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Mirrors > Home > HSE Home > Th. List > lnfn0i | Structured version Visualization version GIF version |
Description: The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnfnl.1 | β’ π β LinFn |
Ref | Expression |
---|---|
lnfn0i | β’ (πβ0β) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 30826 | . . . 4 β’ 0β β β | |
2 | lnfnl.1 | . . . . . 6 β’ π β LinFn | |
3 | 2 | lnfnfi 31864 | . . . . 5 β’ π: ββΆβ |
4 | 3 | ffvelcdmi 7093 | . . . 4 β’ (0β β β β (πβ0β) β β) |
5 | 1, 4 | ax-mp 5 | . . 3 β’ (πβ0β) β β |
6 | 5, 5 | pncan3oi 11507 | . 2 β’ (((πβ0β) + (πβ0β)) β (πβ0β)) = (πβ0β) |
7 | ax-1cn 11197 | . . . . . . 7 β’ 1 β β | |
8 | 2 | lnfnli 31863 | . . . . . . 7 β’ ((1 β β β§ 0β β β β§ 0β β β) β (πβ((1 Β·β 0β) +β 0β)) = ((1 Β· (πβ0β)) + (πβ0β))) |
9 | 7, 1, 1, 8 | mp3an 1458 | . . . . . 6 β’ (πβ((1 Β·β 0β) +β 0β)) = ((1 Β· (πβ0β)) + (πβ0β)) |
10 | 7, 1 | hvmulcli 30837 | . . . . . . . . 9 β’ (1 Β·β 0β) β β |
11 | ax-hvaddid 30827 | . . . . . . . . 9 β’ ((1 Β·β 0β) β β β ((1 Β·β 0β) +β 0β) = (1 Β·β 0β)) | |
12 | 10, 11 | ax-mp 5 | . . . . . . . 8 β’ ((1 Β·β 0β) +β 0β) = (1 Β·β 0β) |
13 | ax-hvmulid 30829 | . . . . . . . . 9 β’ (0β β β β (1 Β·β 0β) = 0β) | |
14 | 1, 13 | ax-mp 5 | . . . . . . . 8 β’ (1 Β·β 0β) = 0β |
15 | 12, 14 | eqtri 2756 | . . . . . . 7 β’ ((1 Β·β 0β) +β 0β) = 0β |
16 | 15 | fveq2i 6900 | . . . . . 6 β’ (πβ((1 Β·β 0β) +β 0β)) = (πβ0β) |
17 | 9, 16 | eqtr3i 2758 | . . . . 5 β’ ((1 Β· (πβ0β)) + (πβ0β)) = (πβ0β) |
18 | 5 | mullidi 11250 | . . . . . 6 β’ (1 Β· (πβ0β)) = (πβ0β) |
19 | 18 | oveq1i 7430 | . . . . 5 β’ ((1 Β· (πβ0β)) + (πβ0β)) = ((πβ0β) + (πβ0β)) |
20 | 17, 19 | eqtr3i 2758 | . . . 4 β’ (πβ0β) = ((πβ0β) + (πβ0β)) |
21 | 20 | oveq1i 7430 | . . 3 β’ ((πβ0β) β (πβ0β)) = (((πβ0β) + (πβ0β)) β (πβ0β)) |
22 | 5 | subidi 11562 | . . 3 β’ ((πβ0β) β (πβ0β)) = 0 |
23 | 21, 22 | eqtr3i 2758 | . 2 β’ (((πβ0β) + (πβ0β)) β (πβ0β)) = 0 |
24 | 6, 23 | eqtr3i 2758 | 1 β’ (πβ0β) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 β wcel 2099 βcfv 6548 (class class class)co 7420 βcc 11137 0cc0 11139 1c1 11140 + caddc 11142 Β· cmul 11144 β cmin 11475 βchba 30742 +β cva 30743 Β·β csm 30744 0βc0v 30747 LinFnclf 30777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-hilex 30822 ax-hv0cl 30826 ax-hvaddid 30827 ax-hfvmul 30828 ax-hvmulid 30829 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-sub 11477 df-lnfn 31671 |
This theorem is referenced by: lnfnmuli 31867 lnfn0 31870 nmbdfnlbi 31872 nmcfnexi 31874 nmcfnlbi 31875 nlelshi 31883 |
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