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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 29669 | . . . . 5 β’ (π β NrmCVec β π· = (π β π)) |
5 | 4 | 3ad2ant1 1134 | . . . 4 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β π· = (π β π)) |
6 | 5 | fveq1d 6845 | . . 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 29629 | . . . . 5 β’ (π β NrmCVec β π:(π Γ π)βΆπ) |
9 | opelxpi 5671 | . . . . 5 β’ ((π΄ β π β§ π΅ β π) β β¨π΄, π΅β© β (π Γ π)) | |
10 | fvco3 6941 | . . . . 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 597 | . . . 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 1116 | . . 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 2773 | . 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 7361 | . 2 β’ (π΄π·π΅) = (π·ββ¨π΄, π΅β©) | |
15 | df-ov 7361 | . . 3 β’ (π΄ππ΅) = (πββ¨π΄, π΅β©) | |
16 | 15 | fveq2i 6846 | . 2 β’ (πβ(π΄ππ΅)) = (πβ(πββ¨π΄, π΅β©)) |
17 | 13, 14, 16 | 3eqtr4g 2798 | 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 397 β§ w3a 1088 = wceq 1542 β wcel 2107 β¨cop 4593 Γ cxp 5632 β ccom 5638 βΆwf 6493 βcfv 6497 (class class class)co 7358 NrmCVeccnv 29568 BaseSetcba 29570 βπ£ cnsb 29573 normCVcnmcv 29574 IndMetcims 29575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-ltxr 11199 df-sub 11392 df-neg 11393 df-grpo 29477 df-gid 29478 df-ginv 29479 df-gdiv 29480 df-ablo 29529 df-vc 29543 df-nv 29576 df-va 29579 df-ba 29580 df-sm 29581 df-0v 29582 df-vs 29583 df-nmcv 29584 df-ims 29585 |
This theorem is referenced by: imsdval2 29671 nvnd 29672 vacn 29678 smcnlem 29681 sspimsval 29722 blometi 29787 blocnilem 29788 ubthlem2 29855 minvecolem2 29859 minvecolem4 29864 minvecolem5 29865 minvecolem6 29866 h2hmetdval 29962 hhssmetdval 30261 |
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