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Mirrors > Home > HSE Home > Th. List > lnopmul | Structured version Visualization version GIF version |
Description: Multiplicative property of a linear Hilbert space operator. (Contributed by NM, 13-Aug-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopmul | ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 ·ℎ (𝑇‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 28786 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | lnopl 29697 | . . . 4 ⊢ (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ (𝐵 ∈ ℋ ∧ 0ℎ ∈ ℋ)) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 0ℎ)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ))) | |
3 | 1, 2 | mpanr2 703 | . . 3 ⊢ (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ 𝐵 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 0ℎ)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ))) |
4 | 3 | 3impa 1107 | . 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 0ℎ)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ))) |
5 | hvmulcl 28796 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | |
6 | ax-hvaddid 28787 | . . . . 5 ⊢ ((𝐴 ·ℎ 𝐵) ∈ ℋ → ((𝐴 ·ℎ 𝐵) +ℎ 0ℎ) = (𝐴 ·ℎ 𝐵)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) +ℎ 0ℎ) = (𝐴 ·ℎ 𝐵)) |
8 | 7 | 3adant1 1127 | . . 3 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) +ℎ 0ℎ) = (𝐴 ·ℎ 𝐵)) |
9 | 8 | fveq2d 6649 | . 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 0ℎ)) = (𝑇‘(𝐴 ·ℎ 𝐵))) |
10 | lnop0 29749 | . . . . 5 ⊢ (𝑇 ∈ LinOp → (𝑇‘0ℎ) = 0ℎ) | |
11 | 10 | oveq2d 7151 | . . . 4 ⊢ (𝑇 ∈ LinOp → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ 0ℎ)) |
12 | 11 | 3ad2ant1 1130 | . . 3 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ 0ℎ)) |
13 | lnopf 29642 | . . . . . . . 8 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) | |
14 | 13 | ffvelrnda 6828 | . . . . . . 7 ⊢ ((𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ) → (𝑇‘𝐵) ∈ ℋ) |
15 | hvmulcl 28796 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℋ) → (𝐴 ·ℎ (𝑇‘𝐵)) ∈ ℋ) | |
16 | 14, 15 | sylan2 595 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ)) → (𝐴 ·ℎ (𝑇‘𝐵)) ∈ ℋ) |
17 | 16 | 3impb 1112 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ (𝑇‘𝐵)) ∈ ℋ) |
18 | 17 | 3com12 1120 | . . . 4 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ (𝑇‘𝐵)) ∈ ℋ) |
19 | ax-hvaddid 28787 | . . . 4 ⊢ ((𝐴 ·ℎ (𝑇‘𝐵)) ∈ ℋ → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ 0ℎ) = (𝐴 ·ℎ (𝑇‘𝐵))) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ 0ℎ) = (𝐴 ·ℎ (𝑇‘𝐵))) |
21 | 12, 20 | eqtrd 2833 | . 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ)) = (𝐴 ·ℎ (𝑇‘𝐵))) |
22 | 4, 9, 21 | 3eqtr3d 2841 | 1 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐵)) = (𝐴 ·ℎ (𝑇‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 0ℎc0v 28707 LinOpclo 28730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-hilex 28782 ax-hfvadd 28783 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvdistr2 28792 ax-hvmul0 28793 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-neg 10862 df-hvsub 28754 df-lnop 29624 |
This theorem is referenced by: lnopmuli 29755 homco2 29760 |
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