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Mirrors > Home > MPE Home > Th. List > cnflddsOLD | Structured version Visualization version GIF version |
Description: Obsolete version of cnfldds 21393 as of 27-Apr-2025. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
cnflddsOLD | ⊢ (abs ∘ − ) = (dist‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absf 15372 | . . . 4 ⊢ abs:ℂ⟶ℝ | |
2 | subf 11507 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
3 | fco 6760 | . . . 4 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
4 | 1, 2, 3 | mp2an 692 | . . 3 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ |
5 | cnex 11233 | . . . 4 ⊢ ℂ ∈ V | |
6 | 5, 5 | xpex 7771 | . . 3 ⊢ (ℂ × ℂ) ∈ V |
7 | reex 11243 | . . 3 ⊢ ℝ ∈ V | |
8 | fex2 7956 | . . 3 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ (ℂ × ℂ) ∈ V ∧ ℝ ∈ V) → (abs ∘ − ) ∈ V) | |
9 | 4, 6, 7, 8 | mp3an 1460 | . 2 ⊢ (abs ∘ − ) ∈ V |
10 | cnfldstrOLD 21398 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
11 | dsid 17431 | . . 3 ⊢ dist = Slot (dist‘ndx) | |
12 | snsstp3 4822 | . . . 4 ⊢ {〈(dist‘ndx), (abs ∘ − )〉} ⊆ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} | |
13 | ssun1 4187 | . . . . 5 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ⊆ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) | |
14 | ssun2 4188 | . . . . . 6 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) ⊆ (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
15 | dfcnfldOLD 21397 | . . . . . 6 ⊢ ℂfld = (({〈(Base‘ndx), ℂ〉, 〈(+g‘ndx), + 〉, 〈(.r‘ndx), · 〉} ∪ {〈(*𝑟‘ndx), ∗〉}) ∪ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉})) | |
16 | 14, 15 | sseqtrri 4032 | . . . . 5 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) ⊆ ℂfld |
17 | 13, 16 | sstri 4004 | . . . 4 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ⊆ ℂfld |
18 | 12, 17 | sstri 4004 | . . 3 ⊢ {〈(dist‘ndx), (abs ∘ − )〉} ⊆ ℂfld |
19 | 10, 11, 18 | strfv 17237 | . 2 ⊢ ((abs ∘ − ) ∈ V → (abs ∘ − ) = (dist‘ℂfld)) |
20 | 9, 19 | ax-mp 5 | 1 ⊢ (abs ∘ − ) = (dist‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 Vcvv 3477 ∪ cun 3960 {csn 4630 {ctp 4634 〈cop 4636 × cxp 5686 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 ℂcc 11150 ℝcr 11151 1c1 11153 + caddc 11155 · cmul 11157 ≤ cle 11293 − cmin 11489 3c3 12319 ;cdc 12730 ∗ccj 15131 abscabs 15269 ndxcnx 17226 Basecbs 17244 +gcplusg 17297 .rcmulr 17298 *𝑟cstv 17299 TopSetcts 17303 lecple 17304 distcds 17306 UnifSetcunif 17307 MetOpencmopn 21371 metUnifcmetu 21372 ℂfldccnfld 21381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-cnfld 21382 |
This theorem is referenced by: (None) |
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