Theorem adjmo 31907

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 3121	. . . . . 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 2757	. . . . . . 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 3106	. . . . . 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 235	. . . . 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 31905	. . . . . 6 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦) ↔ 𝑢 = 𝑣))
6	5	biimpa 476	. . . . 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 593	. . . 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 660	. . 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 1797	. 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 6640	. . . 4 ⊢ (𝑢 = 𝑣 → (𝑢: ℋ⟶ ℋ ↔ 𝑣: ℋ⟶ ℋ))
11		fveq1 6833	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (𝑢‘𝑥) = (𝑣‘𝑥))
12	11	oveq1d 7373	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦))
13	12	eqeq2d 2747	. . . . 5 ⊢ (𝑢 = 𝑣 → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)))
14	13	2ralbidv 3200	. . . 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 632	. . 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 2566	. 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 231	1 ⊢ ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃*wmo 2537  ∀wral 3051  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ℋchba 30994   ·_ih csp 30997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680  ax-resscn 11083  ax-1cn 11084  ax-icn 11085  ax-addcl 11086  ax-addrcl 11087  ax-mulcl 11088  ax-mulrcl 11089  ax-mulcom 11090  ax-addass 11091  ax-mulass 11092  ax-distr 11093  ax-i2m1 11094  ax-1ne0 11095  ax-1rid 11096  ax-rnegex 11097  ax-rrecex 11098  ax-cnre 11099  ax-pre-lttri 11100  ax-pre-lttrn 11101  ax-pre-ltadd 11102  ax-hfvadd 31075  ax-hvcom 31076  ax-hvass 31077  ax-hv0cl 31078  ax-hvaddid 31079  ax-hfvmul 31080  ax-hvmulid 31081  ax-hvdistr2 31084  ax-hvmul0 31085  ax-hfi 31154  ax-his2 31158  ax-his3 31159  ax-his4 31160

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-po 5532  df-so 5533  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-er 8635  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11168  df-mnf 11169  df-ltxr 11171  df-sub 11366  df-neg 11367  df-hvsub 31046

This theorem is referenced by:  funadj  31961  adjeu  31964  cnlnadjeui  32152