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Mirrors > Home > HSE Home > Th. List > hoeq1 | Structured version Visualization version GIF version |
Description: A condition implying that two Hilbert space operators are equal. Lemma 3.2(S9) of [Beran] p. 95. (Contributed by NM, 15-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hoeq1 | ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ 𝑆 = 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffvelrn 6357 | . . . . 5 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑆‘𝑥) ∈ ℋ) | |
2 | ffvelrn 6357 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ) | |
3 | hial2eq 27963 | . . . . 5 ⊢ (((𝑆‘𝑥) ∈ ℋ ∧ (𝑇‘𝑥) ∈ ℋ) → (∀𝑦 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ (𝑆‘𝑥) = (𝑇‘𝑥))) | |
4 | 1, 2, 3 | syl2an 494 | . . . 4 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (∀𝑦 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ (𝑆‘𝑥) = (𝑇‘𝑥))) |
5 | 4 | anandirs 874 | . . 3 ⊢ (((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (∀𝑦 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ (𝑆‘𝑥) = (𝑇‘𝑥))) |
6 | 5 | ralbidva 2985 | . 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ ∀𝑥 ∈ ℋ (𝑆‘𝑥) = (𝑇‘𝑥))) |
7 | ffn 6045 | . . 3 ⊢ (𝑆: ℋ⟶ ℋ → 𝑆 Fn ℋ) | |
8 | ffn 6045 | . . 3 ⊢ (𝑇: ℋ⟶ ℋ → 𝑇 Fn ℋ) | |
9 | eqfnfv 6311 | . . 3 ⊢ ((𝑆 Fn ℋ ∧ 𝑇 Fn ℋ) → (𝑆 = 𝑇 ↔ ∀𝑥 ∈ ℋ (𝑆‘𝑥) = (𝑇‘𝑥))) | |
10 | 7, 8, 9 | syl2an 494 | . 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (𝑆 = 𝑇 ↔ ∀𝑥 ∈ ℋ (𝑆‘𝑥) = (𝑇‘𝑥))) |
11 | 6, 10 | bitr4d 271 | 1 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ ∀𝑦 ∈ ℋ ((𝑆‘𝑥) ·ih 𝑦) = ((𝑇‘𝑥) ·ih 𝑦) ↔ 𝑆 = 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Fn wfn 5883 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℋchil 27776 ·ih csp 27779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-hfvadd 27857 ax-hvcom 27858 ax-hvass 27859 ax-hv0cl 27860 ax-hvaddid 27861 ax-hfvmul 27862 ax-hvmulid 27863 ax-hvdistr2 27866 ax-hvmul0 27867 ax-hfi 27936 ax-his2 27940 ax-his3 27941 ax-his4 27942 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268 df-neg 10269 df-hvsub 27828 |
This theorem is referenced by: hoeq2 28690 adjmo 28691 adjadj 28795 |
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