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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 6624 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
3 | hoeq.2 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6624 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
5 | ax-hvcom 28447 | . . . . 5 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) | |
6 | 2, 4, 5 | syl2anc 579 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆‘𝑥) +ℎ (𝑇‘𝑥)) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) |
7 | hosval 29188 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) | |
8 | 1, 3, 7 | mp3an12 1524 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) |
9 | hosval 29188 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) | |
10 | 3, 1, 9 | mp3an12 1524 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) |
11 | 6, 8, 10 | 3eqtr4d 2824 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)) |
12 | 11 | rgen 3104 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) |
13 | 1, 3 | hoaddcli 29216 | . . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
14 | 3, 1 | hoaddcli 29216 | . . 3 ⊢ (𝑇 +op 𝑆): ℋ⟶ ℋ |
15 | 13, 14 | hoeqi 29209 | . 2 ⊢ (∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) ↔ (𝑆 +op 𝑇) = (𝑇 +op 𝑆)) |
16 | 12, 15 | mpbi 222 | 1 ⊢ (𝑆 +op 𝑇) = (𝑇 +op 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 ∈ wcel 2107 ∀wral 3090 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ℋchba 28365 +ℎ cva 28366 +op chos 28384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-hilex 28445 ax-hfvadd 28446 ax-hvcom 28447 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-map 8144 df-hosum 29178 |
This theorem is referenced by: hoaddcom 29222 hoadd12i 29225 hoadd32i 29226 hoaddsubi 29269 hosd1i 29270 hosubeq0i 29274 |
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