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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 31032 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | lnopl 31943 | . . . 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 704 | . . 3 ⊢ (((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ) ∧ 𝐵 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 0ℎ)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ))) |
4 | 3 | 3impa 1109 | . 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 0ℎ)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ))) |
5 | hvmulcl 31042 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ 𝐵) ∈ ℋ) | |
6 | ax-hvaddid 31033 | . . . . 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 1129 | . . 3 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ 𝐵) +ℎ 0ℎ) = (𝐴 ·ℎ 𝐵)) |
9 | 8 | fveq2d 6911 | . 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝑇‘((𝐴 ·ℎ 𝐵) +ℎ 0ℎ)) = (𝑇‘(𝐴 ·ℎ 𝐵))) |
10 | lnop0 31995 | . . . . 5 ⊢ (𝑇 ∈ LinOp → (𝑇‘0ℎ) = 0ℎ) | |
11 | 10 | oveq2d 7447 | . . . 4 ⊢ (𝑇 ∈ LinOp → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ 0ℎ)) |
12 | 11 | 3ad2ant1 1132 | . . 3 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ)) = ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ 0ℎ)) |
13 | lnopf 31888 | . . . . . . . 8 ⊢ (𝑇 ∈ LinOp → 𝑇: ℋ⟶ ℋ) | |
14 | 13 | ffvelcdmda 7104 | . . . . . . 7 ⊢ ((𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ) → (𝑇‘𝐵) ∈ ℋ) |
15 | hvmulcl 31042 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝐵) ∈ ℋ) → (𝐴 ·ℎ (𝑇‘𝐵)) ∈ ℋ) | |
16 | 14, 15 | sylan2 593 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ)) → (𝐴 ·ℎ (𝑇‘𝐵)) ∈ ℋ) |
17 | 16 | 3impb 1114 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇 ∈ LinOp ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ (𝑇‘𝐵)) ∈ ℋ) |
18 | 17 | 3com12 1122 | . . . 4 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ℎ (𝑇‘𝐵)) ∈ ℋ) |
19 | ax-hvaddid 31033 | . . . 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 2775 | . 2 ⊢ ((𝑇 ∈ LinOp ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·ℎ (𝑇‘𝐵)) +ℎ (𝑇‘0ℎ)) = (𝐴 ·ℎ (𝑇‘𝐵))) |
22 | 4, 9, 21 | 3eqtr3d 2783 | 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℋchba 30948 +ℎ cva 30949 ·ℎ csm 30950 0ℎc0v 30953 LinOpclo 30976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-hilex 31028 ax-hfvadd 31029 ax-hvass 31031 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 ax-hvdistr2 31038 ax-hvmul0 31039 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-neg 11493 df-hvsub 31000 df-lnop 31870 |
This theorem is referenced by: lnopmuli 32001 homco2 32006 |
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