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Theorem isph 29813
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 29805 . 2 (π‘ˆ ∈ CPreHilOLD β†’ π‘ˆ ∈ NrmCVec)
2 isph.2 . . . . 5 𝐺 = ( +𝑣 β€˜π‘ˆ)
3 eqid 2733 . . . . 5 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
4 isph.6 . . . . 5 𝑁 = (normCVβ€˜π‘ˆ)
52, 3, 4nvop 29667 . . . 4 (π‘ˆ ∈ NrmCVec β†’ π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ©)
6 eleq1 2822 . . . . 5 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ (π‘ˆ ∈ CPreHilOLD ↔ ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ CPreHilOLD))
72fvexi 6860 . . . . . . 7 𝐺 ∈ V
8 fvex 6859 . . . . . . 7 ( ·𝑠OLD β€˜π‘ˆ) ∈ V
94fvexi 6860 . . . . . . 7 𝑁 ∈ V
10 isph.1 . . . . . . . . 9 𝑋 = (BaseSetβ€˜π‘ˆ)
1110, 2bafval 29595 . . . . . . . 8 𝑋 = ran 𝐺
1211isphg 29808 . . . . . . 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 1462 . . . . . 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 29633 . . . . . . . . . . . . . . 15 ((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑀𝑦) = (π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))
16153expa 1119 . . . . . . . . . . . . . 14 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘₯𝑀𝑦) = (π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦)))
1716fveq2d 6850 . . . . . . . . . . . . 13 (((π‘ˆ ∈ NrmCVec ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯𝑀𝑦)) = (π‘β€˜(π‘₯𝐺(-1( ·𝑠OLD β€˜π‘ˆ)𝑦))))
1817oveq1d 7376 . . . . . . . . . . . 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 7377 . . . . . . . . . . 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 2735 . . . . . . . . . 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 3169 . . . . . . . . 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 3169 . . . . . . . 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 576 . . . . . . 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 2822 . . . . . . . 8 (π‘ˆ = ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© β†’ (π‘ˆ ∈ NrmCVec ↔ ⟨⟨𝐺, ( ·𝑠OLD β€˜π‘ˆ)⟩, π‘βŸ© ∈ NrmCVec))
2524anbi1d 631 . . . . . . 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 543 . 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3447  βŸ¨cop 4596  β€˜cfv 6500  (class class class)co 7361  1c1 11060   + caddc 11062   Β· cmul 11064  -cneg 11394  2c2 12216  β†‘cexp 13976  NrmCVeccnv 29575   +𝑣 cpv 29576  BaseSetcba 29577   ·𝑠OLD cns 29578   βˆ’π‘£ cnsb 29580  normCVcnmcv 29581  CPreHilOLDccphlo 29803
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-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-po 5549  df-so 5550  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-ltxr 11202  df-sub 11395  df-neg 11396  df-grpo 29484  df-gid 29485  df-ginv 29486  df-gdiv 29487  df-ablo 29536  df-vc 29550  df-nv 29583  df-va 29586  df-ba 29587  df-sm 29588  df-0v 29589  df-vs 29590  df-nmcv 29591  df-ph 29804
This theorem is referenced by:  phpar2  29814
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