ellnop - Hilbert Space Explorer

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Theorem ellnop 31944

Description: Property defining a linear Hilbert space operator. (Contributed by NM, 18-Jan-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
ellnop	⊢ (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))

Distinct variable group: 𝑥,𝑦,𝑧,𝑇

Proof of Theorem ellnop Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 6833	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)))
2		fveq1 6833	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
3	2	oveq2d 7376	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑥 ·_ℎ (𝑡‘𝑦)) = (𝑥 ·_ℎ (𝑇‘𝑦)))
4		fveq1 6833	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑧) = (𝑇‘𝑧))
5	3, 4	oveq12d 7378	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑥 ·_ℎ (𝑡‘𝑦)) +_ℎ (𝑡‘𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧)))
6	1, 5	eqeq12d 2753	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝑡‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑡‘𝑦)) +_ℎ (𝑡‘𝑧)) ↔ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
7	6	ralbidv 3161	. . . 4 ⊢ (𝑡 = 𝑇 → (∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑡‘𝑦)) +_ℎ (𝑡‘𝑧)) ↔ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
8	7	2ralbidv 3202	. . 3 ⊢ (𝑡 = 𝑇 → (∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑡‘𝑦)) +_ℎ (𝑡‘𝑧)) ↔ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
9		df-lnop 31927	. . 3 ⊢ LinOp = {𝑡 ∈ ( ℋ ↑_m ℋ) ∣ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑡‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑡‘𝑦)) +_ℎ (𝑡‘𝑧))}
10	8, 9	elrab2 3638	. 2 ⊢ (𝑇 ∈ LinOp ↔ (𝑇 ∈ ( ℋ ↑_m ℋ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
11		ax-hilex 31085	. . . 4 ⊢ ℋ ∈ V
12	11, 11	elmap 8812	. . 3 ⊢ (𝑇 ∈ ( ℋ ↑_m ℋ) ↔ 𝑇: ℋ⟶ ℋ)
13	12	anbi1i 625	. 2 ⊢ ((𝑇 ∈ ( ℋ ↑_m ℋ) ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))) ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))
14	10, 13	bitri 275	1 ⊢ (𝑇 ∈ LinOp ↔ (𝑇: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (𝑇‘((𝑥 ·_ℎ 𝑦) +_ℎ 𝑧)) = ((𝑥 ·_ℎ (𝑇‘𝑦)) +_ℎ (𝑇‘𝑧))))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⟶wf 6488 ‘cfv 6492 (class class class)co 7360 ↑_m cmap 8766 ℂcc 11027 ℋchba 31005 +_ℎ cva 31006 ·_ℎ csm 31007 LinOpclo 31033

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5231 ax-pow 5302 ax-pr 5370 ax-un 7682 ax-hilex 31085

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fv 6500 df-ov 7363 df-oprab 7364 df-mpo 7365 df-map 8768 df-lnop 31927

This theorem is referenced by: lnopf 31945 lnopl 32000 unoplin 32006 hmoplin 32028 lnopmi 32086 lnophsi 32087 lnopcoi 32089 cnlnadjlem6 32158 adjlnop 32172

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