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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 20541 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
3 | cnfldadd 20542 | . . . 4 ⊢ + = (+g‘ℂfld) | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘ℂfld)) |
5 | cnfldmul 20543 | . . . 4 ⊢ · = (.r‘ℂfld) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → · = (.r‘ℂfld)) |
7 | addcl 10611 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
8 | addass 10616 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
9 | 0cn 10625 | . . . . 5 ⊢ 0 ∈ ℂ | |
10 | addid2 10815 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
11 | negcl 10878 | . . . . 5 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
12 | addcom 10818 | . . . . . . 7 ⊢ ((-𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) | |
13 | 11, 12 | mpancom 686 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
14 | negid 10925 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
15 | 13, 14 | eqtrd 2854 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = 0) |
16 | 1, 3, 7, 8, 9, 10, 11, 15 | isgrpi 18118 | . . . 4 ⊢ ℂfld ∈ Grp |
17 | 16 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Grp) |
18 | mulcl 10613 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
19 | 18 | 3adant1 1125 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
20 | mulass 10617 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) | |
21 | 20 | adantl 484 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 · 𝑦) · 𝑧) = (𝑥 · (𝑦 · 𝑧))) |
22 | adddi 10618 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) | |
23 | 22 | adantl 484 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → (𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧))) |
24 | adddir 10624 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) | |
25 | 24 | adantl 484 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ)) → ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧))) |
26 | 1cnd 10628 | . . 3 ⊢ (⊤ → 1 ∈ ℂ) | |
27 | mulid2 10632 | . . . 4 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
28 | 27 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (1 · 𝑥) = 𝑥) |
29 | mulid1 10631 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 · 1) = 𝑥) | |
30 | 29 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥 · 1) = 𝑥) |
31 | mulcom 10615 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) | |
32 | 31 | 3adant1 1125 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) |
33 | 2, 4, 6, 17, 19, 21, 23, 25, 26, 28, 30, 32 | iscrngd 19328 | . 2 ⊢ (⊤ → ℂfld ∈ CRing) |
34 | 33 | mptru 1538 | 1 ⊢ ℂfld ∈ CRing |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1082 = wceq 1531 ⊤wtru 1532 ∈ wcel 2108 ‘cfv 6348 (class class class)co 7148 ℂcc 10527 0cc0 10529 1c1 10530 + caddc 10532 · cmul 10534 -cneg 10863 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Grpcgrp 18095 CRingccrg 19290 ℂfldccnfld 20537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-addf 10608 ax-mulf 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-cmn 18900 df-mgp 19232 df-ring 19291 df-cring 19292 df-cnfld 20538 |
This theorem is referenced by: cnring 20559 cnmgpabl 20598 zringcrng 20611 zring0 20619 re0g 20748 refld 20755 smadiadetr 21276 plypf1 24794 amgmlem 25559 amgm 25560 wilthlem2 25638 wilthlem3 25639 gzcrng 30905 ccfldextrr 31031 2zrng0 44200 amgmwlem 44894 amgmlemALT 44895 |
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