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Mirrors > Home > HSE Home > Th. List > lnopaddmuli | Structured version Visualization version GIF version |
Description: Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopl.1 | ⊢ 𝑇 ∈ LinOp |
Ref | Expression |
---|---|
lnopaddmuli | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) +ℎ (𝐴 ·ℎ (𝑇‘𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hvmulcl 28951 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·ℎ 𝐶) ∈ ℋ) | |
2 | lnopl.1 | . . . . . 6 ⊢ 𝑇 ∈ LinOp | |
3 | 2 | lnopaddi 29909 | . . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·ℎ 𝐶) ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) +ℎ (𝑇‘(𝐴 ·ℎ 𝐶)))) |
4 | 1, 3 | sylan2 596 | . . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) +ℎ (𝑇‘(𝐴 ·ℎ 𝐶)))) |
5 | 4 | 3impb 1116 | . . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) +ℎ (𝑇‘(𝐴 ·ℎ 𝐶)))) |
6 | 5 | 3com12 1124 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) +ℎ (𝑇‘(𝐴 ·ℎ 𝐶)))) |
7 | 2 | lnopmuli 29910 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐶)) = (𝐴 ·ℎ (𝑇‘𝐶))) |
8 | 7 | 3adant2 1132 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·ℎ 𝐶)) = (𝐴 ·ℎ (𝑇‘𝐶))) |
9 | 8 | oveq2d 7189 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝑇‘𝐵) +ℎ (𝑇‘(𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) +ℎ (𝐴 ·ℎ (𝑇‘𝐶)))) |
10 | 6, 9 | eqtrd 2774 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +ℎ (𝐴 ·ℎ 𝐶))) = ((𝑇‘𝐵) +ℎ (𝐴 ·ℎ (𝑇‘𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ‘cfv 6340 (class class class)co 7173 ℂcc 10616 ℋchba 28857 +ℎ cva 28858 ·ℎ csm 28859 LinOpclo 28885 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-hilex 28937 ax-hfvadd 28938 ax-hvass 28940 ax-hv0cl 28941 ax-hvaddid 28942 ax-hfvmul 28943 ax-hvmulid 28944 ax-hvdistr2 28947 ax-hvmul0 28948 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-id 5430 df-po 5443 df-so 5444 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-er 8323 df-map 8442 df-en 8559 df-dom 8560 df-sdom 8561 df-pnf 10758 df-mnf 10759 df-ltxr 10761 df-sub 10953 df-neg 10954 df-hvsub 28909 df-lnop 29779 |
This theorem is referenced by: lnopsubi 29912 lnopeq0lem1 29943 lnophmlem2 29955 |
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