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Mirrors > Home > MPE Home > Th. List > cnsrng | Structured version Visualization version GIF version |
Description: The complex numbers form a *-ring. (Contributed by Mario Carneiro, 6-Oct-2015.) |
Ref | Expression |
---|---|
cnsrng | ⊢ ℂfld ∈ *-Ring |
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 | cnfldcj 20552 | . . . 4 ⊢ ∗ = (*𝑟‘ℂfld) | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → ∗ = (*𝑟‘ℂfld)) |
9 | cnring 20567 | . . . 4 ⊢ ℂfld ∈ Ring | |
10 | 9 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Ring) |
11 | cjcl 14464 | . . . 4 ⊢ (𝑥 ∈ ℂ → (∗‘𝑥) ∈ ℂ) | |
12 | 11 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (∗‘𝑥) ∈ ℂ) |
13 | cjadd 14500 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 + 𝑦)) = ((∗‘𝑥) + (∗‘𝑦))) | |
14 | 13 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 + 𝑦)) = ((∗‘𝑥) + (∗‘𝑦))) |
15 | mulcom 10623 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) | |
16 | 15 | fveq2d 6674 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 · 𝑦)) = (∗‘(𝑦 · 𝑥))) |
17 | cjmul 14501 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (∗‘(𝑦 · 𝑥)) = ((∗‘𝑦) · (∗‘𝑥))) | |
18 | 17 | ancoms 461 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑦 · 𝑥)) = ((∗‘𝑦) · (∗‘𝑥))) |
19 | 16, 18 | eqtrd 2856 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 · 𝑦)) = ((∗‘𝑦) · (∗‘𝑥))) |
20 | 19 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 · 𝑦)) = ((∗‘𝑦) · (∗‘𝑥))) |
21 | cjcj 14499 | . . . 4 ⊢ (𝑥 ∈ ℂ → (∗‘(∗‘𝑥)) = 𝑥) | |
22 | 21 | adantl 484 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (∗‘(∗‘𝑥)) = 𝑥) |
23 | 2, 4, 6, 8, 10, 12, 14, 20, 22 | issrngd 19632 | . 2 ⊢ (⊤ → ℂfld ∈ *-Ring) |
24 | 23 | mptru 1544 | 1 ⊢ ℂfld ∈ *-Ring |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 + caddc 10540 · cmul 10542 ∗ccj 14455 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 *𝑟cstv 16567 Ringcrg 19297 *-Ringcsr 19615 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-rep 5190 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-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 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-div 11298 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-cj 14458 df-re 14459 df-im 14460 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-mhm 17956 df-grp 18106 df-ghm 18356 df-cmn 18908 df-mgp 19240 df-ur 19252 df-ring 19299 df-cring 19300 df-oppr 19373 df-rnghom 19467 df-staf 19616 df-srng 19617 df-cnfld 20546 |
This theorem is referenced by: (None) |
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