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Mirrors > Home > HSE Home > Th. List > homulcl | Unicode version |
Description: The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
homulcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffvelrn 5831 |
. . . . 5
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2 | hvmulcl 22473 |
. . . . 5
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3 | 1, 2 | sylan2 461 |
. . . 4
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4 | 3 | anassrs 630 |
. . 3
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5 | eqid 2408 |
. . 3
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6 | 4, 5 | fmptd 5856 |
. 2
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7 | hommval 23196 |
. . 3
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8 | 7 | feq1d 5543 |
. 2
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9 | 6, 8 | mpbird 224 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: honegsubi 23256 homulid2 23260 homco1 23261 homulass 23262 hoadddi 23263 hoadddir 23264 hosubneg 23267 hosubdi 23268 honegsubdi 23270 honegsubdi2 23271 hosub4 23273 hosubsub4 23278 hosubeq0i 23286 nmopnegi 23425 homco2 23437 lnopmi 23460 hmopm 23481 nmophmi 23491 adjmul 23552 opsqrlem1 23600 opsqrlem6 23605 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2389 ax-rep 4284 ax-sep 4294 ax-nul 4302 ax-pow 4341 ax-pr 4367 ax-un 4664 ax-hilex 22459 ax-hfvmul 22465 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2262 df-mo 2263 df-clab 2395 df-cleq 2401 df-clel 2404 df-nfc 2533 df-ne 2573 df-ral 2675 df-rex 2676 df-reu 2677 df-rab 2679 df-v 2922 df-sbc 3126 df-csb 3216 df-dif 3287 df-un 3289 df-in 3291 df-ss 3298 df-nul 3593 df-if 3704 df-pw 3765 df-sn 3784 df-pr 3785 df-op 3787 df-uni 3980 df-iun 4059 df-br 4177 df-opab 4231 df-mpt 4232 df-id 4462 df-xp 4847 df-rel 4848 df-cnv 4849 df-co 4850 df-dm 4851 df-rn 4852 df-res 4853 df-ima 4854 df-iota 5381 df-fun 5419 df-fn 5420 df-f 5421 df-f1 5422 df-fo 5423 df-f1o 5424 df-fv 5425 df-ov 6047 df-oprab 6048 df-mpt2 6049 df-map 6983 df-homul 23191 |
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