Theorem ellnop 29241

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 6411	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑡‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = (𝑇‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)))
2		fveq1 6411	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑦) = (𝑇‘𝑦))
3	2	oveq2d 6895	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑥 ·ℎ (𝑡‘𝑦)) = (𝑥 ·ℎ (𝑇‘𝑦)))
4		fveq1 6411	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑡‘𝑧) = (𝑇‘𝑧))
5	3, 4	oveq12d 6897	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝑥 ·ℎ (𝑡‘𝑦)) +ℎ (𝑡‘𝑧)) = ((𝑥 ·ℎ (𝑇‘𝑦)) +ℎ (𝑇‘𝑧)))
6	1, 5	eqeq12d 2815	. . . . 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 3168	. . . 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 3171	. . 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 29224	. . 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 3561	. 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 28380	. . . 4 ⊢ ℋ ∈ V
12	11, 11	elmap 8125	. . 3 ⊢ (𝑇 ∈ ( ℋ ↑𝑚 ℋ) ↔ 𝑇: ℋ⟶ ℋ)
13	12	anbi1i 618	. 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 267	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))))

Syntax hints:   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∀wral 3090  ⟶wf 6098  ‘cfv 6102  (class class class)co 6879   ↑_𝑚 cmap 8096  ℂcc 10223   ℋchba 28300   +_ℎ cva 28301   ·_ℎ csm 28302  LinOpclo 28328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184  ax-hilex 28380

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fv 6110  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-map 8098  df-lnop 29224

This theorem is referenced by:  lnopf  29242  lnopl  29297  unoplin  29303  hmoplin  29325  lnopmi  29383  lnophsi  29384  lnopcoi  29386  cnlnadjlem6  29455  adjlnop  29469