Theorem ipdir 8502

Description: Distributive law for inner product. Equation I3 of [Ponnusamy] p. 362.

Hypotheses

Ref	Expression
ipdir.1	|- X = (Base` U)
ipdir.2	|- G = (+v` U)
ipdir.7	|- P = (.i` U)

Assertion

Ref	Expression
ipdir	|- ((U e. CPreHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)PC) = ((APC) + (BPC)))

Proof of Theorem ipdir

Step	Hyp	Ref	Expression
1		fveq2 3724	. . . . . . 7 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (Base` U) = (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)))
2		ipdir.1	. . . . . . 7 |- X = (Base` U)
3	1, 2	syl5eq 1519	. . . . . 6 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> X = (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)))
4	3	eleq2d 1541	. . . . 5 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (A e. X <-> A e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.))))
5	3	eleq2d 1541	. . . . 5 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (B e. X <-> B e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.))))
6	3	eleq2d 1541	. . . . 5 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (C e. X <-> C e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.))))
7	4, 5, 6	3anbi123d 893	. . . 4 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> ((A e. X /\ B e. X /\ C e. X) <-> (A e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) /\ B e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) /\ C e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)))))
8		fveq2 3724	. . . . . . . . 9 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (+v` U) = (+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.)))
9		ipdir.2	. . . . . . . . 9 |- G = (+v` U)
10	8, 9	syl5eq 1519	. . . . . . . 8 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> G = (+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.)))
11	10	opreqd 3977	. . . . . . 7 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (AGB) = (A(+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.))B))
12	11	opreq1d 3975	. . . . . 6 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> ((AGB)PC) = ((A(+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.))B)PC))
13		fveq2 3724	. . . . . . . 8 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (.i` U) = (.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.)))
14		ipdir.7	. . . . . . . 8 |- P = (.i` U)
15	13, 14	syl5eq 1519	. . . . . . 7 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> P = (.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.)))
16	15	opreqd 3977	. . . . . 6 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> ((A(+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.))B)PC) = ((A(+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.))B)(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C))
17	12, 16	eqtrd 1507	. . . . 5 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> ((AGB)PC) = ((A(+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.))B)(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C))
18	15	opreqd 3977	. . . . . 6 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (APC) = (A(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C))
19	15	opreqd 3977	. . . . . 6 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (BPC) = (B(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C))
20	18, 19	opreq12d 3978	. . . . 5 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> ((APC) + (BPC)) = ((A(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C) + (B(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C)))
21	17, 20	eqeq12d 1489	. . . 4 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (((AGB)PC) = ((APC) + (BPC)) <-> ((A(+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.))B)(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C) = ((A(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C) + (B(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C))))
22	7, 21	imbi12d 626	. . 3 |- (U = if(U e. CPreHil, U, <.<. + , x. >., abs>.) -> (((A e. X /\ B e. X /\ C e. X) -> ((AGB)PC) = ((APC) + (BPC))) <-> ((A e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) /\ B e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) /\ C e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.))) -> ((A(+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.))B)(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C) = ((A(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C) + (B(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C)))))
23		eqid 1475	. . . 4 |- (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) = (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.))
24		eqid 1475	. . . 4 |- (+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) = (+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.))
25		eqid 1475	. . . 4 |- (.s` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) = (.s` if(U e. CPreHil, U, <.<. + , x. >., abs>.))
26		eqid 1475	. . . 4 |- (.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) = (.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))
27		elimphu 8480	. . . 4 |- if(U e. CPreHil, U, <.<. + , x. >., abs>.) e. CPreHil
28	23, 24, 25, 26, 27	ipdiri 8489	. . 3 |- ((A e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) /\ B e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.)) /\ C e. (Base` if(U e. CPreHil, U, <.<. + , x. >., abs>.))) -> ((A(+v` if(U e. CPreHil, U, <.<. + , x. >., abs>.))B)(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C) = ((A(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C) + (B(.i` if(U e. CPreHil, U, <.<. + , x. >., abs>.))C)))
29	22, 28	dedth 2383	. 2 |- (U e. CPreHil -> ((A e. X /\ B e. X /\ C e. X) -> ((AGB)PC) = ((APC) + (BPC))))
30	29	imp 350	1 |- ((U e. CPreHil /\ (A e. X /\ B e. X /\ C e. X)) -> ((AGB)PC) = ((APC) + (BPC)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  ifcif 2361  <.cop 2411  ` cfv 3182  (class class class)co 3963   + caddc 5237   x. cmul 5239  abscabs 6750  +vcpv 8204  Basecba 8205  .scns 8206  .icip 8349  CPreHilcphl 8471

This theorem is referenced by:  ipdi 8503  ip2dii 8504  ipsubdir 8508  ipblnfi 8516  hlipdir 8614

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777