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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 33351 | . . . . 5 ⊢ ℂfld ∈ Field | |
| 2 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → ℂfld ∈ Field) |
| 3 | 2 | flddrngd 20665 | . . 3 ⊢ (⊤ → ℂfld ∈ DivRing) |
| 4 | 3 | drngringd 20661 | . . . 4 ⊢ (⊤ → ℂfld ∈ Ring) |
| 5 | 3 | drnggrpd 20662 | . . . . 5 ⊢ (⊤ → ℂfld ∈ Grp) |
| 6 | simpr 484 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ Constr) | |
| 7 | 6 | constrcn 33845 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ ℂ) |
| 8 | 7 | ex 412 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ Constr → 𝑥 ∈ ℂ)) |
| 9 | 8 | ssrdv 3936 | . . . . 5 ⊢ (⊤ → Constr ⊆ ℂ) |
| 10 | 1zzd 12513 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ) | |
| 11 | 10 | zconstr 33849 | . . . . . 6 ⊢ (⊤ → 1 ∈ Constr) |
| 12 | 11 | ne0d 4291 | . . . . 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 33847 | . . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 + 𝑦) ∈ Constr) |
| 16 | 15 | ralrimiva 3125 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr) |
| 17 | cnfldneg 21341 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((invg‘ℂfld)‘𝑥) = -𝑥) | |
| 18 | 7, 17 | syl 17 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ((invg‘ℂfld)‘𝑥) = -𝑥) |
| 19 | 6 | constrnegcl 33848 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → -𝑥 ∈ Constr) |
| 20 | 18, 19 | eqeltrd 2833 | . . . . . . 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 21304 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
| 24 | cnfldadd 21306 | . . . . . . 7 ⊢ + = (+g‘ℂfld) | |
| 25 | eqid 2733 | . . . . . . 7 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
| 26 | 23, 24, 25 | issubg2 19062 | . . . . . 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 33856 | . . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 · 𝑦) ∈ Constr) |
| 30 | 29 | anasss 466 | . . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ Constr ∧ 𝑦 ∈ Constr)) → (𝑥 · 𝑦) ∈ Constr) |
| 31 | 30 | ralrimivva 3176 | . . . 4 ⊢ (⊤ → ∀𝑥 ∈ Constr ∀𝑦 ∈ Constr (𝑥 · 𝑦) ∈ Constr) |
| 32 | cnfld1 21339 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
| 33 | cnfldmul 21308 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 34 | 23, 32, 33 | issubrg2 20516 | . . . . 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 3910 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ Constr) |
| 39 | 38 | constrcn 33845 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ ℂ) |
| 40 | eldifsni 4743 | . . . . . . 7 ⊢ (𝑥 ∈ (Constr ∖ {0}) → 𝑥 ≠ 0) | |
| 41 | 40 | adantl 481 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ≠ 0) |
| 42 | cnfldinv 21348 | . . . . . 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 33858 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → (1 / 𝑥) ∈ Constr) |
| 45 | 43, 44 | eqeltrd 2833 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → ((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 46 | 45 | ralrimiva 3125 | . . 3 ⊢ (⊤ → ∀𝑥 ∈ (Constr ∖ {0})((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 47 | eqid 2733 | . . . 4 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
| 48 | cnfld0 21338 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
| 49 | 47, 48 | issdrg2 20719 | . . 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 2113 ≠ wne 2929 ∀wral 3048 ∖ cdif 3895 ⊆ wss 3898 ∅c0 4282 {csn 4577 ‘cfv 6489 (class class class)co 7355 ℂcc 11015 0cc0 11017 1c1 11018 + caddc 11020 · cmul 11022 -cneg 11356 / cdiv 11785 Grpcgrp 18854 invgcminusg 18855 SubGrpcsubg 19041 Ringcrg 20159 invrcinvr 20314 SubRingcsubrg 20493 DivRingcdr 20653 Fieldcfield 20654 SubDRingcsdrg 20710 ℂfldccnfld 21300 Constrcconstr 33814 |
| 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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096 ax-mulf 11097 |
| 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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-tpos 8165 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ioc 13257 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-fl 13703 df-mod 13781 df-seq 13916 df-exp 13976 df-fac 14188 df-bc 14217 df-hash 14245 df-shft 14981 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-limsup 15385 df-clim 15402 df-rlim 15403 df-sum 15601 df-ef 15981 df-sin 15983 df-cos 15984 df-pi 15986 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17414 df-qtop 17419 df-imas 17420 df-xps 17422 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-grp 18857 df-minusg 18858 df-mulg 18989 df-subg 19044 df-cntz 19237 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-ring 20161 df-cring 20162 df-oppr 20264 df-dvdsr 20284 df-unit 20285 df-invr 20315 df-dvr 20328 df-subrng 20470 df-subrg 20494 df-drng 20655 df-field 20656 df-sdrg 20711 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-fbas 21297 df-fg 21298 df-cnfld 21301 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-cld 22954 df-ntr 22955 df-cls 22956 df-nei 23033 df-lp 23071 df-perf 23072 df-cn 23162 df-cnp 23163 df-haus 23250 df-tx 23497 df-hmeo 23690 df-fil 23781 df-fm 23873 df-flim 23874 df-flf 23875 df-xms 24255 df-ms 24256 df-tms 24257 df-cncf 24818 df-limc 25814 df-dv 25815 df-log 26512 df-constr 33815 |
| This theorem is referenced by: constrfld 33861 |
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