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Theorem lnopaddmuli 30236

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 29276	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝐴 ·_ℎ 𝐶) ∈ ℋ)
2		lnopl.1	. . . . . 6 ⊢ 𝑇 ∈ LinOp
3	2	lnopaddi 30234	. . . . 5 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ·_ℎ 𝐶) ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
4	1, 3	sylan2 592	. . . 4 ⊢ ((𝐵 ∈ ℋ ∧ (𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ)) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
5	4	3impb 1113	. . 3 ⊢ ((𝐵 ∈ ℋ ∧ 𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
6	5	3com12 1121	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐵 +_ℎ (𝐴 ·_ℎ 𝐶))) = ((𝑇‘𝐵) +_ℎ (𝑇‘(𝐴 ·_ℎ 𝐶))))
7	2	lnopmuli 30235	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐶)) = (𝐴 ·_ℎ (𝑇‘𝐶)))
8	7	3adant2 1129	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝑇‘(𝐴 ·_ℎ 𝐶)) = (𝐴 ·_ℎ (𝑇‘𝐶)))
9	8	oveq2d 7271	. 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 2778	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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℋchba 29182 +_ℎ cva 29183 ·_ℎ csm 29184 LinOpclo 29210

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-hilex 29262 ax-hfvadd 29263 ax-hvass 29265 ax-hv0cl 29266 ax-hvaddid 29267 ax-hfvmul 29268 ax-hvmulid 29269 ax-hvdistr2 29272 ax-hvmul0 29273

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-ltxr 10945 df-sub 11137 df-neg 11138 df-hvsub 29234 df-lnop 30104

This theorem is referenced by: lnopsubi 30237 lnopeq0lem1 30268 lnophmlem2 30280

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