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Mirrors > Home > HSE Home > Th. List > hoaddcomi | Structured version Visualization version GIF version |
Description: Commutativity of sum of Hilbert space operators. (Contributed by NM, 15-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hoeq.1 | ⊢ 𝑆: ℋ⟶ ℋ |
hoeq.2 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
hoaddcomi | ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hoeq.1 | . . . . . 6 ⊢ 𝑆: ℋ⟶ ℋ | |
2 | 1 | ffvelrni 6842 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
3 | hoeq.2 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6842 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
5 | ax-hvcom 28705 | . . . . 5 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) | |
6 | 2, 4, 5 | syl2anc 584 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) |
7 | hosval 29444 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) | |
8 | 1, 3, 7 | mp3an12 1442 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) |
9 | hosval 29444 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) | |
10 | 3, 1, 9 | mp3an12 1442 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) |
11 | 6, 8, 10 | 3eqtr4d 2863 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)) |
12 | 11 | rgen 3145 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) |
13 | 1, 3 | hoaddcli 29472 | . . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
14 | 3, 1 | hoaddcli 29472 | . . 3 ⊢ (𝑇 +op 𝑆): ℋ⟶ ℋ |
15 | 13, 14 | hoeqi 29465 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆)) |
16 | 12, 15 | mpbi 231 | 1 ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ∀wral 3135 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℋchba 28623 +ℎ cva 28624 +op chos 28642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-hilex 28703 ax-hfvadd 28704 ax-hvcom 28705 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-map 8397 df-hosum 29434 |
This theorem is referenced by: hoaddcom 29478 hoadd12i 29481 hoadd32i 29482 hoaddsubi 29525 hosd1i 29526 hosubeq0i 29530 |
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