| Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > HSE Home > Th. List > pjfni | Structured version Visualization version GIF version | ||
| Description: Functionality of a projection. (Contributed by NM, 30-Oct-1999.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| pjfn.1 | ⊢ 𝐻 ∈ Cℋ |
| Ref | Expression |
|---|---|
| pjfni | ⊢ (projℎ‘𝐻) Fn ℋ |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | riotaex 7392 | . 2 ⊢ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)) ∈ V | |
| 2 | pjfn.1 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
| 3 | pjhfval 31415 | . . 3 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧)))) | |
| 4 | 2, 3 | ax-mp 5 | . 2 ⊢ (projℎ‘𝐻) = (𝑥 ∈ ℋ ↦ (℩𝑦 ∈ 𝐻 ∃𝑧 ∈ (⊥‘𝐻)𝑥 = (𝑦 +ℎ 𝑧))) |
| 5 | 1, 4 | fnmpti 6711 | 1 ⊢ (projℎ‘𝐻) Fn ℋ |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ↦ cmpt 5225 Fn wfn 6556 ‘cfv 6561 ℩crio 7387 (class class class)co 7431 ℋchba 30938 +ℎ cva 30939 Cℋ cch 30948 ⊥cort 30949 projℎcpjh 30956 |
| 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-hilex 31018 |
| 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-reu 3381 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-iun 4993 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-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-pjh 31414 |
| This theorem is referenced by: pjrni 31721 pjfoi 31722 pjfi 31723 dfiop2 31772 hmopidmpji 32171 pjssdif2i 32193 pjimai 32195 |
| Copyright terms: Public domain | W3C validator |