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Mirrors > Home > MPE Home > Th. List > cnfldstr | Structured version Visualization version GIF version |
Description: The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) Revise df-cnfld 21287. (Revised by GG, 31-Mar-2025.) |
Ref | Expression |
---|---|
cnfldstr | ⊢ ℂfld Struct ⟨1, ;13⟩ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnfld 21287 | . 2 ⊢ ℂfld = (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) | |
2 | eqid 2728 | . . . 4 ⊢ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) = ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) | |
3 | 2 | srngstr 17297 | . . 3 ⊢ ({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) Struct ⟨1, 4⟩ |
4 | 9nn 12348 | . . . . 5 ⊢ 9 ∈ ℕ | |
5 | tsetndx 17340 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 12846 | . . . . 5 ⊢ 9 < ;10 | |
7 | 10nn 12731 | . . . . 5 ⊢ ;10 ∈ ℕ | |
8 | plendx 17354 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 12526 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 12525 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 12323 | . . . . . 6 ⊢ 2 ∈ ℕ | |
12 | 2pos 12353 | . . . . . 6 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 12743 | . . . . 5 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 12735 | . . . . 5 ⊢ ;12 ∈ ℕ |
15 | dsndx 17373 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 17136 | . . . 4 ⊢ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} Struct ⟨9, ;12⟩ |
17 | 3nn 12329 | . . . . . 6 ⊢ 3 ∈ ℕ | |
18 | 9, 17 | decnncl 12735 | . . . . 5 ⊢ ;13 ∈ ℕ |
19 | unifndx 17383 | . . . . 5 ⊢ (UnifSet‘ndx) = ;13 | |
20 | 18, 19 | strle1 17134 | . . . 4 ⊢ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} Struct ⟨;13, ;13⟩ |
21 | 2nn0 12527 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
22 | 2lt3 12422 | . . . . 5 ⊢ 2 < 3 | |
23 | 9, 21, 17, 22 | declt 12743 | . . . 4 ⊢ ;12 < ;13 |
24 | 16, 20, 23 | strleun 17133 | . . 3 ⊢ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) Struct ⟨9, ;13⟩ |
25 | 4lt9 12453 | . . 3 ⊢ 4 < 9 | |
26 | 3, 24, 25 | strleun 17133 | . 2 ⊢ (({⟨(Base‘ndx), ℂ⟩, ⟨(+g‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 + 𝑣))⟩, ⟨(.r‘ndx), (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣))⟩} ∪ {⟨(*𝑟‘ndx), ∗⟩}) ∪ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩})) Struct ⟨1, ;13⟩ |
27 | 1, 26 | eqbrtri 5173 | 1 ⊢ ℂfld Struct ⟨1, ;13⟩ |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3947 {csn 4632 {ctp 4636 ⟨cop 4638 class class class wbr 5152 ∘ ccom 5686 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 ℂcc 11144 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 ≤ cle 11287 − cmin 11482 2c2 12305 3c3 12306 4c4 12307 9c9 12312 ;cdc 12715 ∗ccj 15083 abscabs 15221 Struct cstr 17122 ndxcnx 17169 Basecbs 17187 +gcplusg 17240 .rcmulr 17241 *𝑟cstv 17242 TopSetcts 17246 lecple 17247 distcds 17249 UnifSetcunif 17250 MetOpencmopn 21276 metUnifcmetu 21277 ℂfldccnfld 21286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-starv 17255 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-cnfld 21287 |
This theorem is referenced by: cnfldbas 21290 mpocnfldadd 21291 mpocnfldmul 21293 cnfldcj 21295 cnfldtset 21296 cnfldle 21297 cnfldds 21298 cnfldunif 21299 cnfldfun 21300 cffldtocusgr 29280 cffldtocusgrOLD 29281 |
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