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Theorem isph 30070
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 30062 . 2 (π‘ˆ ∈ CPreHilOLD β†’ π‘ˆ ∈ NrmCVec)
2 isph.2 . . . . 5 𝐺 = ( +𝑣 β€˜π‘ˆ)
3 eqid 2732 . . . . 5 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
4 isph.6 . . . . 5 𝑁 = (normCVβ€˜π‘ˆ)
52, 3, 4nvop 29924 . . . 4 (π‘ˆ ∈ NrmCVec β†’ π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ©)
6 eleq1 2821 . . . . 5 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ (π‘ˆ ∈ CPreHilOLD ↔ ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ CPreHilOLD))
72fvexi 6905 . . . . . . 7 𝐺 ∈ V
8 fvex 6904 . . . . . . 7 ( ·𝑠OLD β€˜π‘ˆ) ∈ V
94fvexi 6905 . . . . . . 7 𝑁 ∈ V
10 isph.1 . . . . . . . . 9 𝑋 = (BaseSetβ€˜π‘ˆ)
1110, 2bafval 29852 . . . . . . . 8 𝑋 = ran 𝐺
1211isphg 30065 . . . . . . 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 1461 . . . . . 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 29890 . . . . . . . . . . . . . . 15 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑀𝑦) = (π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))
16153expa 1118 . . . . . . . . . . . . . 14 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑀𝑦) = (π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))
1716fveq2d 6895 . . . . . . . . . . . . 13 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯𝑀𝑦)) = (π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦))))
1817oveq1d 7423 . . . . . . . . . . . 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 7424 . . . . . . . . . . 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 2734 . . . . . . . . . 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 3175 . . . . . . . . 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 3175 . . . . . . . 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 575 . . . . . . 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 2821 . . . . . . . 8 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ (π‘ˆ ∈ NrmCVec ↔ ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ NrmCVec))
2524anbi1d 630 . . . . . . 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 283 . . . . . 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 282 . . . . 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 278 . . . 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 542 . 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 820 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474  βŸ¨cop 4634  β€˜cfv 6543  (class class class)co 7408  1c1 11110   + caddc 11112   Β· cmul 11114  -cneg 11444  2c2 12266  β†‘cexp 14026  NrmCVeccnv 29832   +𝑣 cpv 29833  BaseSetcba 29834   ·𝑠OLD cns 29835   βˆ’π‘£ cnsb 29837  normCVcnmcv 29838  CPreHilOLDccphlo 30060
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-ltxr 11252  df-sub 11445  df-neg 11446  df-grpo 29741  df-gid 29742  df-ginv 29743  df-gdiv 29744  df-ablo 29793  df-vc 29807  df-nv 29840  df-va 29843  df-ba 29844  df-sm 29845  df-0v 29846  df-vs 29847  df-nmcv 29848  df-ph 30061
This theorem is referenced by:  phpar2  30071
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