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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 6612 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑆‘𝑥) ∈ ℋ) |
3 | hoeq.2 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ | |
4 | 3 | ffvelrni 6612 | . . . . 5 ⊢ (𝑥 ∈ ℋ → (𝑇‘𝑥) ∈ ℋ) |
5 | ax-hvcom 28409 | . . . . 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 29150 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) | |
8 | 1, 3, 7 | mp3an12 1579 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑆‘𝑥) +ℎ (𝑇‘𝑥))) |
9 | hosval 29150 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) | |
10 | 3, 1, 9 | mp3an12 1579 | . . . 4 ⊢ (𝑥 ∈ ℋ → ((𝑇 +op 𝑆)‘𝑥) = ((𝑇‘𝑥) +ℎ (𝑆‘𝑥))) |
11 | 6, 8, 10 | 3eqtr4d 2871 | . . 3 ⊢ (𝑥 ∈ ℋ → ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥)) |
12 | 11 | rgen 3131 | . 2 ⊢ ∀𝑥 ∈ ℋ ((𝑆 +op 𝑇)‘𝑥) = ((𝑇 +op 𝑆)‘𝑥) |
13 | 1, 3 | hoaddcli 29178 | . . 3 ⊢ (𝑆 +op 𝑇): ℋ⟶ ℋ |
14 | 3, 1 | hoaddcli 29178 | . . 3 ⊢ (𝑇 +op 𝑆): ℋ⟶ ℋ |
15 | 13, 14 | hoeqi 29171 | . 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 1656 ∈ wcel 2164 ∀wral 3117 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 ℋchba 28327 +ℎ cva 28328 +op chos 28346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-hilex 28407 ax-hfvadd 28408 ax-hvcom 28409 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-map 8129 df-hosum 29140 |
This theorem is referenced by: hoaddcom 29184 hoadd12i 29187 hoadd32i 29188 hoaddsubi 29231 hosd1i 29232 hosubeq0i 29236 |
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