![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > imsdval | Structured version Visualization version GIF version |
Description: Value of the induced metric (distance function) of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 11-Sep-2007.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imsdval.1 | β’ π = (BaseSetβπ) |
imsdval.3 | β’ π = ( βπ£ βπ) |
imsdval.6 | β’ π = (normCVβπ) |
imsdval.8 | β’ π· = (IndMetβπ) |
Ref | Expression |
---|---|
imsdval | β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄ππ΅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imsdval.3 | . . . . . 6 β’ π = ( βπ£ βπ) | |
2 | imsdval.6 | . . . . . 6 β’ π = (normCVβπ) | |
3 | imsdval.8 | . . . . . 6 β’ π· = (IndMetβπ) | |
4 | 1, 2, 3 | imsval 30515 | . . . . 5 β’ (π β NrmCVec β π· = (π β π)) |
5 | 4 | 3ad2ant1 1130 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β π· = (π β π)) |
6 | 5 | fveq1d 6904 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π·ββ¨π΄, π΅β©) = ((π β π)ββ¨π΄, π΅β©)) |
7 | imsdval.1 | . . . . . 6 β’ π = (BaseSetβπ) | |
8 | 7, 1 | nvmf 30475 | . . . . 5 β’ (π β NrmCVec β π:(π Γ π)βΆπ) |
9 | opelxpi 5719 | . . . . 5 β’ ((π΄ β π β§ π΅ β π) β β¨π΄, π΅β© β (π Γ π)) | |
10 | fvco3 7002 | . . . . 5 β’ ((π:(π Γ π)βΆπ β§ β¨π΄, π΅β© β (π Γ π)) β ((π β π)ββ¨π΄, π΅β©) = (πβ(πββ¨π΄, π΅β©))) | |
11 | 8, 9, 10 | syl2an 594 | . . . 4 β’ ((π β NrmCVec β§ (π΄ β π β§ π΅ β π)) β ((π β π)ββ¨π΄, π΅β©) = (πβ(πββ¨π΄, π΅β©))) |
12 | 11 | 3impb 1112 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β ((π β π)ββ¨π΄, π΅β©) = (πβ(πββ¨π΄, π΅β©))) |
13 | 6, 12 | eqtrd 2768 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π·ββ¨π΄, π΅β©) = (πβ(πββ¨π΄, π΅β©))) |
14 | df-ov 7429 | . 2 β’ (π΄π·π΅) = (π·ββ¨π΄, π΅β©) | |
15 | df-ov 7429 | . . 3 β’ (π΄ππ΅) = (πββ¨π΄, π΅β©) | |
16 | 15 | fveq2i 6905 | . 2 β’ (πβ(π΄ππ΅)) = (πβ(πββ¨π΄, π΅β©)) |
17 | 13, 14, 16 | 3eqtr4g 2793 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄ππ΅))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β¨cop 4638 Γ cxp 5680 β ccom 5686 βΆwf 6549 βcfv 6553 (class class class)co 7426 NrmCVeccnv 30414 BaseSetcba 30416 βπ£ cnsb 30419 normCVcnmcv 30420 IndMetcims 30421 |
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-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 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 |
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-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 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-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-1st 7999 df-2nd 8000 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-ltxr 11291 df-sub 11484 df-neg 11485 df-grpo 30323 df-gid 30324 df-ginv 30325 df-gdiv 30326 df-ablo 30375 df-vc 30389 df-nv 30422 df-va 30425 df-ba 30426 df-sm 30427 df-0v 30428 df-vs 30429 df-nmcv 30430 df-ims 30431 |
This theorem is referenced by: imsdval2 30517 nvnd 30518 vacn 30524 smcnlem 30527 sspimsval 30568 blometi 30633 blocnilem 30634 ubthlem2 30701 minvecolem2 30705 minvecolem4 30710 minvecolem5 30711 minvecolem6 30712 h2hmetdval 30808 hhssmetdval 31107 |
Copyright terms: Public domain | W3C validator |