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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 21241. (Revised by GG, 31-Mar-2025.) |
Ref | Expression |
---|---|
cnfldstr | ⊢ ℂfld Struct ⟨1, ;13⟩ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnfld 21241 | . 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 2726 | . . . 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 17263 | . . 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 12314 | . . . . 5 ⊢ 9 ∈ ℕ | |
5 | tsetndx 17306 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 12812 | . . . . 5 ⊢ 9 < ;10 | |
7 | 10nn 12697 | . . . . 5 ⊢ ;10 ∈ ℕ | |
8 | plendx 17320 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 12492 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 12491 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 12289 | . . . . . 6 ⊢ 2 ∈ ℕ | |
12 | 2pos 12319 | . . . . . 6 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 12709 | . . . . 5 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 12701 | . . . . 5 ⊢ ;12 ∈ ℕ |
15 | dsndx 17339 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 17102 | . . . 4 ⊢ {⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} Struct ⟨9, ;12⟩ |
17 | 3nn 12295 | . . . . . 6 ⊢ 3 ∈ ℕ | |
18 | 9, 17 | decnncl 12701 | . . . . 5 ⊢ ;13 ∈ ℕ |
19 | unifndx 17349 | . . . . 5 ⊢ (UnifSet‘ndx) = ;13 | |
20 | 18, 19 | strle1 17100 | . . . 4 ⊢ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩} Struct ⟨;13, ;13⟩ |
21 | 2nn0 12493 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
22 | 2lt3 12388 | . . . . 5 ⊢ 2 < 3 | |
23 | 9, 21, 17, 22 | declt 12709 | . . . 4 ⊢ ;12 < ;13 |
24 | 16, 20, 23 | strleun 17099 | . . 3 ⊢ ({⟨(TopSet‘ndx), (MetOpen‘(abs ∘ − ))⟩, ⟨(le‘ndx), ≤ ⟩, ⟨(dist‘ndx), (abs ∘ − )⟩} ∪ {⟨(UnifSet‘ndx), (metUnif‘(abs ∘ − ))⟩}) Struct ⟨9, ;13⟩ |
25 | 4lt9 12419 | . . 3 ⊢ 4 < 9 | |
26 | 3, 24, 25 | strleun 17099 | . 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 5162 | 1 ⊢ ℂfld Struct ⟨1, ;13⟩ |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3941 {csn 4623 {ctp 4627 ⟨cop 4629 class class class wbr 5141 ∘ ccom 5673 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 ℂcc 11110 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 ≤ cle 11253 − cmin 11448 2c2 12271 3c3 12272 4c4 12273 9c9 12278 ;cdc 12681 ∗ccj 15049 abscabs 15187 Struct cstr 17088 ndxcnx 17135 Basecbs 17153 +gcplusg 17206 .rcmulr 17207 *𝑟cstv 17208 TopSetcts 17212 lecple 17213 distcds 17215 UnifSetcunif 17216 MetOpencmopn 21230 metUnifcmetu 21231 ℂfldccnfld 21240 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-plusg 17219 df-mulr 17220 df-starv 17221 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-cnfld 21241 |
This theorem is referenced by: cnfldbas 21244 mpocnfldadd 21245 mpocnfldmul 21247 cnfldcj 21249 cnfldtset 21250 cnfldle 21251 cnfldds 21252 cnfldunif 21253 cnfldfun 21254 cffldtocusgr 29212 cffldtocusgrOLD 29213 |
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