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| Description: Equality of Hilbert space operators. (Contributed by NM, 14-Nov-2000.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| hoeq.1 | ⊢ 𝑆: ℋ⟶ ℋ | 
| hoeq.2 | ⊢ 𝑇: ℋ⟶ ℋ | 
| Ref | Expression | 
|---|---|
| hoeqi | ⊢ (∀𝑥 ∈ ℋ (𝑆‘𝑥) = (𝑇‘𝑥) ↔ 𝑆 = 𝑇) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hoeq.1 | . 2 ⊢ 𝑆: ℋ⟶ ℋ | |
| 2 | hoeq.2 | . 2 ⊢ 𝑇: ℋ⟶ ℋ | |
| 3 | hoeq 31779 | . 2 ⊢ ((𝑆: ℋ⟶ ℋ ∧ 𝑇: ℋ⟶ ℋ) → (∀𝑥 ∈ ℋ (𝑆‘𝑥) = (𝑇‘𝑥) ↔ 𝑆 = 𝑇)) | |
| 4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (∀𝑥 ∈ ℋ (𝑆‘𝑥) = (𝑇‘𝑥) ↔ 𝑆 = 𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∀wral 3061 ⟶wf 6557 ‘cfv 6561 ℋchba 30938 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 | 
| This theorem is referenced by: hoaddcomi 31791 hodsi 31794 hoaddassi 31795 hocadddiri 31798 hocsubdiri 31799 hoaddridi 31805 ho0coi 31807 hoid1i 31808 hoid1ri 31809 honegsubi 31815 hoddii 32008 pjsdii 32174 pjddii 32175 pjss1coi 32182 pjss2coi 32183 pjorthcoi 32188 pjscji 32189 pjtoi 32198 pjclem4 32218 pj3si 32226 pj3cor1i 32228 | 
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