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Mirrors > Home > MPE Home > Th. List > cnfldds | Structured version Visualization version GIF version |
Description: The metric of the field of complex numbers. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.) |
Ref | Expression |
---|---|
cnfldds | ⊢ (abs ∘ − ) = (dist‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absf 14697 | . . . 4 ⊢ abs:ℂ⟶ℝ | |
2 | subf 10888 | . . . 4 ⊢ − :(ℂ × ℂ)⟶ℂ | |
3 | fco 6531 | . . . 4 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
4 | 1, 2, 3 | mp2an 690 | . . 3 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ |
5 | cnex 10618 | . . . 4 ⊢ ℂ ∈ V | |
6 | 5, 5 | xpex 7476 | . . 3 ⊢ (ℂ × ℂ) ∈ V |
7 | reex 10628 | . . 3 ⊢ ℝ ∈ V | |
8 | fex2 7638 | . . 3 ⊢ (((abs ∘ − ):(ℂ × ℂ)⟶ℝ ∧ (ℂ × ℂ) ∈ V ∧ ℝ ∈ V) → (abs ∘ − ) ∈ V) | |
9 | 4, 6, 7, 8 | mp3an 1457 | . 2 ⊢ (abs ∘ − ) ∈ V |
10 | cnfldstr 20547 | . . 3 ⊢ ℂfld Struct 〈1, ;13〉 | |
11 | dsid 16676 | . . 3 ⊢ dist = Slot (dist‘ndx) | |
12 | snsstp3 4751 | . . . 4 ⊢ {〈(dist‘ndx), (abs ∘ − )〉} ⊆ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} | |
13 | ssun1 4148 | . . . . 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 4149 | . . . . . 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 | df-cnfld 20546 | . . . . . 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 4004 | . . . . 5 ⊢ ({〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ∪ {〈(UnifSet‘ndx), (metUnif‘(abs ∘ − ))〉}) ⊆ ℂfld |
17 | 13, 16 | sstri 3976 | . . . 4 ⊢ {〈(TopSet‘ndx), (MetOpen‘(abs ∘ − ))〉, 〈(le‘ndx), ≤ 〉, 〈(dist‘ndx), (abs ∘ − )〉} ⊆ ℂfld |
18 | 12, 17 | sstri 3976 | . . 3 ⊢ {〈(dist‘ndx), (abs ∘ − )〉} ⊆ ℂfld |
19 | 10, 11, 18 | strfv 16531 | . 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 1537 ∈ wcel 2114 Vcvv 3494 ∪ cun 3934 {csn 4567 {ctp 4571 〈cop 4573 × cxp 5553 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 ℂcc 10535 ℝcr 10536 1c1 10538 + caddc 10540 · cmul 10542 ≤ cle 10676 − cmin 10870 3c3 11694 ;cdc 12099 ∗ccj 14455 abscabs 14593 ndxcnx 16480 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 *𝑟cstv 16567 TopSetcts 16571 lecple 16572 distcds 16574 UnifSetcunif 16575 MetOpencmopn 20535 metUnifcmetu 20536 ℂfldccnfld 20545 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-cnfld 20546 |
This theorem is referenced by: reds 20760 cnfldms 23384 cnfldnm 23387 cnngp 23388 cncms 23958 cnfldcusp 23960 qqhcn 31232 qqhucn 31233 cnrrext 31251 cnpwstotbnd 35090 repwsmet 35127 rrnequiv 35128 |
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