Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  phssip Structured version   Visualization version   GIF version

Theorem phssip 19822
 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 2610 . . . 4 (Base‘𝑋) = (Base‘𝑋)
2 eqid 2610 . . . 4 (·𝑖𝑋) = (·𝑖𝑋)
3 phssip.p . . . 4 𝑃 = (·if𝑋)
41, 2, 3ipffval 19812 . . 3 𝑃 = (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑋)𝑦))
5 phllmod 19794 . . . . . . 7 (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6 phssip.s . . . . . . . 8 𝑆 = (LSubSp‘𝑊)
76lsssubg 18778 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
85, 7sylan 487 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
9 phssip.x . . . . . . 7 𝑋 = (𝑊s 𝑈)
109subgbas 17421 . . . . . 6 (𝑈 ∈ (SubGrp‘𝑊) → 𝑈 = (Base‘𝑋))
118, 10syl 17 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 = (Base‘𝑋))
12 eqidd 2611 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥(·𝑖𝑊)𝑦) = (𝑥(·𝑖𝑊)𝑦))
1311, 11, 12mpt2eq123dv 6615 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥𝑈, 𝑦𝑈 ↦ (𝑥(·𝑖𝑊)𝑦)) = (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑊)𝑦)))
14 eqid 2610 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
1514subgss 17418 . . . . . 6 (𝑈 ∈ (SubGrp‘𝑊) → 𝑈 ⊆ (Base‘𝑊))
168, 15syl 17 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 ⊆ (Base‘𝑊))
17 resmpt2 6656 . . . . 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 691 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑥(·𝑖𝑊)𝑦)))
19 eqid 2610 . . . . . . . 8 (·𝑖𝑊) = (·𝑖𝑊)
209, 19, 2ssipeq 19820 . . . . . . 7 (𝑈𝑆 → (·𝑖𝑋) = (·𝑖𝑊))
2120adantl 481 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑊))
2221oveqd 6566 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥(·𝑖𝑋)𝑦) = (𝑥(·𝑖𝑊)𝑦))
2322mpt2eq3dv 6619 . . . 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 2655 . . 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 2656 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)))
26 phssip.i . . . . 5 · = (·if𝑊)
2714, 19, 26ipffval 19812 . . . 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 5316 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ( · ↾ (𝑈 × 𝑈)) = ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)))
3025, 29eqtr4d 2647 1 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695   ↾s cress 15696  ·𝑖cip 15773  SubGrpcsubg 17411  LModclmod 18686  LSubSpclss 18753  PreHilcphl 19788  ·ifcipf 19789 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-ip 15786  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lss 18754  df-lvec 18924  df-phl 19790  df-ipf 19791 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator