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| Mirrors > Home > MPE Home > Th. List > blof | Structured version Visualization version GIF version | ||
| Description: A bounded operator is an operator. (Contributed by NM, 8-Dec-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| blof.1 | β’ π = (BaseSetβπ) |
| blof.2 | β’ π = (BaseSetβπ) |
| blof.5 | β’ π΅ = (π BLnOp π) |
| Ref | Expression |
|---|---|
| blof | β’ ((π β NrmCVec β§ π β NrmCVec β§ π β π΅) β π:πβΆπ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2730 | . . 3 β’ (π LnOp π) = (π LnOp π) | |
| 2 | blof.5 | . . 3 β’ π΅ = (π BLnOp π) | |
| 3 | 1, 2 | bloln 30302 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π β π΅) β π β (π LnOp π)) |
| 4 | blof.1 | . . 3 β’ π = (BaseSetβπ) | |
| 5 | blof.2 | . . 3 β’ π = (BaseSetβπ) | |
| 6 | 4, 5, 1 | lnof 30273 | . 2 β’ ((π β NrmCVec β§ π β NrmCVec β§ π β (π LnOp π)) β π:πβΆπ) |
| 7 | 3, 6 | syld3an3 1407 | 1 β’ ((π β NrmCVec β§ π β NrmCVec β§ π β π΅) β π:πβΆπ) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ w3a 1085 = wceq 1539 β wcel 2104 βΆwf 6540 βcfv 6544 (class class class)co 7413 NrmCVeccnv 30102 BaseSetcba 30104 LnOp clno 30258 BLnOp cblo 30260 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7416 df-oprab 7417 df-mpo 7418 df-map 8826 df-lno 30262 df-blo 30264 |
| This theorem is referenced by: nmblore 30304 nmblolbii 30317 blometi 30321 ubthlem3 30390 htthlem 30435 |
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