hoeq2 - Hilbert Space Explorer

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Theorem hoeq2 31578

Description: A condition implying that two Hilbert space operators are equal. Lemma 3.2(S11) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.)

**Assertion**
Ref	Expression
hoeq2	⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)) ↔ 𝑆 = 𝑇))

Distinct variable groups: 𝑥,𝑦,𝑆 𝑥,𝑇,𝑦

**Proof of Theorem hoeq2**
Step	Hyp	Ref	Expression
1		ralcom 3278	. . 3 ⊢ (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)))
2	1	a1i 11	. 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦))))
3		ffvelcdm 7074	. . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑆‘𝑦) ∈ ℋ)
4		ffvelcdm 7074	. . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) → (𝑇‘𝑦) ∈ ℋ)
5		hial2eq2 30854	. . . . . 6 ⊢ (((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)) ↔ (𝑆‘𝑦) = (𝑇‘𝑦)))
6		hial2eq 30853	. . . . . 6 ⊢ (((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (∀𝑥 ∈ ℋ ((𝑆‘𝑦) ·_ih 𝑥) = ((𝑇‘𝑦) ·_ih 𝑥) ↔ (𝑆‘𝑦) = (𝑇‘𝑦)))
7	5, 6	bitr4d 282	. . . . 5 ⊢ (((𝑆‘𝑦) ∈ ℋ ∧ (𝑇‘𝑦) ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)) ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑦) ·_ih 𝑥) = ((𝑇‘𝑦) ·_ih 𝑥)))
8	3, 4, 7	syl2an 595	. . . 4 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑦 ∈ ℋ)) → (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)) ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑦) ·_ih 𝑥) = ((𝑇‘𝑦) ·_ih 𝑥)))
9	8	anandirs 676	. . 3 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑦 ∈ ℋ) → (∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)) ↔ ∀𝑥 ∈ ℋ ((𝑆‘𝑦) ·_ih 𝑥) = ((𝑇‘𝑦) ·_ih 𝑥)))
10	9	ralbidva 3167	. 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)) ↔ ∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑆‘𝑦) ·_ih 𝑥) = ((𝑇‘𝑦) ·_ih 𝑥)))
11		hoeq1 31577	. 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑦 ∈ ℋ ∀𝑥 ∈ ℋ ((𝑆‘𝑦) ·_ih 𝑥) = ((𝑇‘𝑦) ·_ih 𝑥) ↔ 𝑆 = 𝑇))
12	2, 10, 11	3bitrd 305	1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ (𝑥 ·_ih (𝑆‘𝑦)) = (𝑥 ·_ih (𝑇‘𝑦)) ↔ 𝑆 = 𝑇))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 ℋchba 30666 ·_ih csp 30669

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 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-hfvadd 30747 ax-hvcom 30748 ax-hvass 30749 ax-hv0cl 30750 ax-hvaddid 30751 ax-hfvmul 30752 ax-hvmulid 30753 ax-hvdistr2 30756 ax-hvmul0 30757 ax-hfi 30826 ax-his1 30829 ax-his2 30830 ax-his3 30831 ax-his4 30832

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-po 5579 df-so 5580 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-2 12274 df-cj 15048 df-re 15049 df-im 15050 df-hvsub 30718

This theorem is referenced by: adjcoi 31847

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