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Mirrors > Home > MPE Home > Th. List > ipcl | Structured version Visualization version GIF version |
Description: Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.) |
Ref | Expression |
---|---|
phlsrng.f | ⊢ 𝐹 = (Scalar‘𝑊) |
phllmhm.h | ⊢ , = (·𝑖‘𝑊) |
phllmhm.v | ⊢ 𝑉 = (Base‘𝑊) |
ipcl.f | ⊢ 𝐾 = (Base‘𝐹) |
Ref | Expression |
---|---|
ipcl | ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phlsrng.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑊) | |
2 | phllmhm.h | . . . . . 6 ⊢ , = (·𝑖‘𝑊) | |
3 | phllmhm.v | . . . . . 6 ⊢ 𝑉 = (Base‘𝑊) | |
4 | eqid 2820 | . . . . . 6 ⊢ (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐵)) = (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐵)) | |
5 | 1, 2, 3, 4 | phllmhm 20771 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐵)) ∈ (𝑊 LMHom (ringLMod‘𝐹))) |
6 | ipcl.f | . . . . . . 7 ⊢ 𝐾 = (Base‘𝐹) | |
7 | rlmbas 19962 | . . . . . . 7 ⊢ (Base‘𝐹) = (Base‘(ringLMod‘𝐹)) | |
8 | 6, 7 | eqtri 2843 | . . . . . 6 ⊢ 𝐾 = (Base‘(ringLMod‘𝐹)) |
9 | 3, 8 | lmhmf 19801 | . . . . 5 ⊢ ((𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐵)) ∈ (𝑊 LMHom (ringLMod‘𝐹)) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐵)):𝑉⟶𝐾) |
10 | 5, 9 | syl 17 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉) → (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐵)):𝑉⟶𝐾) |
11 | 4 | fmpt 6867 | . . . 4 ⊢ (∀𝑥 ∈ 𝑉 (𝑥 , 𝐵) ∈ 𝐾 ↔ (𝑥 ∈ 𝑉 ↦ (𝑥 , 𝐵)):𝑉⟶𝐾) |
12 | 10, 11 | sylibr 236 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉) → ∀𝑥 ∈ 𝑉 (𝑥 , 𝐵) ∈ 𝐾) |
13 | oveq1 7156 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 , 𝐵) = (𝐴 , 𝐵)) | |
14 | 13 | eleq1d 2896 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 , 𝐵) ∈ 𝐾 ↔ (𝐴 , 𝐵) ∈ 𝐾)) |
15 | 14 | rspccva 3619 | . . 3 ⊢ ((∀𝑥 ∈ 𝑉 (𝑥 , 𝐵) ∈ 𝐾 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐵) ∈ 𝐾) |
16 | 12, 15 | stoic3 1776 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐵) ∈ 𝐾) |
17 | 16 | 3com23 1121 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 , 𝐵) ∈ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ↦ cmpt 5139 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 Scalarcsca 16563 ·𝑖cip 16565 LMHom clmhm 19786 ringLModcrglmod 19936 PreHilcphl 20763 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-sca 16576 df-vsca 16577 df-ip 16578 df-ghm 18351 df-lmhm 19789 df-sra 19939 df-rgmod 19940 df-phl 20765 |
This theorem is referenced by: iporthcom 20774 ipdi 20779 ip2di 20780 ipsubdir 20781 ipsubdi 20782 ip2subdi 20783 ipassr 20785 phlipf 20791 ip2eq 20792 phlssphl 20798 lsmcss 20831 cphipcl 23790 cphnmf 23794 cphsubdir 23807 cphsubdi 23808 cph2subdi 23809 tcphcphlem3 23831 ipcau2 23832 tcphcphlem1 23833 tcphcph 23835 nmparlem 23837 pjthlem1 24035 |
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