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Mirrors > Home > HSE Home > Th. List > specval | Structured version Visualization version GIF version |
Description: The value of the spectrum of an operator. (Contributed by NM, 11-Apr-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
specval | β’ (π: ββΆ β β (Lambdaβπ) = {π₯ β β β£ Β¬ (π βop (π₯ Β·op ( I βΎ β))): ββ1-1β β}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 11139 | . . 3 β’ β β V | |
2 | 1 | rabex 5294 | . 2 β’ {π₯ β β β£ Β¬ (π βop (π₯ Β·op ( I βΎ β))): ββ1-1β β} β V |
3 | ax-hilex 29983 | . 2 β’ β β V | |
4 | oveq1 7369 | . . . . 5 β’ (π‘ = π β (π‘ βop (π₯ Β·op ( I βΎ β))) = (π βop (π₯ Β·op ( I βΎ β)))) | |
5 | f1eq1 6738 | . . . . 5 β’ ((π‘ βop (π₯ Β·op ( I βΎ β))) = (π βop (π₯ Β·op ( I βΎ β))) β ((π‘ βop (π₯ Β·op ( I βΎ β))): ββ1-1β β β (π βop (π₯ Β·op ( I βΎ β))): ββ1-1β β)) | |
6 | 4, 5 | syl 17 | . . . 4 β’ (π‘ = π β ((π‘ βop (π₯ Β·op ( I βΎ β))): ββ1-1β β β (π βop (π₯ Β·op ( I βΎ β))): ββ1-1β β)) |
7 | 6 | notbid 318 | . . 3 β’ (π‘ = π β (Β¬ (π‘ βop (π₯ Β·op ( I βΎ β))): ββ1-1β β β Β¬ (π βop (π₯ Β·op ( I βΎ β))): ββ1-1β β)) |
8 | 7 | rabbidv 3418 | . 2 β’ (π‘ = π β {π₯ β β β£ Β¬ (π‘ βop (π₯ Β·op ( I βΎ β))): ββ1-1β β} = {π₯ β β β£ Β¬ (π βop (π₯ Β·op ( I βΎ β))): ββ1-1β β}) |
9 | df-spec 30839 | . 2 β’ Lambda = (π‘ β ( β βm β) β¦ {π₯ β β β£ Β¬ (π‘ βop (π₯ Β·op ( I βΎ β))): ββ1-1β β}) | |
10 | 2, 3, 3, 8, 9 | fvmptmap 8826 | 1 β’ (π: ββΆ β β (Lambdaβπ) = {π₯ β β β£ Β¬ (π βop (π₯ Β·op ( I βΎ β))): ββ1-1β β}) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 = wceq 1542 {crab 3410 I cid 5535 βΎ cres 5640 βΆwf 6497 β1-1βwf1 6498 βcfv 6501 (class class class)co 7362 βcc 11056 βchba 29903 Β·op chot 29923 βop chod 29924 Lambdacspc 29945 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-hilex 29983 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 df-map 8774 df-spec 30839 |
This theorem is referenced by: speccl 30883 |
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