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Theorem vcz 26814
 Description: Anything times the zero vector is the zero vector. Equation 1b of [Kreyszig] p. 51. (Contributed by NM, 24-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vc0.1 𝐺 = (1st𝑊)
vc0.2 𝑆 = (2nd𝑊)
vc0.3 𝑋 = ran 𝐺
vc0.4 𝑍 = (GId‘𝐺)
Assertion
Ref Expression
vcz ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)

Proof of Theorem vcz
StepHypRef Expression
1 vc0.1 . . . . . 6 𝐺 = (1st𝑊)
2 vc0.3 . . . . . 6 𝑋 = ran 𝐺
3 vc0.4 . . . . . 6 𝑍 = (GId‘𝐺)
41, 2, 3vczcl 26811 . . . . 5 (𝑊 ∈ CVecOLD𝑍𝑋)
54anim2i 591 . . . 4 ((𝐴 ∈ ℂ ∧ 𝑊 ∈ CVecOLD) → (𝐴 ∈ ℂ ∧ 𝑍𝑋))
65ancoms 468 . . 3 ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ) → (𝐴 ∈ ℂ ∧ 𝑍𝑋))
7 0cn 9911 . . . 4 0 ∈ ℂ
8 vc0.2 . . . . 5 𝑆 = (2nd𝑊)
91, 8, 2vcass 26806 . . . 4 ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 0 ∈ ℂ ∧ 𝑍𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
107, 9mp3anr2 1414 . . 3 ((𝑊 ∈ CVecOLD ∧ (𝐴 ∈ ℂ ∧ 𝑍𝑋)) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
116, 10syldan 486 . 2 ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = (𝐴𝑆(0𝑆𝑍)))
12 mul01 10094 . . . 4 (𝐴 ∈ ℂ → (𝐴 · 0) = 0)
1312oveq1d 6564 . . 3 (𝐴 ∈ ℂ → ((𝐴 · 0)𝑆𝑍) = (0𝑆𝑍))
141, 8, 2, 3vc0 26813 . . . 4 ((𝑊 ∈ CVecOLD𝑍𝑋) → (0𝑆𝑍) = 𝑍)
154, 14mpdan 699 . . 3 (𝑊 ∈ CVecOLD → (0𝑆𝑍) = 𝑍)
1613, 15sylan9eqr 2666 . 2 ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ) → ((𝐴 · 0)𝑆𝑍) = 𝑍)
1715oveq2d 6565 . . 3 (𝑊 ∈ CVecOLD → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍))
1817adantr 480 . 2 ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ) → (𝐴𝑆(0𝑆𝑍)) = (𝐴𝑆𝑍))
1911, 16, 183eqtr3rd 2653 1 ((𝑊 ∈ CVecOLD𝐴 ∈ ℂ) → (𝐴𝑆𝑍) = 𝑍)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  ℂcc 9813  0cc0 9815   · cmul 9820  GIdcgi 26728  CVecOLDcvc 26797 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-ltxr 9958  df-grpo 26731  df-gid 26732  df-ginv 26733  df-ablo 26783  df-vc 26798 This theorem is referenced by:  nvsz  26877
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