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Theorem tcphval 24735
Description: Define a function to augment a subcomplex pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
tcphval.n 𝐺 = (toβ„‚PreHilβ€˜π‘Š)
tcphval.v 𝑉 = (Baseβ€˜π‘Š)
tcphval.h , = (Β·π‘–β€˜π‘Š)
Assertion
Ref Expression
tcphval 𝐺 = (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
Distinct variable groups:   π‘₯, ,   π‘₯,𝐺   π‘₯,𝑉   π‘₯,π‘Š

Proof of Theorem tcphval
Dummy variable 𝑀 is distinct from all other variables.
StepHypRef Expression
1 tcphval.n . 2 𝐺 = (toβ„‚PreHilβ€˜π‘Š)
2 id 22 . . . . 5 (𝑀 = π‘Š β†’ 𝑀 = π‘Š)
3 fveq2 6892 . . . . . . 7 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = (Baseβ€˜π‘Š))
4 tcphval.v . . . . . . 7 𝑉 = (Baseβ€˜π‘Š)
53, 4eqtr4di 2791 . . . . . 6 (𝑀 = π‘Š β†’ (Baseβ€˜π‘€) = 𝑉)
6 fveq2 6892 . . . . . . . . 9 (𝑀 = π‘Š β†’ (Β·π‘–β€˜π‘€) = (Β·π‘–β€˜π‘Š))
7 tcphval.h . . . . . . . . 9 , = (Β·π‘–β€˜π‘Š)
86, 7eqtr4di 2791 . . . . . . . 8 (𝑀 = π‘Š β†’ (Β·π‘–β€˜π‘€) = , )
98oveqd 7426 . . . . . . 7 (𝑀 = π‘Š β†’ (π‘₯(Β·π‘–β€˜π‘€)π‘₯) = (π‘₯ , π‘₯))
109fveq2d 6896 . . . . . 6 (𝑀 = π‘Š β†’ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)) = (βˆšβ€˜(π‘₯ , π‘₯)))
115, 10mpteq12dv 5240 . . . . 5 (𝑀 = π‘Š β†’ (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯))) = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
122, 11oveq12d 7427 . . . 4 (𝑀 = π‘Š β†’ (𝑀 toNrmGrp (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)))) = (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))
13 df-tcph 24686 . . . 4 toβ„‚PreHil = (𝑀 ∈ V ↦ (𝑀 toNrmGrp (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)))))
14 ovex 7442 . . . 4 (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))) ∈ V
1512, 13, 14fvmpt 6999 . . 3 (π‘Š ∈ V β†’ (toβ„‚PreHilβ€˜π‘Š) = (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))
16 fvprc 6884 . . . 4 (Β¬ π‘Š ∈ V β†’ (toβ„‚PreHilβ€˜π‘Š) = βˆ…)
17 reldmtng 24147 . . . . 5 Rel dom toNrmGrp
1817ovprc1 7448 . . . 4 (Β¬ π‘Š ∈ V β†’ (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))) = βˆ…)
1916, 18eqtr4d 2776 . . 3 (Β¬ π‘Š ∈ V β†’ (toβ„‚PreHilβ€˜π‘Š) = (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))
2015, 19pm2.61i 182 . 2 (toβ„‚PreHilβ€˜π‘Š) = (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
211, 20eqtri 2761 1 𝐺 = (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   = wceq 1542   ∈ wcel 2107  Vcvv 3475  βˆ…c0 4323   ↦ cmpt 5232  β€˜cfv 6544  (class class class)co 7409  βˆšcsqrt 15180  Basecbs 17144  Β·π‘–cip 17202   toNrmGrp ctng 24087  toβ„‚PreHilctcph 24684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-tng 24093  df-tcph 24686
This theorem is referenced by:  tcphbas  24736  tchplusg  24737  tcphmulr  24739  tcphsca  24740  tcphvsca  24741  tcphip  24742  tcphtopn  24743  tchnmfval  24745  tcphds  24748  tcphcph  24754  rrxsca  24913  rrx0  24914  rrxdim  32699
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