Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > lnopl | Structured version Visualization version Unicode version |
Description: Basic linearity property of a linear Hilbert space operator. (Contributed by NM, 22-Jan-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopl |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ellnop 28717 | . . . . . 6 | |
2 | 1 | simprbi 480 | . . . . 5 |
3 | oveq1 6657 | . . . . . . . . 9 | |
4 | 3 | oveq1d 6665 | . . . . . . . 8 |
5 | 4 | fveq2d 6195 | . . . . . . 7 |
6 | oveq1 6657 | . . . . . . . 8 | |
7 | 6 | oveq1d 6665 | . . . . . . 7 |
8 | 5, 7 | eqeq12d 2637 | . . . . . 6 |
9 | oveq2 6658 | . . . . . . . . 9 | |
10 | 9 | oveq1d 6665 | . . . . . . . 8 |
11 | 10 | fveq2d 6195 | . . . . . . 7 |
12 | fveq2 6191 | . . . . . . . . 9 | |
13 | 12 | oveq2d 6666 | . . . . . . . 8 |
14 | 13 | oveq1d 6665 | . . . . . . 7 |
15 | 11, 14 | eqeq12d 2637 | . . . . . 6 |
16 | oveq2 6658 | . . . . . . . 8 | |
17 | 16 | fveq2d 6195 | . . . . . . 7 |
18 | fveq2 6191 | . . . . . . . 8 | |
19 | 18 | oveq2d 6666 | . . . . . . 7 |
20 | 17, 19 | eqeq12d 2637 | . . . . . 6 |
21 | 8, 15, 20 | rspc3v 3325 | . . . . 5 |
22 | 2, 21 | syl5 34 | . . . 4 |
23 | 22 | 3expb 1266 | . . 3 |
24 | 23 | impcom 446 | . 2 |
25 | 24 | anassrs 680 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 wf 5884 cfv 5888 (class class class)co 6650 cc 9934 chil 27776 cva 27777 csm 27778 clo 27804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-lnop 28700 |
This theorem is referenced by: lnop0 28825 lnopmul 28826 lnopli 28827 |
Copyright terms: Public domain | W3C validator |