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Theorem adjmo 31072

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 3138	. . . . . 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 2756	. . . . . . 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 3123	. . . . . 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 31070	. . . . . 6 ⊢ ((𝑢: ℋ⟶ ℋ ∧ 𝑣: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑢‘𝑥) ·_ih 𝑦) = ((𝑣‘𝑥) ·_ih 𝑦) ↔ 𝑢 = 𝑣))
6	5	biimpa 477	. . . . 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 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 1798	. 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 6695	. . . 4 ⊢ (𝑢 = 𝑣 → (𝑢: ℋ⟶ ℋ ↔ 𝑣: ℋ⟶ ℋ))
11		fveq1 6887	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (𝑢‘𝑥) = (𝑣‘𝑥))
12	11	oveq1d 7420	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((𝑢‘𝑥) ·_ih 𝑦) = ((𝑣‘𝑥) ·_ih 𝑦))
13	12	eqeq2d 2743	. . . . 5 ⊢ (𝑢 = 𝑣 → ((𝑥 ·_ih (𝑇‘𝑦)) = ((𝑢‘𝑥) ·_ih 𝑦) ↔ (𝑥 ·_ih (𝑇‘𝑦)) = ((𝑣‘𝑥) ·_ih 𝑦)))
14	13	2ralbidv 3218	. . . 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 631	. . 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 2560	. 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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∀wal 1539 = wceq 1541 ∃*wmo 2532 ∀wral 3061 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℋchba 30159 ·_ih csp 30162

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-hfvadd 30240 ax-hvcom 30241 ax-hvass 30242 ax-hv0cl 30243 ax-hvaddid 30244 ax-hfvmul 30245 ax-hvmulid 30246 ax-hvdistr2 30249 ax-hvmul0 30250 ax-hfi 30319 ax-his2 30323 ax-his3 30324 ax-his4 30325

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-ltxr 11249 df-sub 11442 df-neg 11443 df-hvsub 30211

This theorem is referenced by: funadj 31126 adjeu 31129 cnlnadjeui 31317

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