ax-pre-sup 11079 ax-addf 11080 ax-mulf 11081 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4279 df-if 4471 df-pw 4547 df-sn 4572 df-pr 4574 df-tp 4576 df-op 4578 df-uni 4855 df-int 4893 df-iun 4938 df-iin 4939 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5506 df-eprel 5511 df-po 5519 df-so 5520 df-fr 5564 df-se 5565 df-we 5566 df-xp 5617 df-rel 5618 df-cnv 5619 df-co 5620 df-dm 5621 df-rn 5622 df-res 5623 df-ima 5624 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-of 7605 df-om 7792 df-1st 7916 df-2nd 7917 df-supp 8086 df-tpos 8151 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8617 df-map 8747 df-pm 8748 df-ixp 8817 df-en 8865 df-dom 8866 df-sdom 8867 df-fin 8868 df-fsupp 9241 df-fi 9290 df-sup 9321 df-inf 9322 df-oi 9391 df-card 9827 df-pnf 11143 df-mnf 11144 df-xr 11145 df-ltxr 11146 df-le 11147 df-sub 11341 df-neg 11342 df-div 11770 df-nn 12121 df-2 12183 df-3 12184 df-4 12185 df-5 12186 df-6 12187 df-7 12188 df-8 12189 df-9 12190 df-n0 12377 df-z 12464 df-dec 12584 df-uz 12728 df-q 12842 df-rp 12886 df-xneg 13006 df-xadd 13007 df-xmul 13008 df-ioo 13244 df-ioc 13245 df-ico 13246 df-icc 13247 df-fz 13403 df-fzo 13550 df-fl 13691 df-mod 13769 df-seq 13904 df-exp 13964 df-fac 14176 df-bc 14205 df-hash 14233 df-shft 14969 df-cj 15001 df-re 15002 df-im 15003 df-sqrt 15137 df-abs 15138 df-limsup 15373 df-clim 15390 df-rlim 15391 df-sum 15589 df-ef 15969 df-sin 15971 df-cos 15972 df-pi 15974 df-struct 17053 df-sets 17070 df-slot 17088 df-ndx 17100 df-base 17116 df-ress 17137 df-plusg 17169 df-mulr 17170 df-starv 17171 df-sca 17172 df-vsca 17173 df-ip 17174 df-tset 17175 df-ple 17176 df-ds 17178 df-unif 17179 df-hom 17180 df-cco 17181 df-rest 17321 df-topn 17322 df-0g 17340 df-gsum 17341 df-topgen 17342 df-pt 17343 df-prds 17346 df-xrs 17401 df-qtop 17406 df-imas 17407 df-xps 17409 df-mre 17483 df-mrc 17484 df-acs 17486 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 19224 df-cmn 19689 df-abl 19690 df-mgp 20054 df-rng 20066 df-ur 20095 df-ring 20148 df-cring 20149 df-oppr 20250 df-dvdsr 20270 df-unit 20271 df-invr 20301 df-dvr 20314 df-subrng 20456 df-subrg 20480 df-drng 20641 df-field 20642 df-sdrg 20697 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-fbas 21283 df-fg 21284 df-cnfld 21287 df-top 22804 df-topon 22821 df-topsp 22843 df-bases 22856 df-cld 22929 df-ntr 22930 df-cls 22931 df-nei 23008 df-lp 23046 df-perf 23047 df-cn 23137 df-cnp 23138 df-haus 23225 df-tx 23472 df-hmeo 23665 df-fil 23756 df-fm 23848 df-flim 23849 df-flf 23850 df-xms 24230 df-ms 24231 df-tms 24232 df-cncf 24793 df-limc 25789 df-dv 25790 df-log 26487 df-constr 33735 |
| This theorem is referenced by: constrfld 33781 |
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