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Mirrors > Home > MPE Home > Th. List > cnfld1 | Structured version Visualization version GIF version |
Description: One is the unity element of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) |
Ref | Expression |
---|---|
cnfld1 | ⊢ 1 = (1r‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 11164 | . . . 4 ⊢ 1 ∈ ℂ | |
2 | mullid 11209 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (1 · 𝑥) = 𝑥) | |
3 | mulrid 11208 | . . . . . 6 ⊢ (𝑥 ∈ ℂ → (𝑥 · 1) = 𝑥) | |
4 | 2, 3 | jca 513 | . . . . 5 ⊢ (𝑥 ∈ ℂ → ((1 · 𝑥) = 𝑥 ∧ (𝑥 · 1) = 𝑥)) |
5 | 4 | rgen 3064 | . . . 4 ⊢ ∀𝑥 ∈ ℂ ((1 · 𝑥) = 𝑥 ∧ (𝑥 · 1) = 𝑥) |
6 | 1, 5 | pm3.2i 472 | . . 3 ⊢ (1 ∈ ℂ ∧ ∀𝑥 ∈ ℂ ((1 · 𝑥) = 𝑥 ∧ (𝑥 · 1) = 𝑥)) |
7 | cnring 20952 | . . . 4 ⊢ ℂfld ∈ Ring | |
8 | cnfldbas 20933 | . . . . 5 ⊢ ℂ = (Base‘ℂfld) | |
9 | cnfldmul 20935 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
10 | eqid 2733 | . . . . 5 ⊢ (1r‘ℂfld) = (1r‘ℂfld) | |
11 | 8, 9, 10 | isringid 20078 | . . . 4 ⊢ (ℂfld ∈ Ring → ((1 ∈ ℂ ∧ ∀𝑥 ∈ ℂ ((1 · 𝑥) = 𝑥 ∧ (𝑥 · 1) = 𝑥)) ↔ (1r‘ℂfld) = 1)) |
12 | 7, 11 | ax-mp 5 | . . 3 ⊢ ((1 ∈ ℂ ∧ ∀𝑥 ∈ ℂ ((1 · 𝑥) = 𝑥 ∧ (𝑥 · 1) = 𝑥)) ↔ (1r‘ℂfld) = 1) |
13 | 6, 12 | mpbi 229 | . 2 ⊢ (1r‘ℂfld) = 1 |
14 | 13 | eqcomi 2742 | 1 ⊢ 1 = (1r‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ‘cfv 6540 (class class class)co 7404 ℂcc 11104 1c1 11107 · cmul 11111 1rcur 19996 Ringcrg 20047 ℂfldccnfld 20929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185 ax-mulf 11186 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-cmn 19643 df-mgp 19980 df-ur 19997 df-ring 20049 df-cring 20050 df-cnfld 20930 |
This theorem is referenced by: cndrng 20959 cnfldinv 20961 cnfldexp 20963 cnsubrglem 20980 cnsubdrglem 20981 zsssubrg 20988 cnmgpid 20992 gzrngunitlem 20995 expmhm 20999 nn0srg 21000 rge0srg 21001 zring1 21013 re1r 21150 clm1 24571 isclmp 24595 cnlmod 24638 cphsubrglem 24676 taylply2 25862 efsubm 26042 amgmlem 26474 amgm 26475 wilthlem2 26553 wilthlem3 26554 dchrelbas3 26721 dchrzrh1 26727 dchrmulcl 26732 dchrn0 26733 dchrinvcl 26736 dchrfi 26738 dchrabs 26743 sumdchr2 26753 rpvmasum2 26995 qrng1 27105 psgnid 32234 cnmsgn0g 32283 altgnsg 32286 xrge0slmod 32432 fermltlchr 32447 znfermltl 32448 iistmd 32820 xrge0iifmhm 32857 cnsrexpcl 41840 rngunsnply 41848 proot1ex 41876 amgmwlem 47751 amgmlemALT 47752 |
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