ipsubdir

Description: Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007) (Revised by Mario Carneiro, 7-Oct-2015)

	Ref	Expression
Hypotheses	phlsrng.f	$⊢ F = Scalar (W)$
	phllmhm.h	$⊢ \underset{˙}{,} = \cdot_{𝑖} (W)$
	phllmhm.v	$⊢ V = {Base}_{W}$
	ipsubdir.m	$⊢ \underset{˙}{-} = -_{W}$
	ipsubdir.s	$⊢ S = -_{F}$
Assertion	ipsubdir	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to (A \underset{˙}{-} B) \underset{˙}{,} C = (A \underset{˙}{,} C) S (B \underset{˙}{,} C)$

Proof

Step	Hyp	Ref	Expression
1		phlsrng.f	$⊢ F = Scalar (W)$
2		phllmhm.h	$⊢ \underset{˙}{,} = \cdot_{𝑖} (W)$
3		phllmhm.v	$⊢ V = {Base}_{W}$
4		ipsubdir.m	$⊢ \underset{˙}{-} = -_{W}$
5		ipsubdir.s	$⊢ S = -_{F}$
6		simpl	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to W \in PreHil$
7		phllmod	$⊢ W \in PreHil \to W \in LMod$
8	7	adantr	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to W \in LMod$
9		lmodgrp	$⊢ W \in LMod \to W \in Grp$
10	8 9	syl	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to W \in Grp$
11		simpr1	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to A \in V$
12		simpr2	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to B \in V$
13	3 4	grpsubcl	$⊢ (W \in Grp \land A \in V \land B \in V) \to A \underset{˙}{-} B \in V$
14	10 11 12 13	syl3anc	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to A \underset{˙}{-} B \in V$
15		simpr3	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to C \in V$
16		eqid	$⊢ +_{W} = +_{W}$
17		eqid	$⊢ +_{F} = +_{F}$
18	1 2 3 16 17	ipdir	$⊢ (W \in PreHil \land (A \underset{˙}{-} B \in V \land B \in V \land C \in V)) \to ((A \underset{˙}{-} B) +_{W} B) \underset{˙}{,} C = ((A \underset{˙}{-} B) \underset{˙}{,} C) +_{F} (B \underset{˙}{,} C)$
19	6 14 12 15 18	syl13anc	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to ((A \underset{˙}{-} B) +_{W} B) \underset{˙}{,} C = ((A \underset{˙}{-} B) \underset{˙}{,} C) +_{F} (B \underset{˙}{,} C)$
20	3 16 4	grpnpcan	$⊢ (W \in Grp \land A \in V \land B \in V) \to (A \underset{˙}{-} B) +_{W} B = A$
21	10 11 12 20	syl3anc	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to (A \underset{˙}{-} B) +_{W} B = A$
22	21	oveq1d	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to ((A \underset{˙}{-} B) +_{W} B) \underset{˙}{,} C = A \underset{˙}{,} C$
23	19 22	eqtr3d	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to ((A \underset{˙}{-} B) \underset{˙}{,} C) +_{F} (B \underset{˙}{,} C) = A \underset{˙}{,} C$
24	1	lmodfgrp	$⊢ W \in LMod \to F \in Grp$
25	8 24	syl	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to F \in Grp$
26		eqid	$⊢ {Base}_{F} = {Base}_{F}$
27	1 2 3 26	ipcl	$⊢ (W \in PreHil \land A \in V \land C \in V) \to A \underset{˙}{,} C \in {Base}_{F}$
28	6 11 15 27	syl3anc	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to A \underset{˙}{,} C \in {Base}_{F}$
29	1 2 3 26	ipcl	$⊢ (W \in PreHil \land B \in V \land C \in V) \to B \underset{˙}{,} C \in {Base}_{F}$
30	6 12 15 29	syl3anc	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to B \underset{˙}{,} C \in {Base}_{F}$
31	1 2 3 26	ipcl	$⊢ (W \in PreHil \land A \underset{˙}{-} B \in V \land C \in V) \to (A \underset{˙}{-} B) \underset{˙}{,} C \in {Base}_{F}$
32	6 14 15 31	syl3anc	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to (A \underset{˙}{-} B) \underset{˙}{,} C \in {Base}_{F}$
33	26 17 5	grpsubadd	$⊢ (F \in Grp \land (A \underset{˙}{,} C \in {Base}_{F} \land B \underset{˙}{,} C \in {Base}_{F} \land (A \underset{˙}{-} B) \underset{˙}{,} C \in {Base}_{F})) \to ((A \underset{˙}{,} C) S (B \underset{˙}{,} C) = (A \underset{˙}{-} B) \underset{˙}{,} C \leftrightarrow ((A \underset{˙}{-} B) \underset{˙}{,} C) +_{F} (B \underset{˙}{,} C) = A \underset{˙}{,} C)$
34	25 28 30 32 33	syl13anc	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to ((A \underset{˙}{,} C) S (B \underset{˙}{,} C) = (A \underset{˙}{-} B) \underset{˙}{,} C \leftrightarrow ((A \underset{˙}{-} B) \underset{˙}{,} C) +_{F} (B \underset{˙}{,} C) = A \underset{˙}{,} C)$
35	23 34	mpbird	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to (A \underset{˙}{,} C) S (B \underset{˙}{,} C) = (A \underset{˙}{-} B) \underset{˙}{,} C$
36	35	eqcomd	$⊢ (W \in PreHil \land (A \in V \land B \in V \land C \in V)) \to (A \underset{˙}{-} B) \underset{˙}{,} C = (A \underset{˙}{,} C) S (B \underset{˙}{,} C)$

Metamath Proof Explorer

Theorem ipsubdir

Proof