Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > pjhval | Structured version Visualization version GIF version |
Description: Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjhval | ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjhfval 28255 | . . 3 ⊢ (𝐻 ∈ Cℋ → (projℎ‘𝐻) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))) | |
2 | 1 | fveq1d 6193 | . 2 ⊢ (𝐻 ∈ Cℋ → ((projℎ‘𝐻)‘𝐴) = ((𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))‘𝐴)) |
3 | eqeq1 2626 | . . . . 5 ⊢ (𝑧 = 𝐴 → (𝑧 = (𝑥 +ℎ 𝑦) ↔ 𝐴 = (𝑥 +ℎ 𝑦))) | |
4 | 3 | rexbidv 3052 | . . . 4 ⊢ (𝑧 = 𝐴 → (∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
5 | 4 | riotabidv 6613 | . . 3 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
6 | eqid 2622 | . . 3 ⊢ (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))) = (𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦))) | |
7 | riotaex 6615 | . . 3 ⊢ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦)) ∈ V | |
8 | 5, 6, 7 | fvmpt 6282 | . 2 ⊢ (𝐴 ∈ ℋ → ((𝑧 ∈ ℋ ↦ (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 +ℎ 𝑦)))‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
9 | 2, 8 | sylan9eq 2676 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) = (℩𝑥 ∈ 𝐻 ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 +ℎ 𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ↦ cmpt 4729 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 ℋchil 27776 +ℎ cva 27777 Cℋ cch 27786 ⊥cort 27787 projℎcpjh 27794 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-hilex 27856 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-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-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-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-pjh 28254 |
This theorem is referenced by: pjpreeq 28257 |
Copyright terms: Public domain | W3C validator |