Theorem vecax6 14801

Description: 6th "axiom" of a vector space or module. Relation between scalar multiplication and vector multiplication.

Hypotheses

Ref	Expression
vecax6.1	|- X = ran (1st` (1st` R))
vecax6.3	|- .t = (2nd` (1st` R))
vecax6.5	|- .w = (2nd` (2nd` R))
vecax6.6	|- W = ran (1st` (2nd` R))

Assertion

Ref	Expression
vecax6	|- (R e. Vec -> A.u e. W A.x e. X A.y e. X ((x.t y).w u) = (x.w (y.w u)))

Distinct variable group:   u,R,x,y

Proof of Theorem vecax6

Step	Hyp	Ref	Expression
1		vecax6.1	. . . 4 |- X = ran (1st` (1st` R))
2		eqid 1884	. . . 4 |- (1st` (1st` R)) = (1st` (1st` R))
3		vecax6.3	. . . 4 |- .t = (2nd` (1st` R))
4		eqid 1884	. . . 4 |- (1st` (2nd` R)) = (1st` (2nd` R))
5		vecax6.5	. . . 4 |- .w = (2nd` (2nd` R))
6		vecax6.6	. . . 4 |- W = ran (1st` (2nd` R))
7	1, 2, 3, 4, 5, 6	vecval3b 14795	. . 3 |- (R e. Vec -> ((1st` (2nd` R)) e. Abel /\ .w :(X X. W)-->W /\ A.u e. W (((Id` .t ).w u) = u /\ A.x e. X (A.z e. W (x.w (u(1st` (2nd` R))z)) = ((x.w u)(1st` (2nd` R))(x.w z)) /\ A.y e. X (((x(1st` (1st` R))y).w u) = ((x.w u)(1st` (2nd` R))(y.w u)) /\ ((x.t y).w u) = (x.w (y.w u)))))))
8	7	simp3d 890	. 2 |- (R e. Vec -> A.u e. W (((Id` .t ).w u) = u /\ A.x e. X (A.z e. W (x.w (u(1st` (2nd` R))z)) = ((x.w u)(1st` (2nd` R))(x.w z)) /\ A.y e. X (((x(1st` (1st` R))y).w u) = ((x.w u)(1st` (2nd` R))(y.w u)) /\ ((x.t y).w u) = (x.w (y.w u))))))
9		simpr 350	. . . . . . 7 |- ((((x(1st` (1st` R))y).w u) = ((x.w u)(1st` (2nd` R))(y.w u)) /\ ((x.t y).w u) = (x.w (y.w u))) -> ((x.t y).w u) = (x.w (y.w u)))
10	9	ralimi 2168	. . . . . 6 |- (A.y e. X (((x(1st` (1st` R))y).w u) = ((x.w u)(1st` (2nd` R))(y.w u)) /\ ((x.t y).w u) = (x.w (y.w u))) -> A.y e. X ((x.t y).w u) = (x.w (y.w u)))
11	10	adantl 424	. . . . 5 |- ((A.z e. W (x.w (u(1st` (2nd` R))z)) = ((x.w u)(1st` (2nd` R))(x.w z)) /\ A.y e. X (((x(1st` (1st` R))y).w u) = ((x.w u)(1st` (2nd` R))(y.w u)) /\ ((x.t y).w u) = (x.w (y.w u)))) -> A.y e. X ((x.t y).w u) = (x.w (y.w u)))
12	11	ralimi 2168	. . . 4 |- (A.x e. X (A.z e. W (x.w (u(1st` (2nd` R))z)) = ((x.w u)(1st` (2nd` R))(x.w z)) /\ A.y e. X (((x(1st` (1st` R))y).w u) = ((x.w u)(1st` (2nd` R))(y.w u)) /\ ((x.t y).w u) = (x.w (y.w u)))) -> A.x e. X A.y e. X ((x.t y).w u) = (x.w (y.w u)))
13	12	adantl 424	. . 3 |- ((((Id` .t ).w u) = u /\ A.x e. X (A.z e. W (x.w (u(1st` (2nd` R))z)) = ((x.w u)(1st` (2nd` R))(x.w z)) /\ A.y e. X (((x(1st` (1st` R))y).w u) = ((x.w u)(1st` (2nd` R))(y.w u)) /\ ((x.t y).w u) = (x.w (y.w u))))) -> A.x e. X A.y e. X ((x.t y).w u) = (x.w (y.w u)))
14	13	ralimi 2168	. 2 |- (A.u e. W (((Id` .t ).w u) = u /\ A.x e. X (A.z e. W (x.w (u(1st` (2nd` R))z)) = ((x.w u)(1st` (2nd` R))(x.w z)) /\ A.y e. X (((x(1st` (1st` R))y).w u) = ((x.w u)(1st` (2nd` R))(y.w u)) /\ ((x.t y).w u) = (x.w (y.w u))))) -> A.u e. W A.x e. X A.y e. X ((x.t y).w u) = (x.w (y.w u)))
15	8, 14	syl 12	1 |- (R e. Vec -> A.u e. W A.x e. X A.y e. X ((x.t y).w u) = (x.w (y.w u)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105   X. cxp 3984  ran crn 3987  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  Idcgi 9312  Abelcabl 9407  Veccvec 14792

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-vec 14793

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