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Theorem isph 30626
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 30618 . 2 (π‘ˆ ∈ CPreHilOLD β†’ π‘ˆ ∈ NrmCVec)
2 isph.2 . . . . 5 𝐺 = ( +𝑣 β€˜π‘ˆ)
3 eqid 2728 . . . . 5 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
4 isph.6 . . . . 5 𝑁 = (normCVβ€˜π‘ˆ)
52, 3, 4nvop 30480 . . . 4 (π‘ˆ ∈ NrmCVec β†’ π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ©)
6 eleq1 2817 . . . . 5 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ (π‘ˆ ∈ CPreHilOLD ↔ ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ CPreHilOLD))
72fvexi 6906 . . . . . . 7 𝐺 ∈ V
8 fvex 6905 . . . . . . 7 ( ·𝑠OLD β€˜π‘ˆ) ∈ V
94fvexi 6906 . . . . . . 7 𝑁 ∈ V
10 isph.1 . . . . . . . . 9 𝑋 = (BaseSetβ€˜π‘ˆ)
1110, 2bafval 30408 . . . . . . . 8 𝑋 = ran 𝐺
1211isphg 30621 . . . . . . 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 1458 . . . . . 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 30446 . . . . . . . . . . . . . . 15 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑀𝑦) = (π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))
16153expa 1116 . . . . . . . . . . . . . 14 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑀𝑦) = (π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))
1716fveq2d 6896 . . . . . . . . . . . . 13 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯𝑀𝑦)) = (π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦))))
1817oveq1d 7430 . . . . . . . . . . . 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 7431 . . . . . . . . . . 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 2730 . . . . . . . . . 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 3171 . . . . . . . . 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 3171 . . . . . . . 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 2817 . . . . . . . 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 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 821 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 1534   ∈ wcel 2099  βˆ€wral 3057  Vcvv 3470  βŸ¨cop 4631  β€˜cfv 6543  (class class class)co 7415  1c1 11134   + caddc 11136   Β· cmul 11138  -cneg 11470  2c2 12292  β†‘cexp 14053  NrmCVeccnv 30388   +𝑣 cpv 30389  BaseSetcba 30390   ·𝑠OLD cns 30391   βˆ’π‘£ cnsb 30393  normCVcnmcv 30394  CPreHilOLDccphlo 30616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-po 5585  df-so 5586  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  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 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7988  df-2nd 7989  df-er 8719  df-en 8959  df-dom 8960  df-sdom 8961  df-pnf 11275  df-mnf 11276  df-ltxr 11278  df-sub 11471  df-neg 11472  df-grpo 30297  df-gid 30298  df-ginv 30299  df-gdiv 30300  df-ablo 30349  df-vc 30363  df-nv 30396  df-va 30399  df-ba 30400  df-sm 30401  df-0v 30402  df-vs 30403  df-nmcv 30404  df-ph 30617
This theorem is referenced by:  phpar2  30627
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