Theorem vcsm 30287

Description: Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
vciOLD.1	⊢ 𝐺 = (1st ‘𝑊)
vciOLD.2	⊢ 𝑆 = (2nd ‘𝑊)
vciOLD.3	⊢ 𝑋 = ran 𝐺

Assertion

Ref	Expression
vcsm	⊢ (𝑊 ∈ CVecOLD → 𝑆:(ℂ × 𝑋)⟶𝑋)

Proof of Theorem vcsm

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vciOLD.1	. . 3 ⊢ 𝐺 = (1st ‘𝑊)
2		vciOLD.2	. . 3 ⊢ 𝑆 = (2nd ‘𝑊)
3		vciOLD.3	. . 3 ⊢ 𝑋 = ran 𝐺
4	1, 2, 3	vciOLD 30286	. 2 ⊢ (𝑊 ∈ CVecOLD → (𝐺 ∈ AbelOp ∧ 𝑆:(ℂ × 𝑋)⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ((1𝑆𝑥) = 𝑥 ∧ ∀𝑦 ∈ ℂ (∀𝑧 ∈ 𝑋 (𝑦𝑆(𝑥𝐺𝑧)) = ((𝑦𝑆𝑥)𝐺(𝑦𝑆𝑧)) ∧ ∀𝑧 ∈ ℂ (((𝑦 + 𝑧)𝑆𝑥) = ((𝑦𝑆𝑥)𝐺(𝑧𝑆𝑥)) ∧ ((𝑦 · 𝑧)𝑆𝑥) = (𝑦𝑆(𝑧𝑆𝑥)))))))
5	4	simp2d 1140	1 ⊢ (𝑊 ∈ CVecOLD → 𝑆:(ℂ × 𝑋)⟶𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053   × cxp 5665  ran crn 5668  ⟶wf 6530  ‘cfv 6534  (class class class)co 7402  1^st c1st 7967  2^nd c2nd 7968  ℂcc 11105  1c1 11108   + caddc 11110   · cmul 11112  AbelOpcablo 30269  CVec_OLDcvc 30283

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-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-1st 7969  df-2nd 7970  df-vc 30284

This theorem is referenced by:  vccl  30288  nvsf  30344