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Theorem nvscom 29668
Description: Commutative law for the scalar product of a normed complex vector space. (Contributed by NM, 14-Feb-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvscl.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nvscl.4 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
Assertion
Ref Expression
nvscom ((π‘ˆ ∈ NrmCVec ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝑆(𝐡𝑆𝐢)) = (𝐡𝑆(𝐴𝑆𝐢)))
Proof of Theorem nvscom
StepHypRef Expression
1 mulcom 11161 . . . . 5 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ (𝐴 Β· 𝐡) = (𝐡 Β· 𝐴))
21oveq1d 7392 . . . 4 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚) β†’ ((𝐴 Β· 𝐡)𝑆𝐢) = ((𝐡 Β· 𝐴)𝑆𝐢))
323adant3 1132 . . 3 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ 𝑋) β†’ ((𝐴 Β· 𝐡)𝑆𝐢) = ((𝐡 Β· 𝐴)𝑆𝐢))
43adantl 482 . 2 ((π‘ˆ ∈ NrmCVec ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴 Β· 𝐡)𝑆𝐢) = ((𝐡 Β· 𝐴)𝑆𝐢))
5 nvscl.1 . . 3 𝑋 = (BaseSetβ€˜π‘ˆ)
6 nvscl.4 . . 3 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
75, 6nvsass 29667 . 2 ((π‘ˆ ∈ NrmCVec ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴 Β· 𝐡)𝑆𝐢) = (𝐴𝑆(𝐡𝑆𝐢)))
8 3ancoma 1098 . . 3 ((𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ 𝑋) ↔ (𝐡 ∈ β„‚ ∧ 𝐴 ∈ β„‚ ∧ 𝐢 ∈ 𝑋))
95, 6nvsass 29667 . . 3 ((π‘ˆ ∈ NrmCVec ∧ (𝐡 ∈ β„‚ ∧ 𝐴 ∈ β„‚ ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐡 Β· 𝐴)𝑆𝐢) = (𝐡𝑆(𝐴𝑆𝐢)))
108, 9sylan2b 594 . 2 ((π‘ˆ ∈ NrmCVec ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐡 Β· 𝐴)𝑆𝐢) = (𝐡𝑆(𝐴𝑆𝐢)))
114, 7, 103eqtr3d 2779 1 ((π‘ˆ ∈ NrmCVec ∧ (𝐴 ∈ β„‚ ∧ 𝐡 ∈ β„‚ ∧ 𝐢 ∈ 𝑋)) β†’ (𝐴𝑆(𝐡𝑆𝐢)) = (𝐡𝑆(𝐴𝑆𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  β€˜cfv 6516  (class class class)co 7377  β„‚cc 11073   Β· cmul 11080  NrmCVeccnv 29623  BaseSetcba 29625   ·𝑠OLD cns 29626
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 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pr 5404  ax-un 7692  ax-mulcom 11139
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-ov 7380  df-oprab 7381  df-1st 7941  df-2nd 7942  df-vc 29598  df-nv 29631  df-va 29634  df-ba 29635  df-sm 29636  df-0v 29637  df-nmcv 29639
This theorem is referenced by:  nvmdi  29687
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