Theorem adjmo 32184

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 3150	. . . . . 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 2784	. . . . . . 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 3135	. . . . . 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 238	. . . . 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 32182	. . . . . 6 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦) ↔ 𝑢 = 𝑣))
6	5	biimpa 481	. . . . 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 604	. . . 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 672	. . 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 1826	. 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 6683	. . . 4 ⊢ (𝑢 = 𝑣 → (𝑢: ℋ⟶ ℋ ↔ 𝑣: ℋ⟶ ℋ))
11		fveq1 6880	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (𝑢‘𝑥) = (𝑣‘𝑥))
12	11	oveq1d 7425	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦))
13	12	eqeq2d 2774	. . . . 5 ⊢ (𝑢 = 𝑣 → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)))
14	13	2ralbidv 3229	. . . 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 643	. . 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 2594	. 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 234	1 ⊢ ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃*wmo 2565  ∀wral 3079  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ℋchba 31271   ·_ih csp 31274

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-hfvadd 31352  ax-hvcom 31353  ax-hvass 31354  ax-hv0cl 31355  ax-hvaddid 31356  ax-hfvmul 31357  ax-hvmulid 31358  ax-hvdistr2 31361  ax-hvmul0 31362  ax-hfi 31431  ax-his2 31435  ax-his3 31436  ax-his4 31437

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-po 5569  df-so 5570  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-ltxr 11243  df-sub 11438  df-neg 11439  df-hvsub 31323

This theorem is referenced by:  funadj  32238  adjeu  32241  cnlnadjeui  32429