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Theorem vcablo 29853
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 29845 . 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 3062   Γ— cxp 5675  ran crn 5678  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  2nd c2nd 7974  β„‚cc 11108  1c1 11111   + caddc 11113   Β· cmul 11115  AbelOpcablo 29828  CVecOLDcvc 29842
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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-ov 7412  df-1st 7975  df-2nd 7976  df-vc 29843
This theorem is referenced by:  vcgrp  29854  nvablo  29900  ip0i  30109  ipdirilem  30113
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