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Theorem phssip 20220
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 2771 . . . 4 (Base‘𝑋) = (Base‘𝑋)
2 eqid 2771 . . . 4 (·𝑖𝑋) = (·𝑖𝑋)
3 phssip.p . . . 4 𝑃 = (·if𝑋)
41, 2, 3ipffval 20210 . . 3 𝑃 = (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑋)𝑦))
5 phllmod 20192 . . . . . . 7 (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6 phssip.s . . . . . . . 8 𝑆 = (LSubSp‘𝑊)
76lsssubg 19170 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
85, 7sylan 569 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
9 phssip.x . . . . . . 7 𝑋 = (𝑊s 𝑈)
109subgbas 17806 . . . . . 6 (𝑈 ∈ (SubGrp‘𝑊) → 𝑈 = (Base‘𝑋))
118, 10syl 17 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 = (Base‘𝑋))
12 eqidd 2772 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥(·𝑖𝑊)𝑦) = (𝑥(·𝑖𝑊)𝑦))
1311, 11, 12mpt2eq123dv 6864 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥𝑈, 𝑦𝑈 ↦ (𝑥(·𝑖𝑊)𝑦)) = (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑊)𝑦)))
14 eqid 2771 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
1514subgss 17803 . . . . . 6 (𝑈 ∈ (SubGrp‘𝑊) → 𝑈 ⊆ (Base‘𝑊))
168, 15syl 17 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 ⊆ (Base‘𝑊))
17 resmpt2 6905 . . . . 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 573 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑥(·𝑖𝑊)𝑦)))
19 eqid 2771 . . . . . . . 8 (·𝑖𝑊) = (·𝑖𝑊)
209, 19, 2ssipeq 20218 . . . . . . 7 (𝑈𝑆 → (·𝑖𝑋) = (·𝑖𝑊))
2120adantl 467 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑊))
2221oveqd 6810 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥(·𝑖𝑋)𝑦) = (𝑥(·𝑖𝑊)𝑦))
2322mpt2eq3dv 6868 . . . 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 2816 . . 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 2817 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)))
26 phssip.i . . . . 5 · = (·if𝑊)
2714, 19, 26ipffval 20210 . . . 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 5533 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ( · ↾ (𝑈 × 𝑈)) = ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)))
3025, 29eqtr4d 2808 1 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wss 3723   × cxp 5247  cres 5251  cfv 6031  (class class class)co 6793  cmpt2 6795  Basecbs 16064  s cress 16065  ·𝑖cip 16154  SubGrpcsubg 17796  LModclmod 19073  LSubSpclss 19142  PreHilcphl 20186  ·ifcipf 20187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-3 11282  df-4 11283  df-5 11284  df-6 11285  df-7 11286  df-8 11287  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-ip 16167  df-0g 16310  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-grp 17633  df-minusg 17634  df-sbg 17635  df-subg 17799  df-mgp 18698  df-ur 18710  df-ring 18757  df-lmod 19075  df-lss 19143  df-lvec 19316  df-phl 20188  df-ipf 20189
This theorem is referenced by: (None)
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