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Mirrors > Home > MPE Home > Th. List > cnfldcj | Structured version Visualization version GIF version |
Description: The conjugation operation of the field of complex numbers. (Contributed by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Revised by Thierry Arnoux, 17-Dec-2017.) |
Ref | Expression |
---|---|
cnfldcj | ⊢ ∗ = (*𝑟‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjf 15048 | . . 3 ⊢ ∗:ℂ⟶ℂ | |
2 | cnex 11188 | . . 3 ⊢ ℂ ∈ V | |
3 | fex2 7921 | . . 3 ⊢ ((∗:ℂ⟶ℂ ∧ ℂ ∈ V ∧ ℂ ∈ V) → ∗ ∈ V) | |
4 | 1, 2, 2, 3 | mp3an 1462 | . 2 ⊢ ∗ ∈ V |
5 | cnfldstr 20939 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
6 | starvid 17245 | . . 3 ⊢ *𝑟 = Slot (*𝑟‘ndx) | |
7 | ssun2 4173 | . . . 4 ⊢ {〈(*𝑟‘ndx), ∗〉} ⊆ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
8 | ssun1 4172 | . . . . 5 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
9 | df-cnfld 20938 | . . . . 5 ⊢ ℂfld = (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
10 | 8, 9 | sseqtrri 4019 | . . . 4 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ⊆ ℂfld |
11 | 7, 10 | sstri 3991 | . . 3 ⊢ {〈(*𝑟‘ndx), ∗〉} ⊆ ℂfld |
12 | 5, 6, 11 | strfv 17134 | . 2 ⊢ (∗ ∈ V → ∗ = (*𝑟‘ℂfld)) |
13 | 4, 12 | ax-mp 5 | 1 ⊢ ∗ = (*𝑟‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3946 {csn 4628 {ctp 4632 〈cop 4634 ∘ ccom 5680 ⟶wf 6537 ‘cfv 6541 ℂcc 11105 1c1 11108 + caddc 11110 · cmul 11112 ≤ cle 11246 − cmin 11441 3c3 12265 ;cdc 12674 ∗ccj 15040 abscabs 15178 ndxcnx 17123 Basecbs 17141 +gcplusg 17194 .rcmulr 17195 *𝑟cstv 17196 TopSetcts 17200 lecple 17201 distcds 17203 UnifSetcunif 17204 MetOpencmopn 20927 metUnifcmetu 20928 ℂfldccnfld 20937 |
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 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-cj 15043 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-mulr 17208 df-starv 17209 df-tset 17213 df-ple 17214 df-ds 17216 df-unif 17217 df-cnfld 20938 |
This theorem is referenced by: cnsrng 20972 refldcj 21165 clmcj 24584 cphcjcl 24692 ipcau2 24743 |
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