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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 20549 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
3 | cnfldadd 20550 | . . . 4 ⊢ + = (+g‘ℂfld) | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘ℂfld)) |
5 | cnfldmul 20551 | . . . 4 ⊢ · = (.r‘ℂfld) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → · = (.r‘ℂfld)) |
7 | addcl 10619 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 + 𝑦) ∈ ℂ) | |
8 | addass 10624 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) | |
9 | 0cn 10633 | . . . . 5 ⊢ 0 ∈ ℂ | |
10 | addid2 10823 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (0 + 𝑥) = 𝑥) | |
11 | negcl 10886 | . . . . 5 ⊢ (𝑥 ∈ ℂ → -𝑥 ∈ ℂ) | |
12 | addcom 10826 | . . . . . . 7 ⊢ ((-𝑥 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) | |
13 | 11, 12 | mpancom 686 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = (𝑥 + -𝑥)) |
14 | negid 10933 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (𝑥 + -𝑥) = 0) | |
15 | 13, 14 | eqtrd 2856 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (-𝑥 + 𝑥) = 0) |
16 | 1, 3, 7, 8, 9, 10, 11, 15 | isgrpi 18126 | . . . 4 ⊢ ℂfld ∈ Grp |
17 | 16 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Grp) |
18 | mulcl 10621 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) | |
19 | 18 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) ∈ ℂ) |
20 | mulass 10625 | . . . 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 10626 | . . . 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 10632 | . . . 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 10636 | . . 3 ⊢ (⊤ → 1 ∈ ℂ) | |
27 | mulid2 10640 | . . . 4 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
28 | 27 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (1 · 𝑥) = 𝑥) |
29 | mulid1 10639 | . . . 4 ⊢ (𝑥 ∈ ℂ → (𝑥 · 1) = 𝑥) | |
30 | 29 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (𝑥 · 1) = 𝑥) |
31 | mulcom 10623 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) | |
32 | 31 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) |
33 | 2, 4, 6, 17, 19, 21, 23, 25, 26, 28, 30, 32 | iscrngd 19336 | . 2 ⊢ (⊤ → ℂfld ∈ CRing) |
34 | 33 | mptru 1544 | 1 ⊢ ℂfld ∈ CRing |
Colors of variables: wff setvar class |
Syntax hints: ∧ w3a 1083 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 -cneg 10871 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 Grpcgrp 18103 CRingccrg 19298 ℂfldccnfld 20545 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-cmn 18908 df-mgp 19240 df-ring 19299 df-cring 19300 df-cnfld 20546 |
This theorem is referenced by: cnring 20567 cnmgpabl 20606 zringcrng 20619 zring0 20627 re0g 20756 refld 20763 smadiadetr 21284 plypf1 24802 amgmlem 25567 amgm 25568 wilthlem2 25646 wilthlem3 25647 gzcrng 30912 ccfldextrr 31038 2zrng0 44229 amgmwlem 44923 amgmlemALT 44924 |
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