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Theorem phssip 20795
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 2824 . . . 4 (Base‘𝑋) = (Base‘𝑋)
2 eqid 2824 . . . 4 (·𝑖𝑋) = (·𝑖𝑋)
3 phssip.p . . . 4 𝑃 = (·if𝑋)
41, 2, 3ipffval 20785 . . 3 𝑃 = (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑋)𝑦))
5 phllmod 20767 . . . . . . 7 (𝑊 ∈ PreHil → 𝑊 ∈ LMod)
6 phssip.s . . . . . . . 8 𝑆 = (LSubSp‘𝑊)
76lsssubg 19722 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
85, 7sylan 583 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
9 phssip.x . . . . . . 7 𝑋 = (𝑊s 𝑈)
109subgbas 18279 . . . . . 6 (𝑈 ∈ (SubGrp‘𝑊) → 𝑈 = (Base‘𝑋))
118, 10syl 17 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 = (Base‘𝑋))
12 eqidd 2825 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥(·𝑖𝑊)𝑦) = (𝑥(·𝑖𝑊)𝑦))
1311, 11, 12mpoeq123dv 7218 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥𝑈, 𝑦𝑈 ↦ (𝑥(·𝑖𝑊)𝑦)) = (𝑥 ∈ (Base‘𝑋), 𝑦 ∈ (Base‘𝑋) ↦ (𝑥(·𝑖𝑊)𝑦)))
14 eqid 2824 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
1514subgss 18276 . . . . . 6 (𝑈 ∈ (SubGrp‘𝑊) → 𝑈 ⊆ (Base‘𝑊))
168, 15syl 17 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑈 ⊆ (Base‘𝑊))
17 resmpo 7261 . . . . 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 587 . . . 4 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑥(·𝑖𝑊)𝑦)))
19 eqid 2824 . . . . . . . 8 (·𝑖𝑊) = (·𝑖𝑊)
209, 19, 2ssipeq 20793 . . . . . . 7 (𝑈𝑆 → (·𝑖𝑋) = (·𝑖𝑊))
2120adantl 485 . . . . . 6 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (·𝑖𝑋) = (·𝑖𝑊))
2221oveqd 7162 . . . . 5 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → (𝑥(·𝑖𝑋)𝑦) = (𝑥(·𝑖𝑊)𝑦))
2322mpoeq3dv 7222 . . . 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 2870 . . 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 2871 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)))
26 phssip.i . . . . 5 · = (·if𝑊)
2714, 19, 26ipffval 20785 . . . 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 5839 . 2 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → ( · ↾ (𝑈 × 𝑈)) = ((𝑥 ∈ (Base‘𝑊), 𝑦 ∈ (Base‘𝑊) ↦ (𝑥(·𝑖𝑊)𝑦)) ↾ (𝑈 × 𝑈)))
3025, 29eqtr4d 2862 1 ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wss 3919   × cxp 5540  cres 5544  cfv 6343  (class class class)co 7145  cmpo 7147  Basecbs 16479  s cress 16480  ·𝑖cip 16566  SubGrpcsubg 18269  LModclmod 19627  LSubSpclss 19696  PreHilcphl 20761  ·ifcipf 20762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-tp 4554  df-op 4556  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7571  df-1st 7679  df-2nd 7680  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-er 8279  df-en 8500  df-dom 8501  df-sdom 8502  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11693  df-3 11694  df-4 11695  df-5 11696  df-6 11697  df-7 11698  df-8 11699  df-ndx 16482  df-slot 16483  df-base 16485  df-sets 16486  df-ress 16487  df-plusg 16574  df-ip 16579  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-grp 18102  df-minusg 18103  df-sbg 18104  df-subg 18272  df-mgp 19236  df-ur 19248  df-ring 19295  df-lmod 19629  df-lss 19697  df-lvec 19868  df-phl 20763  df-ipf 20764
This theorem is referenced by: (None)
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