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Mirrors > Home > HSE Home > Th. List > 0lnfn | Structured version Unicode version |
Description: The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0lnfn |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 9490 |
. . 3
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2 | 1 | fconst6 5709 |
. 2
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3 | hvmulcl 24568 |
. . . . . . 7
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4 | hvaddcl 24567 |
. . . . . . 7
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5 | 3, 4 | sylan 471 |
. . . . . 6
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6 | c0ex 9492 |
. . . . . . 7
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7 | 6 | fvconst2 6043 |
. . . . . 6
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8 | 5, 7 | syl 16 |
. . . . 5
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9 | 6 | fvconst2 6043 |
. . . . . . . . 9
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10 | 9 | oveq2d 6217 |
. . . . . . . 8
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11 | mul01 9660 |
. . . . . . . 8
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12 | 10, 11 | sylan9eqr 2517 |
. . . . . . 7
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13 | 6 | fvconst2 6043 |
. . . . . . 7
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14 | 12, 13 | oveqan12d 6220 |
. . . . . 6
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15 | 00id 9656 |
. . . . . 6
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16 | 14, 15 | syl6eq 2511 |
. . . . 5
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17 | 8, 16 | eqtr4d 2498 |
. . . 4
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18 | 17 | 3impa 1183 |
. . 3
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19 | 18 | rgen3 2919 |
. 2
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20 | ellnfn 25440 |
. 2
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21 | 2, 19, 20 | mpbir2an 911 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-cnex 9450 ax-resscn 9451 ax-1cn 9452 ax-icn 9453 ax-addcl 9454 ax-addrcl 9455 ax-mulcl 9456 ax-mulrcl 9457 ax-mulcom 9458 ax-addass 9459 ax-mulass 9460 ax-distr 9461 ax-i2m1 9462 ax-1ne0 9463 ax-1rid 9464 ax-rnegex 9465 ax-rrecex 9466 ax-cnre 9467 ax-pre-lttri 9468 ax-pre-lttrn 9469 ax-pre-ltadd 9470 ax-hilex 24554 ax-hfvadd 24555 ax-hfvmul 24560 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-op 3993 df-uni 4201 df-iun 4282 df-br 4402 df-opab 4460 df-mpt 4461 df-id 4745 df-po 4750 df-so 4751 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-ov 6204 df-oprab 6205 df-mpt2 6206 df-er 7212 df-map 7327 df-en 7422 df-dom 7423 df-sdom 7424 df-pnf 9532 df-mnf 9533 df-ltxr 9535 df-lnfn 25405 |
This theorem is referenced by: nmfn0 25544 lnfn0 25604 lnfnmul 25605 nmbdfnlb 25607 nmcfnex 25610 nmcfnlb 25611 lnfncon 25613 riesz4 25621 riesz1 25622 |
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