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Mirrors > Home > MPE Home > Th. List > imsdval2 | Structured version Visualization version GIF version |
Description: Value of the distance function of the induced metric of a normed complex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
imsdval2.1 | β’ π = (BaseSetβπ) |
imsdval2.2 | β’ πΊ = ( +π£ βπ) |
imsdval2.4 | β’ π = ( Β·π OLD βπ) |
imsdval2.6 | β’ π = (normCVβπ) |
imsdval2.8 | β’ π· = (IndMetβπ) |
Ref | Expression |
---|---|
imsdval2 | β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄πΊ(-1ππ΅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imsdval2.1 | . . 3 β’ π = (BaseSetβπ) | |
2 | eqid 2726 | . . 3 β’ ( βπ£ βπ) = ( βπ£ βπ) | |
3 | imsdval2.6 | . . 3 β’ π = (normCVβπ) | |
4 | imsdval2.8 | . . 3 β’ π· = (IndMetβπ) | |
5 | 1, 2, 3, 4 | imsdval 30448 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄( βπ£ βπ)π΅))) |
6 | imsdval2.2 | . . . 4 β’ πΊ = ( +π£ βπ) | |
7 | imsdval2.4 | . . . 4 β’ π = ( Β·π OLD βπ) | |
8 | 1, 6, 7, 2 | nvmval 30404 | . . 3 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄( βπ£ βπ)π΅) = (π΄πΊ(-1ππ΅))) |
9 | 8 | fveq2d 6889 | . 2 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (πβ(π΄( βπ£ βπ)π΅)) = (πβ(π΄πΊ(-1ππ΅)))) |
10 | 5, 9 | eqtrd 2766 | 1 β’ ((π β NrmCVec β§ π΄ β π β§ π΅ β π) β (π΄π·π΅) = (πβ(π΄πΊ(-1ππ΅)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βcfv 6537 (class class class)co 7405 1c1 11113 -cneg 11449 NrmCVeccnv 30346 +π£ cpv 30347 BaseSetcba 30348 Β·π OLD cns 30349 βπ£ cnsb 30351 normCVcnmcv 30352 IndMetcims 30353 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-ltxr 11257 df-sub 11450 df-neg 11451 df-grpo 30255 df-gid 30256 df-ginv 30257 df-gdiv 30258 df-ablo 30307 df-vc 30321 df-nv 30354 df-va 30357 df-ba 30358 df-sm 30359 df-0v 30360 df-vs 30361 df-nmcv 30362 df-ims 30363 |
This theorem is referenced by: imsmetlem 30452 nmcvcn 30457 smcnlem 30459 |
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