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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 33320 | . . . . 5 ⊢ ℂfld ∈ Field | |
| 2 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → ℂfld ∈ Field) |
| 3 | 2 | flddrngd 20656 | . . 3 ⊢ (⊤ → ℂfld ∈ DivRing) |
| 4 | 3 | drngringd 20652 | . . . 4 ⊢ (⊤ → ℂfld ∈ Ring) |
| 5 | 3 | drnggrpd 20653 | . . . . 5 ⊢ (⊤ → ℂfld ∈ Grp) |
| 6 | simpr 484 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ Constr) | |
| 7 | 6 | constrcn 33756 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ ℂ) |
| 8 | 7 | ex 412 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ Constr → 𝑥 ∈ ℂ)) |
| 9 | 8 | ssrdv 3954 | . . . . 5 ⊢ (⊤ → Constr ⊆ ℂ) |
| 10 | 1zzd 12570 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ) | |
| 11 | 10 | zconstr 33760 | . . . . . 6 ⊢ (⊤ → 1 ∈ Constr) |
| 12 | 11 | ne0d 4307 | . . . . 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 33758 | . . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 + 𝑦) ∈ Constr) |
| 16 | 15 | ralrimiva 3126 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr) |
| 17 | cnfldneg 21313 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((invg‘ℂfld)‘𝑥) = -𝑥) | |
| 18 | 7, 17 | syl 17 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ((invg‘ℂfld)‘𝑥) = -𝑥) |
| 19 | 6 | constrnegcl 33759 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → -𝑥 ∈ Constr) |
| 20 | 18, 19 | eqeltrd 2829 | . . . . . . 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 3126 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ Constr (∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr ∧ ((invg‘ℂfld)‘𝑥) ∈ Constr)) |
| 23 | cnfldbas 21274 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
| 24 | cnfldadd 21276 | . . . . . . 7 ⊢ + = (+g‘ℂfld) | |
| 25 | eqid 2730 | . . . . . . 7 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
| 26 | 23, 24, 25 | issubg2 19079 | . . . . . 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 33767 | . . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 · 𝑦) ∈ Constr) |
| 30 | 29 | anasss 466 | . . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ Constr ∧ 𝑦 ∈ Constr)) → (𝑥 · 𝑦) ∈ Constr) |
| 31 | 30 | ralrimivva 3181 | . . . 4 ⊢ (⊤ → ∀𝑥 ∈ Constr ∀𝑦 ∈ Constr (𝑥 · 𝑦) ∈ Constr) |
| 32 | cnfld1 21311 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
| 33 | cnfldmul 21278 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 34 | 23, 32, 33 | issubrg2 20507 | . . . . 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 3928 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ Constr) |
| 39 | 38 | constrcn 33756 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ ℂ) |
| 40 | eldifsni 4756 | . . . . . . 7 ⊢ (𝑥 ∈ (Constr ∖ {0}) → 𝑥 ≠ 0) | |
| 41 | 40 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ≠ 0) |
| 42 | cnfldinv 21320 | . . . . . 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 33769 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → (1 / 𝑥) ∈ Constr) |
| 45 | 43, 44 | eqeltrd 2829 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → ((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 46 | 45 | ralrimiva 3126 | . . 3 ⊢ (⊤ → ∀𝑥 ∈ (Constr ∖ {0})((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 47 | eqid 2730 | . . . 4 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
| 48 | cnfld0 21310 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
| 49 | 47, 48 | issdrg2 20710 | . . 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 2926 ∀wral 3045 ∖ cdif 3913 ⊆ wss 3916 ∅c0 4298 {csn 4591 ‘cfv 6513 (class class class)co 7389 ℂcc 11072 0cc0 11074 1c1 11075 + caddc 11077 · cmul 11079 -cneg 11412 / cdiv 11841 Grpcgrp 18871 invgcminusg 18872 SubGrpcsubg 19058 Ringcrg 20148 invrcinvr 20302 SubRingcsubrg 20484 DivRingcdr 20644 Fieldcfield 20645 SubDRingcsdrg 20701 ℂfldccnfld 21270 Constrcconstr 33725 |
| 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 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-inf2 9600 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152 ax-addf 11153 ax-mulf 11154 |
| 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 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-er 8673 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-fi 9368 df-sup 9399 df-inf 9400 df-oi 9469 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-z 12536 df-dec 12656 df-uz 12800 df-q 12914 df-rp 12958 df-xneg 13078 df-xadd 13079 df-xmul 13080 df-ioo 13316 df-ioc 13317 df-ico 13318 df-icc 13319 df-fz 13475 df-fzo 13622 df-fl 13760 df-mod 13838 df-seq 13973 df-exp 14033 df-fac 14245 df-bc 14274 df-hash 14302 df-shft 15039 df-cj 15071 df-re 15072 df-im 15073 df-sqrt 15207 df-abs 15208 df-limsup 15443 df-clim 15460 df-rlim 15461 df-sum 15659 df-ef 16039 df-sin 16041 df-cos 16042 df-pi 16044 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-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-xrs 17471 df-qtop 17476 df-imas 17477 df-xps 17479 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18717 df-grp 18874 df-minusg 18875 df-mulg 19006 df-subg 19061 df-cntz 19255 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-ring 20150 df-cring 20151 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-dvr 20316 df-subrng 20461 df-subrg 20485 df-drng 20646 df-field 20647 df-sdrg 20702 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-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 25773 df-dv 25774 df-log 26471 df-constr 33726 |
| This theorem is referenced by: constrfld 33772 |
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