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Theorem tcphval 24605
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 6846 . . . . . . 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 6846 . . . . . . . . 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 7378 . . . . . . 7 (𝑀 = π‘Š β†’ (π‘₯(Β·π‘–β€˜π‘€)π‘₯) = (π‘₯ , π‘₯))
109fveq2d 6850 . . . . . 6 (𝑀 = π‘Š β†’ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)) = (βˆšβ€˜(π‘₯ , π‘₯)))
115, 10mpteq12dv 5200 . . . . 5 (𝑀 = π‘Š β†’ (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯))) = (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯))))
122, 11oveq12d 7379 . . . 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 24556 . . . 4 toβ„‚PreHil = (𝑀 ∈ V ↦ (𝑀 toNrmGrp (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (βˆšβ€˜(π‘₯(Β·π‘–β€˜π‘€)π‘₯)))))
14 ovex 7394 . . . 4 (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))) ∈ V
1512, 13, 14fvmpt 6952 . . 3 (π‘Š ∈ V β†’ (toβ„‚PreHilβ€˜π‘Š) = (π‘Š toNrmGrp (π‘₯ ∈ 𝑉 ↦ (βˆšβ€˜(π‘₯ , π‘₯)))))
16 fvprc 6838 . . . 4 (Β¬ π‘Š ∈ V β†’ (toβ„‚PreHilβ€˜π‘Š) = βˆ…)
17 reldmtng 24017 . . . . 5 Rel dom toNrmGrp
1817ovprc1 7400 . . . 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 3447  βˆ…c0 4286   ↦ cmpt 5192  β€˜cfv 6500  (class class class)co 7361  βˆšcsqrt 15127  Basecbs 17091  Β·π‘–cip 17146   toNrmGrp ctng 23957  toβ„‚PreHilctcph 24554
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 5260  ax-nul 5267  ax-pr 5388
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-tng 23963  df-tcph 24556
This theorem is referenced by:  tcphbas  24606  tchplusg  24607  tcphmulr  24609  tcphsca  24610  tcphvsca  24611  tcphip  24612  tcphtopn  24613  tchnmfval  24615  tcphds  24618  tcphcph  24624  rrxsca  24783  rrx0  24784  rrxdim  32373
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