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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 33429 | . . . . 5 ⊢ ℂfld ∈ Field | |
| 2 | 1 | a1i 11 | . . . 4 ⊢ (⊤ → ℂfld ∈ Field) |
| 3 | 2 | flddrngd 20717 | . . 3 ⊢ (⊤ → ℂfld ∈ DivRing) |
| 4 | 3 | drngringd 20713 | . . . 4 ⊢ (⊤ → ℂfld ∈ Ring) |
| 5 | 3 | drnggrpd 20714 | . . . . 5 ⊢ (⊤ → ℂfld ∈ Grp) |
| 6 | simpr 486 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ Constr) | |
| 7 | 6 | constrcn 33956 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → 𝑥 ∈ ℂ) |
| 8 | 7 | ex 414 | . . . . . 6 ⊢ (⊤ → (𝑥 ∈ Constr → 𝑥 ∈ ℂ)) |
| 9 | 8 | ssrdv 3923 | . . . . 5 ⊢ (⊤ → Constr ⊆ ℂ) |
| 10 | 1zzd 12553 | . . . . . . 7 ⊢ (⊤ → 1 ∈ ℤ) | |
| 11 | 10 | zconstr 33960 | . . . . . 6 ⊢ (⊤ → 1 ∈ Constr) |
| 12 | 11 | ne0d 4273 | . . . . 5 ⊢ (⊤ → Constr ≠ ∅) |
| 13 | simplr 775 | . . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → 𝑥 ∈ Constr) | |
| 14 | simpr 486 | . . . . . . . . 9 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → 𝑦 ∈ Constr) | |
| 15 | 13, 14 | constraddcl 33958 | . . . . . . . 8 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 + 𝑦) ∈ Constr) |
| 16 | 15 | ralrimiva 3133 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr) |
| 17 | cnfldneg 21377 | . . . . . . . . 9 ⊢ (𝑥 ∈ ℂ → ((invg‘ℂfld)‘𝑥) = -𝑥) | |
| 18 | 7, 17 | syl 17 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ((invg‘ℂfld)‘𝑥) = -𝑥) |
| 19 | 6 | constrnegcl 33959 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → -𝑥 ∈ Constr) |
| 20 | 18, 19 | eqeltrd 2841 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → ((invg‘ℂfld)‘𝑥) ∈ Constr) |
| 21 | 16, 20 | jca 517 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ Constr) → (∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr ∧ ((invg‘ℂfld)‘𝑥) ∈ Constr)) |
| 22 | 21 | ralrimiva 3133 | . . . . 5 ⊢ (⊤ → ∀𝑥 ∈ Constr (∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr ∧ ((invg‘ℂfld)‘𝑥) ∈ Constr)) |
| 23 | cnfldbas 21355 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
| 24 | cnfldadd 21357 | . . . . . . 7 ⊢ + = (+g‘ℂfld) | |
| 25 | eqid 2741 | . . . . . . 7 ⊢ (invg‘ℂfld) = (invg‘ℂfld) | |
| 26 | 23, 24, 25 | issubg2 19112 | . . . . . 6 ⊢ (ℂfld ∈ Grp → (Constr ∈ (SubGrp‘ℂfld) ↔ (Constr ⊆ ℂ ∧ Constr ≠ ∅ ∧ ∀𝑥 ∈ Constr (∀𝑦 ∈ Constr (𝑥 + 𝑦) ∈ Constr ∧ ((invg‘ℂfld)‘𝑥) ∈ Constr)))) |
| 27 | 26 | biimpar 479 | . . . . 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 1381 | . . . 4 ⊢ (⊤ → Constr ∈ (SubGrp‘ℂfld)) |
| 29 | 13, 14 | constrmulcl 33967 | . . . . . 6 ⊢ (((⊤ ∧ 𝑥 ∈ Constr) ∧ 𝑦 ∈ Constr) → (𝑥 · 𝑦) ∈ Constr) |
| 30 | 29 | anasss 468 | . . . . 5 ⊢ ((⊤ ∧ (𝑥 ∈ Constr ∧ 𝑦 ∈ Constr)) → (𝑥 · 𝑦) ∈ Constr) |
| 31 | 30 | ralrimivva 3184 | . . . 4 ⊢ (⊤ → ∀𝑥 ∈ Constr ∀𝑦 ∈ Constr (𝑥 · 𝑦) ∈ Constr) |
| 32 | cnfld1 21376 | . . . . . 6 ⊢ 1 = (1r‘ℂfld) | |
| 33 | cnfldmul 21359 | . . . . . 6 ⊢ · = (.r‘ℂfld) | |
| 34 | 23, 32, 33 | issubrg2 20568 | . . . . 5 ⊢ (ℂfld ∈ Ring → (Constr ∈ (SubRing‘ℂfld) ↔ (Constr ∈ (SubGrp‘ℂfld) ∧ 1 ∈ Constr ∧ ∀𝑥 ∈ Constr ∀𝑦 ∈ Constr (𝑥 · 𝑦) ∈ Constr))) |
| 35 | 34 | biimpar 479 | . . . 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 1381 | . . 3 ⊢ (⊤ → Constr ∈ (SubRing‘ℂfld)) |
| 37 | simpr 486 | . . . . . . . 8 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ (Constr ∖ {0})) | |
| 38 | 37 | eldifad 3897 | . . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ Constr) |
| 39 | 38 | constrcn 33956 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ∈ ℂ) |
| 40 | eldifsni 4726 | . . . . . . 7 ⊢ (𝑥 ∈ (Constr ∖ {0}) → 𝑥 ≠ 0) | |
| 41 | 40 | adantl 483 | . . . . . 6 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → 𝑥 ≠ 0) |
| 42 | cnfldinv 21382 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) | |
| 43 | 39, 41, 42 | syl2anc 591 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → ((invr‘ℂfld)‘𝑥) = (1 / 𝑥)) |
| 44 | 38, 41 | constrinvcl 33969 | . . . . 5 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → (1 / 𝑥) ∈ Constr) |
| 45 | 43, 44 | eqeltrd 2841 | . . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ (Constr ∖ {0})) → ((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 46 | 45 | ralrimiva 3133 | . . 3 ⊢ (⊤ → ∀𝑥 ∈ (Constr ∖ {0})((invr‘ℂfld)‘𝑥) ∈ Constr) |
| 47 | eqid 2741 | . . . 4 ⊢ (invr‘ℂfld) = (invr‘ℂfld) | |
| 48 | cnfld0 21375 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
| 49 | 47, 48 | issdrg2 20771 | . . 3 ⊢ (Constr ∈ (SubDRing‘ℂfld) ↔ (ℂfld ∈ DivRing ∧ Constr ∈ (SubRing‘ℂfld) ∧ ∀𝑥 ∈ (Constr ∖ {0})((invr‘ℂfld)‘𝑥) ∈ Constr)) |
| 50 | 3, 36, 46, 49 | syl3anbrc 1351 | . 2 ⊢ (⊤ → Constr ∈ (SubDRing‘ℂfld)) |
| 51 | 50 | mptru 1555 | 1 ⊢ Constr ∈ (SubDRing‘ℂfld) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 397 ∧ w3a 1093 = wceq 1548 ⊤wtru 1549 ∈ wcel 2121 ≠ wne 2936 ∀wral 3055 ∖ cdif 3882 ⊆ wss 3885 ∅c0 4264 {csn 4558 ‘cfv 6489 (class class class)co 7360 ℂcc 11031 0cc0 11033 1c1 11034 + caddc 11036 · cmul 11038 -cneg 11373 / cdiv 11802 Grpcgrp 18904 invgcminusg 18905 SubGrpcsubg 19091 Ringcrg 20209 invrcinvr 20362 SubRingcsubrg 20545 DivRingcdr 20705 Fieldcfield 20706 SubDRingcsdrg 20762 ℂfldccnfld 21351 Constrcconstr 33925 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111 ax-addf 11112 ax-mulf 11113 |
| This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 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 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8105 df-tpos 8170 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-fi 9318 df-sup 9349 df-inf 9350 df-oi 9419 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-z 12520 df-dec 12640 df-uz 12784 df-q 12894 df-rp 12938 df-xneg 13058 df-xadd 13059 df-xmul 13060 df-ioo 13297 df-ioc 13298 df-ico 13299 df-icc 13300 df-fz 13457 df-fzo 13604 df-fl 13746 df-mod 13824 df-seq 13959 df-exp 14019 df-fac 14231 df-bc 14260 df-hash 14288 df-shft 15024 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-sum 15644 df-ef 16027 df-sin 16029 df-cos 16030 df-pi 16032 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-rest 17380 df-topn 17381 df-0g 17399 df-gsum 17400 df-topgen 17401 df-pt 17402 df-prds 17405 df-xrs 17461 df-qtop 17466 df-imas 17467 df-xps 17469 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-grp 18907 df-minusg 18908 df-mulg 19039 df-subg 19094 df-cntz 19287 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-cring 20212 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-subrng 20522 df-subrg 20546 df-drng 20707 df-field 20708 df-sdrg 20763 df-psmet 21343 df-xmet 21344 df-met 21345 df-bl 21346 df-mopn 21347 df-fbas 21348 df-fg 21349 df-cnfld 21352 df-top 22881 df-topon 22898 df-topsp 22920 df-bases 22933 df-cld 23006 df-ntr 23007 df-cls 23008 df-nei 23085 df-lp 23123 df-perf 23124 df-cn 23214 df-cnp 23215 df-haus 23302 df-tx 23549 df-hmeo 23742 df-fil 23833 df-fm 23925 df-flim 23926 df-flf 23927 df-xms 24307 df-ms 24308 df-tms 24309 df-cncf 24867 df-limc 25855 df-dv 25856 df-log 26542 df-constr 33926 |
| This theorem is referenced by: constrfld 33972 |
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