Theorem adjmo 29215

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 3247	. . . . . 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 2820	. . . . . . 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 227	. . . . 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 29213	. . . . . 6 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦) ↔ 𝑢 = 𝑣))
6	5	biimpa 469	. . . . 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 587	. . . 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 651	. . 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 1892	. 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 6238	. . . 4 ⊢ (𝑢 = 𝑣 → (𝑢: ℋ⟶ ℋ ↔ 𝑣: ℋ⟶ ℋ))
11		fveq1 6411	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (𝑢‘𝑥) = (𝑣‘𝑥))
12	11	oveq1d 6894	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((𝑢‘𝑥) ·ih 𝑦) = ((𝑣‘𝑥) ·ih 𝑦))
13	12	eqeq2d 2810	. . . . 5 ⊢ (𝑢 = 𝑣 → ((𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦) ↔ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·ih 𝑦)))
14	13	2ralbidv 3171	. . . 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 625	. . 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 2672	. 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 223	1 ⊢ ∃*𝑢(𝑢: ℋ⟶ ℋ ∧ ∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·ih 𝑦))

Syntax hints:   → wi 4   ∧ wa 385  ∀wal 1651   = wceq 1653  ∃*wmo 2590  ∀wral 3090  ⟶wf 6098  ‘cfv 6102  (class class class)co 6879   ℋchba 28300   ·_ih csp 28303

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-resscn 10282  ax-1cn 10283  ax-icn 10284  ax-addcl 10285  ax-addrcl 10286  ax-mulcl 10287  ax-mulrcl 10288  ax-mulcom 10289  ax-addass 10290  ax-mulass 10291  ax-distr 10292  ax-i2m1 10293  ax-1ne0 10294  ax-1rid 10295  ax-rnegex 10296  ax-rrecex 10297  ax-cnre 10298  ax-pre-lttri 10299  ax-pre-lttrn 10300  ax-pre-ltadd 10301  ax-hfvadd 28381  ax-hvcom 28382  ax-hvass 28383  ax-hv0cl 28384  ax-hvaddid 28385  ax-hfvmul 28386  ax-hvmulid 28387  ax-hvdistr2 28390  ax-hvmul0 28391  ax-hfi 28460  ax-his2 28464  ax-his3 28465  ax-his4 28466

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  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-ne 2973  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  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-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-po 5234  df-so 5235  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-er 7983  df-en 8197  df-dom 8198  df-sdom 8199  df-pnf 10366  df-mnf 10367  df-ltxr 10369  df-sub 10559  df-neg 10560  df-hvsub 28352

This theorem is referenced by:  funadj  29269  adjeu  29272  cnlnadjeui  29460