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Mirrors > Home > MPE Home > Th. List > cncrng | Structured version Visualization version GIF version |
Description: The complex numbers form a commutative ring. (Contributed by Mario Carneiro, 8-Jan-2015.) |
Ref | Expression |
---|---|
cncrng | ⊢ ℂfld ∈ CRing |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnfldbas 20321 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
3 | cnfldadd 20322 | . . . 4 ⊢ + = (+g‘ℂfld) | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘ℂfld)) |
5 | cnfldmul 20323 | . . . 4 ⊢ · = (.r‘ℂfld) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → · = (.r‘ℂfld)) |
7 | addcl 10776 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
8 | addass 10781 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
9 | 0cn 10790 | . . . . 5 ⊢ 0 ∈ ℂ | |
10 | addid2 10980 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
11 | negcl 11043 | . . . . 5 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
12 | addcom 10983 | . . . . . . 7 ⊢ ((-𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) | |
13 | 11, 12 | mpancom 688 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
14 | negid 11090 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
15 | 13, 14 | eqtrd 2771 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = 0) |
16 | 1, 3, 7, 8, 9, 10, 11, 15 | isgrpi 18344 | . . . 4 ⊢ ℂfld ∈ Grp |
17 | 16 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Grp) |
18 | mulcl 10778 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
19 | 18 | 3adant1 1132 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
20 | mulass 10782 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) | |
21 | 20 | adantl 485 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
22 | adddi 10783 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) | |
23 | 22 | adantl 485 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
24 | adddir 10789 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) | |
25 | 24 | adantl 485 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
26 | 1cnd 10793 | . . 3 ⊢ (⊤ → 1 ∈ ℂ) | |
27 | mulid2 10797 | . . . 4 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
28 | 27 | adantl 485 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (1 · 𝑥) = 𝑥) |
29 | mulid1 10796 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 · 1) = 𝑥) | |
30 | 29 | adantl 485 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥 · 1) = 𝑥) |
31 | mulcom 10780 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) | |
32 | 31 | 3adant1 1132 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) |
33 | 2, 4, 6, 17, 19, 21, 23, 25, 26, 28, 30, 32 | iscrngd 19558 | . 2 ⊢ (⊤ → ℂfld ∈ CRing) |
34 | 33 | mptru 1550 | 1 ⊢ ℂfld ∈ CRing |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1089 = wceq 1543 ⊤wtru 1544 ∈ wcel 2112 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 1c1 10695 + caddc 10697 · cmul 10699 -cneg 11028 Basecbs 16666 +gcplusg 16749 .rcmulr 16750 Grpcgrp 18319 CRingccrg 19517 ℂfldccnfld 20317 |
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 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-plusg 16762 df-mulr 16763 df-starv 16764 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-0g 16900 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-grp 18322 df-cmn 19126 df-mgp 19459 df-ring 19518 df-cring 19519 df-cnfld 20318 |
This theorem is referenced by: cnring 20339 cnmgpabl 20378 zringcrng 20391 zring0 20399 re0g 20528 refld 20535 smadiadetr 21526 plypf1 25060 amgmlem 25826 amgm 25827 wilthlem2 25905 wilthlem3 25906 gzcrng 31211 ccfldextrr 31391 2zrng0 45112 amgmwlem 46120 amgmlemALT 46121 |
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