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Theorem phssip 19917
Description: The inner product (as a function) on a subspace is a restriction of the inner product (as a function) on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.)
Hypotheses
Ref Expression
phssip.x 𝑋 = (𝑊s 𝑈)
phssip.s 𝑆 = (LSubSp‘𝑊)
phssip.i · = (·if𝑊)
phssip.p 𝑃 = (·if𝑋)
Assertion
Ref Expression
phssip ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈)))

Proof of Theorem phssip
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2626 . . . 4 (Base‘𝑋) = (Base‘𝑋)
2 eqid 2626 . . . 4 (·𝑖𝑋) = (·𝑖𝑋)
3 phssip.p . . . 4 𝑃 = (·if𝑋)
41, 2, 3ipffval 19907 . . 3 𝑃 = (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑋)𝑦))
5 phllmod 19889 . . . . . . 7 (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6 phssip.s . . . . . . . 8 𝑆 = (LSubSp‘𝑊)
76lsssubg 18871 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
85, 7sylan 488 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
9 phssip.x . . . . . . 7 𝑋 = (𝑊s 𝑈)
109subgbas 17514 . . . . . 6 (𝑈 ∈ (SubGrp‘𝑊) → 𝑈 = (Base‘𝑋))
118, 10syl 17 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 = (Base‘𝑋))
12 eqidd 2627 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥(·𝑖𝑊)𝑦) = (𝑥(·𝑖𝑊)𝑦))
1311, 11, 12mpt2eq123dv 6671 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥𝑈, 𝑦𝑈 ↦ (𝑥(·𝑖𝑊)𝑦)) = (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑊)𝑦)))
14 eqid 2626 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
1514subgss 17511 . . . . . 6 (𝑈 ∈ (SubGrp‘𝑊) → 𝑈 ⊆ (Base‘𝑊))
168, 15syl 17 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 ⊆ (Base‘𝑊))
17 resmpt2 6712 . . . . 5 ((𝑈 ⊆ (Base‘𝑊) ∧ 𝑈 ⊆ (Base‘𝑊)) → ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑥(·𝑖𝑊)𝑦)))
1816, 16, 17syl2anc 692 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑥(·𝑖𝑊)𝑦)))
19 eqid 2626 . . . . . . . 8 (·𝑖𝑊) = (·𝑖𝑊)
209, 19, 2ssipeq 19915 . . . . . . 7 (𝑈𝑆 → (·𝑖𝑋) = (·𝑖𝑊))
2120adantl 482 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑊))
2221oveqd 6622 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥(·𝑖𝑋)𝑦) = (𝑥(·𝑖𝑊)𝑦))
2322mpt2eq3dv 6675 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑋)𝑦)) = (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑊)𝑦)))
2413, 18, 233eqtr4rd 2671 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑋)𝑦)) = ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)))
254, 24syl5eq 2672 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)))
26 phssip.i . . . . 5 · = (·if𝑊)
2714, 19, 26ipffval 19907 . . . 4 · = (𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦))
2827a1i 11 . . 3 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → · = (𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)))
2928reseq1d 5359 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ( · ↾ (𝑈 × 𝑈)) = ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)))
3025, 29eqtr4d 2663 1 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wss 3560   × cxp 5077  cres 5081  cfv 5850  (class class class)co 6605  cmpt2 6607  Basecbs 15776  s cress 15777  ·𝑖cip 15862  SubGrpcsubg 17504  LModclmod 18779  LSubSpclss 18846  PreHilcphl 19883  ·ifcipf 19884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-mulcom 9945  ax-addass 9946  ax-mulass 9947  ax-distr 9948  ax-i2m1 9949  ax-1ne0 9950  ax-1rid 9951  ax-rnegex 9952  ax-rrecex 9953  ax-cnre 9954  ax-pre-lttri 9955  ax-pre-lttrn 9956  ax-pre-ltadd 9957  ax-pre-mulgt0 9958
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-nel 2900  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-riota 6566  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-pnf 10021  df-mnf 10022  df-xr 10023  df-ltxr 10024  df-le 10025  df-sub 10213  df-neg 10214  df-nn 10966  df-2 11024  df-3 11025  df-4 11026  df-5 11027  df-6 11028  df-7 11029  df-8 11030  df-ndx 15779  df-slot 15780  df-base 15781  df-sets 15782  df-ress 15783  df-plusg 15870  df-ip 15875  df-0g 16018  df-mgm 17158  df-sgrp 17200  df-mnd 17211  df-grp 17341  df-minusg 17342  df-sbg 17343  df-subg 17507  df-mgp 18406  df-ur 18418  df-ring 18465  df-lmod 18781  df-lss 18847  df-lvec 19017  df-phl 19885  df-ipf 19886
This theorem is referenced by: (None)
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