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Mirrors > Home > HSE Home > Th. List > hoaddassi | Structured version Visualization version Unicode version |
Description: Associativity of sum of Hilbert space operators. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hods.1 | |
hods.2 | |
hods.3 |
Ref | Expression |
---|---|
hoaddassi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hods.1 | . . . . . 6 | |
2 | hods.2 | . . . . . 6 | |
3 | hosval 28599 | . . . . . 6 | |
4 | 1, 2, 3 | mp3an12 1414 | . . . . 5 |
5 | 4 | oveq1d 6665 | . . . 4 |
6 | 1, 2 | hoaddcli 28627 | . . . . 5 |
7 | hods.3 | . . . . 5 | |
8 | hosval 28599 | . . . . 5 | |
9 | 6, 7, 8 | mp3an12 1414 | . . . 4 |
10 | hosval 28599 | . . . . . . 7 | |
11 | 2, 7, 10 | mp3an12 1414 | . . . . . 6 |
12 | 11 | oveq2d 6666 | . . . . 5 |
13 | 2, 7 | hoaddcli 28627 | . . . . . 6 |
14 | hosval 28599 | . . . . . 6 | |
15 | 1, 13, 14 | mp3an12 1414 | . . . . 5 |
16 | 1 | ffvelrni 6358 | . . . . . 6 |
17 | 2 | ffvelrni 6358 | . . . . . 6 |
18 | 7 | ffvelrni 6358 | . . . . . 6 |
19 | ax-hvass 27859 | . . . . . 6 | |
20 | 16, 17, 18, 19 | syl3anc 1326 | . . . . 5 |
21 | 12, 15, 20 | 3eqtr4d 2666 | . . . 4 |
22 | 5, 9, 21 | 3eqtr4d 2666 | . . 3 |
23 | 22 | rgen 2922 | . 2 |
24 | 6, 7 | hoaddcli 28627 | . . 3 |
25 | 1, 13 | hoaddcli 28627 | . . 3 |
26 | 24, 25 | hoeqi 28620 | . 2 |
27 | 23, 26 | mpbi 220 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 wcel 1990 wral 2912 wf 5884 cfv 5888 (class class class)co 6650 chil 27776 cva 27777 chos 27795 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-hilex 27856 ax-hfvadd 27857 ax-hvass 27859 |
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-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-hosum 28589 |
This theorem is referenced by: hoadd12i 28636 hoadd32i 28637 hoaddass 28641 hosubeq0i 28685 |
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