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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.) |
Ref | Expression |
---|---|
cnfldstr | ⊢ ℂfld Struct 〈1, ;13〉 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnfld 21235 | . 2 ⊢ ℂfld = (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
2 | eqid 2731 | . . . 4 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) = ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) | |
3 | 2 | srngstr 17261 | . . 3 ⊢ ({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) Struct 〈1, 4〉 |
4 | 9nn 12317 | . . . . 5 ⊢ 9 ∈ ℕ | |
5 | tsetndx 17304 | . . . . 5 ⊢ (TopSet‘ndx) = 9 | |
6 | 9lt10 12815 | . . . . 5 ⊢ 9 < ;10 | |
7 | 10nn 12700 | . . . . 5 ⊢ ;10 ∈ ℕ | |
8 | plendx 17318 | . . . . 5 ⊢ (le‘ndx) = ;10 | |
9 | 1nn0 12495 | . . . . . 6 ⊢ 1 ∈ ℕ0 | |
10 | 0nn0 12494 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
11 | 2nn 12292 | . . . . . 6 ⊢ 2 ∈ ℕ | |
12 | 2pos 12322 | . . . . . 6 ⊢ 0 < 2 | |
13 | 9, 10, 11, 12 | declt 12712 | . . . . 5 ⊢ ;10 < ;12 |
14 | 9, 11 | decnncl 12704 | . . . . 5 ⊢ ;12 ∈ ℕ |
15 | dsndx 17337 | . . . . 5 ⊢ (dist‘ndx) = ;12 | |
16 | 4, 5, 6, 7, 8, 13, 14, 15 | strle3 17100 | . . . 4 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} Struct 〈9, ;12〉 |
17 | 3nn 12298 | . . . . . 6 ⊢ 3 ∈ ℕ | |
18 | 9, 17 | decnncl 12704 | . . . . 5 ⊢ ;13 ∈ ℕ |
19 | unifndx 17347 | . . . . 5 ⊢ (UnifSet‘ndx) = ;13 | |
20 | 18, 19 | strle1 17098 | . . . 4 ⊢ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉} Struct 〈;13, ;13〉 |
21 | 2nn0 12496 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
22 | 2lt3 12391 | . . . . 5 ⊢ 2 < 3 | |
23 | 9, 21, 17, 22 | declt 12712 | . . . 4 ⊢ ;12 < ;13 |
24 | 16, 20, 23 | strleun 17097 | . . 3 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) Struct 〈9, ;13〉 |
25 | 4lt9 12422 | . . 3 ⊢ 4 < 9 | |
26 | 3, 24, 25 | strleun 17097 | . 2 ⊢ (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) Struct 〈1, ;13〉 |
27 | 1, 26 | eqbrtri 5169 | 1 ⊢ ℂfld Struct 〈1, ;13〉 |
Colors of variables: wff setvar class |
Syntax hints: ∪ cun 3946 {csn 4628 {ctp 4632 〈cop 4634 class class class wbr 5148 ∘ ccom 5680 ‘cfv 6543 ℂcc 11114 0cc0 11116 1c1 11117 + caddc 11119 · cmul 11121 ≤ cle 11256 − cmin 11451 2c2 12274 3c3 12275 4c4 12276 9c9 12281 ;cdc 12684 ∗ccj 15050 abscabs 15188 Struct cstr 17086 ndxcnx 17133 Basecbs 17151 +gcplusg 17204 .rcmulr 17205 *𝑟cstv 17206 TopSetcts 17210 lecple 17211 distcds 17213 UnifSetcunif 17214 MetOpencmopn 21224 metUnifcmetu 21225 ℂfldccnfld 21234 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-1o 8472 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12480 df-z 12566 df-dec 12685 df-uz 12830 df-fz 13492 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mulr 17218 df-starv 17219 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-cnfld 21235 |
This theorem is referenced by: cnfldex 21237 cnfldbas 21238 cnfldadd 21239 cnfldmul 21240 cnfldcj 21241 cnfldtset 21242 cnfldle 21243 cnfldds 21244 cnfldunif 21245 cnfldfun 21246 cffldtocusgr 29139 gg-cnfldbas 35639 mpocnfldadd 35640 mpocnfldmul 35641 gg-cnfldcj 35642 gg-cnfldtset 35643 gg-cnfldle 35644 gg-cnfldds 35645 gg-cnfldunif 35646 gg-cnfldfun 35647 gg-cffldtocusgr 35649 |
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