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Theorem isph 30547
Description: The predicate "is an inner product space." (Contributed by NM, 1-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
isph.1 𝑋 = (BaseSetβ€˜π‘ˆ)
isph.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
isph.3 𝑀 = ( βˆ’π‘£ β€˜π‘ˆ)
isph.6 𝑁 = (normCVβ€˜π‘ˆ)
Assertion
Ref Expression
isph (π‘ˆ ∈ CPreHilOLD ↔ (π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))))
Distinct variable groups:   π‘₯,𝑦,𝐺   π‘₯,𝑀,𝑦   π‘₯,𝑁,𝑦   π‘₯,π‘ˆ,𝑦   π‘₯,𝑋,𝑦

Proof of Theorem isph
StepHypRef Expression
1 phnv 30539 . 2 (π‘ˆ ∈ CPreHilOLD β†’ π‘ˆ ∈ NrmCVec)
2 isph.2 . . . . 5 𝐺 = ( +𝑣 β€˜π‘ˆ)
3 eqid 2724 . . . . 5 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
4 isph.6 . . . . 5 𝑁 = (normCVβ€˜π‘ˆ)
52, 3, 4nvop 30401 . . . 4 (π‘ˆ ∈ NrmCVec β†’ π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ©)
6 eleq1 2813 . . . . 5 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ (π‘ˆ ∈ CPreHilOLD ↔ ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ CPreHilOLD))
72fvexi 6896 . . . . . . 7 𝐺 ∈ V
8 fvex 6895 . . . . . . 7 ( ·𝑠OLD β€˜π‘ˆ) ∈ V
94fvexi 6896 . . . . . . 7 𝑁 ∈ V
10 isph.1 . . . . . . . . 9 𝑋 = (BaseSetβ€˜π‘ˆ)
1110, 2bafval 30329 . . . . . . . 8 𝑋 = ran 𝐺
1211isphg 30542 . . . . . . 7 ((𝐺 ∈ V ∧ ( ·𝑠OLD β€˜π‘ˆ) ∈ V ∧ 𝑁 ∈ V) β†’ (⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ CPreHilOLD ↔ (⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))))
137, 8, 9, 12mp3an 1457 . . . . . 6 (⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ CPreHilOLD ↔ (⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))))
14 isph.3 . . . . . . . . . . . . . . . 16 𝑀 = ( βˆ’π‘£ β€˜π‘ˆ)
1510, 2, 3, 14nvmval 30367 . . . . . . . . . . . . . . 15 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑀𝑦) = (π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))
16153expa 1115 . . . . . . . . . . . . . 14 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑀𝑦) = (π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))
1716fveq2d 6886 . . . . . . . . . . . . 13 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯𝑀𝑦)) = (π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦))))
1817oveq1d 7417 . . . . . . . . . . . 12 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ ((π‘β€˜(π‘₯𝑀𝑦))↑2) = ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2))
1918oveq2d 7418 . . . . . . . . . . 11 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)))
2019eqeq1d 2726 . . . . . . . . . 10 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ ((((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) ↔ (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))))
2120ralbidva 3167 . . . . . . . . 9 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) β†’ (βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) ↔ βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))))
2221ralbidva 3167 . . . . . . . 8 (π‘ˆ ∈ NrmCVec β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))))
2322pm5.32i 574 . . . . . . 7 ((π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))) ↔ (π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))))
24 eleq1 2813 . . . . . . . 8 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ (π‘ˆ ∈ NrmCVec ↔ ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ NrmCVec))
2524anbi1d 629 . . . . . . 7 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ ((π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))) ↔ (⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))))
2623, 25bitr2id 284 . . . . . 6 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ ((⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))) ↔ (π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))))
2713, 26bitrid 283 . . . . 5 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ (⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ CPreHilOLD ↔ (π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))))
286, 27bitrd 279 . . . 4 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ (π‘ˆ ∈ CPreHilOLD ↔ (π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))))
295, 28syl 17 . . 3 (π‘ˆ ∈ NrmCVec β†’ (π‘ˆ ∈ CPreHilOLD ↔ (π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2))))))
3029bianabs 541 . 2 (π‘ˆ ∈ NrmCVec β†’ (π‘ˆ ∈ CPreHilOLD ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))))
311, 30biadanii 819 1 (π‘ˆ ∈ CPreHilOLD ↔ (π‘ˆ ∈ NrmCVec ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (((π‘β€˜(π‘₯𝐺𝑦))↑2) + ((π‘β€˜(π‘₯𝑀𝑦))↑2)) = (2 Β· (((π‘β€˜π‘₯)↑2) + ((π‘β€˜π‘¦)↑2)))))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  Vcvv 3466  βŸ¨cop 4627  β€˜cfv 6534  (class class class)co 7402  1c1 11108   + caddc 11110   Β· cmul 11112  -cneg 11443  2c2 12265  β†‘cexp 14025  NrmCVeccnv 30309   +𝑣 cpv 30310  BaseSetcba 30311   ·𝑠OLD cns 30312   βˆ’π‘£ cnsb 30314  normCVcnmcv 30315  CPreHilOLDccphlo 30537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-po 5579  df-so 5580  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-1st 7969  df-2nd 7970  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11248  df-mnf 11249  df-ltxr 11251  df-sub 11444  df-neg 11445  df-grpo 30218  df-gid 30219  df-ginv 30220  df-gdiv 30221  df-ablo 30270  df-vc 30284  df-nv 30317  df-va 30320  df-ba 30321  df-sm 30322  df-0v 30323  df-vs 30324  df-nmcv 30325  df-ph 30538
This theorem is referenced by:  phpar2  30548
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