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Theorem vcablo 29553
Description: Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
Hypothesis
Ref Expression
vcabl.1 𝐺 = (1st β€˜π‘Š)
Assertion
Ref Expression
vcablo (π‘Š ∈ CVecOLD β†’ 𝐺 ∈ AbelOp)

Proof of Theorem vcablo
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vcabl.1 . . 3 𝐺 = (1st β€˜π‘Š)
2 eqid 2733 . . 3 (2nd β€˜π‘Š) = (2nd β€˜π‘Š)
3 eqid 2733 . . 3 ran 𝐺 = ran 𝐺
41, 2, 3vciOLD 29545 . 2 (π‘Š ∈ CVecOLD β†’ (𝐺 ∈ AbelOp ∧ (2nd β€˜π‘Š):(β„‚ Γ— ran 𝐺)⟢ran 𝐺 ∧ βˆ€π‘₯ ∈ ran 𝐺((1(2nd β€˜π‘Š)π‘₯) = π‘₯ ∧ βˆ€π‘¦ ∈ β„‚ (βˆ€π‘§ ∈ ran 𝐺(𝑦(2nd β€˜π‘Š)(π‘₯𝐺𝑧)) = ((𝑦(2nd β€˜π‘Š)π‘₯)𝐺(𝑦(2nd β€˜π‘Š)𝑧)) ∧ βˆ€π‘§ ∈ β„‚ (((𝑦 + 𝑧)(2nd β€˜π‘Š)π‘₯) = ((𝑦(2nd β€˜π‘Š)π‘₯)𝐺(𝑧(2nd β€˜π‘Š)π‘₯)) ∧ ((𝑦 Β· 𝑧)(2nd β€˜π‘Š)π‘₯) = (𝑦(2nd β€˜π‘Š)(𝑧(2nd β€˜π‘Š)π‘₯)))))))
54simp1d 1143 1 (π‘Š ∈ CVecOLD β†’ 𝐺 ∈ AbelOp)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   Γ— cxp 5632  ran crn 5635  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  β„‚cc 11054  1c1 11057   + caddc 11059   Β· cmul 11061  AbelOpcablo 29528  CVecOLDcvc 29542
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-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-1st 7922  df-2nd 7923  df-vc 29543
This theorem is referenced by:  vcgrp  29554  nvablo  29600  ip0i  29809  ipdirilem  29813
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