lnopaddmuli - Hilbert Space Explorer

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Theorem lnopaddmuli 28960

Description: Sum/product property of a linear Hilbert space operator. (Contributed by NM, 1-Jul-2005.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
lnopl.1	⊢ 𝑇 ∈ LinOp

**Assertion**
Ref	Expression
lnopaddmuli	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝐴 ·_ℎ (𝑇‘𝐶))))

**Proof of Theorem lnopaddmuli**
Step	Hyp	Ref	Expression
1		hvmulcl 27998	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		lnopl.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopaddi 28958	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
4	1, 3	sylan2 490	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
5	4	3impb 1279	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
6	5	3com12 1288	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
7	2	lnopmuli 28959	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐶)) = (𝐴 ·_ℎ (𝑇‘𝐶)))
8	7	3adant2 1100	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐶)) = (𝐴 ·_ℎ (𝑇‘𝐶)))
9	8	oveq2d 6706	. 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 2685	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 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℋchil 27904 +_ℎ cva 27905 ·_ℎ csm 27906 LinOpclo 27932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-hilex 27984 ax-hfvadd 27985 ax-hvass 27987 ax-hv0cl 27988 ax-hvaddid 27989 ax-hfvmul 27990 ax-hvmulid 27991 ax-hvdistr2 27994 ax-hvmul0 27995

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-po 5064 df-so 5065 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-ltxr 10117 df-sub 10306 df-neg 10307 df-hvsub 27956 df-lnop 28828

This theorem is referenced by: lnopsubi 28961 lnopeq0lem1 28992 lnophmlem2 29004

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