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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 6903 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
3 | hoeq.2 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6903 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
5 | ax-hvcom 29082 | . . . . 5 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) | |
6 | 2, 4, 5 | syl2anc 587 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) |
7 | hosval 29821 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) | |
8 | 1, 3, 7 | mp3an12 1453 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) |
9 | hosval 29821 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) | |
10 | 3, 1, 9 | mp3an12 1453 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) |
11 | 6, 8, 10 | 3eqtr4d 2787 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)) |
12 | 11 | rgen 3071 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) |
13 | 1, 3 | hoaddcli 29849 | . . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
14 | 3, 1 | hoaddcli 29849 | . . 3 ⊢ (𝑇 +op 𝑆): ℋ⟶ ℋ |
15 | 13, 14 | hoeqi 29842 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆)) |
16 | 12, 15 | mpbi 233 | 1 ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2110 ∀wral 3061 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℋchba 29000 +ℎ cva 29001 +op chos 29019 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-hilex 29080 ax-hfvadd 29081 ax-hvcom 29082 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-map 8510 df-hosum 29811 |
This theorem is referenced by: hoaddcom 29855 hoadd12i 29858 hoadd32i 29859 hoaddsubi 29902 hosd1i 29903 hosubeq0i 29907 |
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