Theorem adjmo 31714

Description: Every Hilbert space operator has at most one adjoint. (Contributed by NM, 18-Feb-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
adjmo	⊢ ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))

Distinct variable group:   𝑥,𝑦,𝑢,𝑇

Proof of Theorem adjmo

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.26-2 3127	. . . . . 6 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)) ↔ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)))
2		eqtr2 2749	. . . . . . 7 ⊢ (((𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)) → ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦))
3	2	2ralimi 3112	. . . . . 6 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ∧ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦))
4	1, 3	sylbir 234	. . . . 5 ⊢ ((∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)) → ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦))
5		hoeq1 31712	. . . . . 6 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦) ↔ 𝑢 = 𝑣))
6	5	biimpa 475	. . . . 5 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦)) → 𝑢 = 𝑣)
7	4, 6	sylan2 591	. . . 4 ⊢ (((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) ∧ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦))) → 𝑢 = 𝑣)
8	7	an4s 658	. . 3 ⊢ (((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦)) ∧ (𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦))) → 𝑢 = 𝑣)
9	8	gen2 1790	. 2 ⊢ ∀𝑢∀𝑣(((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦)) ∧ (𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦))) → 𝑢 = 𝑣)
10		feq1 6704	. . . 4 ⊢ (𝑢 = 𝑣 → (𝑢: ℋ⟶ ℋ ↔ 𝑣: ℋ⟶ ℋ))
11		fveq1 6895	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (𝑢‘𝑥) = (𝑣‘𝑥))
12	11	oveq1d 7434	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦))
13	12	eqeq2d 2736	. . . . 5 ⊢ (𝑢 = 𝑣 → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)))
14	13	2ralbidv 3208	. . . 4 ⊢ (𝑢 = 𝑣 → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)))
15	10, 14	anbi12d 630	. . 3 ⊢ (𝑢 = 𝑣 → ((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦)) ↔ (𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦))))
16	15	mo4 2554	. 2 ⊢ (∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦)) ↔ ∀𝑢∀𝑣(((𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦)) ∧ (𝑣: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦))) → 𝑢 = 𝑣))
17	9, 16	mpbir 230	1 ⊢ ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))

Syntax hints:   → wi 4   ∧ wa 394  ∀wal 1531   = wceq 1533  ∃*wmo 2526  ∀wral 3050  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419   ℋchba 30801   ·_ih csp 30804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-hfvadd 30882  ax-hvcom 30883  ax-hvass 30884  ax-hv0cl 30885  ax-hvaddid 30886  ax-hfvmul 30887  ax-hvmulid 30888  ax-hvdistr2 30891  ax-hvmul0 30892  ax-hfi 30961  ax-his2 30965  ax-his3 30966  ax-his4 30967

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-po 5590  df-so 5591  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-er 8725  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11282  df-mnf 11283  df-ltxr 11285  df-sub 11478  df-neg 11479  df-hvsub 30853

This theorem is referenced by:  funadj  31768  adjeu  31771  cnlnadjeui  31959