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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 19873 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
2 | 1 | a1i 11 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
3 | cnfldadd 19874 | . . . 4 ⊢ + = (+g‘ℂfld) | |
4 | 3 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘ℂfld)) |
5 | cnfldmul 19875 | . . . 4 ⊢ · = (.r‘ℂfld) | |
6 | 5 | a1i 11 | . . 3 ⊢ (⊤ → · = (.r‘ℂfld)) |
7 | cnfldcj 19876 | . . . 4 ⊢ ∗ = (*𝑟‘ℂfld) | |
8 | 7 | a1i 11 | . . 3 ⊢ (⊤ → ∗ = (*𝑟‘ℂfld)) |
9 | cnring 19891 | . . . 4 ⊢ ℂfld ∈ Ring | |
10 | 9 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Ring) |
11 | cjcl 13965 | . . . 4 ⊢ (𝑥 ∈ ℂ → (∗‘𝑥) ∈ ℂ) | |
12 | 11 | adantl 473 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (∗‘𝑥) ∈ ℂ) |
13 | cjadd 14001 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 + 𝑦)) = ((∗‘𝑥) + (∗‘𝑦))) | |
14 | 13 | 3adant1 1122 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 + 𝑦)) = ((∗‘𝑥) + (∗‘𝑦))) |
15 | mulcom 10135 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 · 𝑦) = (𝑦 · 𝑥)) | |
16 | 15 | fveq2d 6308 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 · 𝑦)) = (∗‘(𝑦 · 𝑥))) |
17 | cjmul 14002 | . . . . . 6 ⊢ ((𝑦 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (∗‘(𝑦 · 𝑥)) = ((∗‘𝑦) · (∗‘𝑥))) | |
18 | 17 | ancoms 468 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑦 · 𝑥)) = ((∗‘𝑦) · (∗‘𝑥))) |
19 | 16, 18 | eqtrd 2758 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 · 𝑦)) = ((∗‘𝑦) · (∗‘𝑥))) |
20 | 19 | 3adant1 1122 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (∗‘(𝑥 · 𝑦)) = ((∗‘𝑦) · (∗‘𝑥))) |
21 | cjcj 14000 | . . . 4 ⊢ (𝑥 ∈ ℂ → (∗‘(∗‘𝑥)) = 𝑥) | |
22 | 21 | adantl 473 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ) → (∗‘(∗‘𝑥)) = 𝑥) |
23 | 2, 4, 6, 8, 10, 12, 14, 20, 22 | issrngd 18984 | . 2 ⊢ (⊤ → ℂfld ∈ *-Ring) |
24 | 23 | trud 1606 | 1 ⊢ ℂfld ∈ *-Ring |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1596 ⊤wtru 1597 ∈ wcel 2103 ‘cfv 6001 (class class class)co 6765 ℂcc 10047 + caddc 10052 · cmul 10054 ∗ccj 13956 Basecbs 15980 +gcplusg 16064 .rcmulr 16065 *𝑟cstv 16066 Ringcrg 18668 *-Ringcsr 18967 ℂfldccnfld 19869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 ax-addf 10128 ax-mulf 10129 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-int 4584 df-iun 4630 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-om 7183 df-1st 7285 df-2nd 7286 df-tpos 7472 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-1o 7680 df-oadd 7684 df-er 7862 df-map 7976 df-en 8073 df-dom 8074 df-sdom 8075 df-fin 8076 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-nn 11134 df-2 11192 df-3 11193 df-4 11194 df-5 11195 df-6 11196 df-7 11197 df-8 11198 df-9 11199 df-n0 11406 df-z 11491 df-dec 11607 df-uz 11801 df-fz 12441 df-cj 13959 df-re 13960 df-im 13961 df-struct 15982 df-ndx 15983 df-slot 15984 df-base 15986 df-sets 15987 df-plusg 16077 df-mulr 16078 df-starv 16079 df-tset 16083 df-ple 16084 df-ds 16087 df-unif 16088 df-0g 16225 df-mgm 17364 df-sgrp 17406 df-mnd 17417 df-mhm 17457 df-grp 17547 df-ghm 17780 df-cmn 18316 df-mgp 18611 df-ur 18623 df-ring 18670 df-cring 18671 df-oppr 18744 df-rnghom 18838 df-staf 18968 df-srng 18969 df-cnfld 19870 |
This theorem is referenced by: (None) |
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