Theorem eqlkr 38797

Description: Two functionals with the same kernel are the same up to a constant. (Contributed by NM, 18-Apr-2014.)

Hypotheses

Ref	Expression
eqlkr.d	⊢ 𝐷 = (Scalar‘𝑊)
eqlkr.k	⊢ 𝐾 = (Base‘𝐷)
eqlkr.t	⊢ · = (.r‘𝐷)
eqlkr.v	⊢ 𝑉 = (Base‘𝑊)
eqlkr.f	⊢ 𝐹 = (LFnl‘𝑊)
eqlkr.l	⊢ 𝐿 = (LKer‘𝑊)

Assertion

Ref	Expression
eqlkr	⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))

Distinct variable groups:   𝑥,𝑟,𝐷   𝑥,𝐹   𝐺,𝑟,𝑥   𝐻,𝑟,𝑥   𝑉,𝑟,𝑥   𝐾,𝑟   𝑥,𝐿   · ,𝑟   𝑥,𝑊

Allowed substitution hints:   · (𝑥)   𝐹(𝑟)   𝐾(𝑥)   𝐿(𝑟)   𝑊(𝑟)

Proof of Theorem eqlkr

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1188	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → 𝑊 ∈ LVec)
2		lveclmod 21084	. . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
3		eqlkr.d	. . . . . . 7 ⊢ 𝐷 = (Scalar‘𝑊)
4	3	lmodring 20844	. . . . . 6 ⊢ (𝑊 ∈ LMod → 𝐷 ∈ Ring)
5	2, 4	syl 17	. . . . 5 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ Ring)
6	1, 5	syl 17	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → 𝐷 ∈ Ring)
7		eqlkr.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐷)
8		eqid 2726	. . . . 5 ⊢ (1r‘𝐷) = (1r‘𝐷)
9	7, 8	ringidcl 20245	. . . 4 ⊢ (𝐷 ∈ Ring → (1r‘𝐷) ∈ 𝐾)
10	6, 9	syl 17	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → (1r‘𝐷) ∈ 𝐾)
11		simp11 1200	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ LVec)
12	11, 5	syl 17	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐷 ∈ Ring)
13		simp12l 1283	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐺 ∈ 𝐹)
14		simp3 1135	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
15		eqlkr.v	. . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑊)
16		eqlkr.f	. . . . . . . . 9 ⊢ 𝐹 = (LFnl‘𝑊)
17	3, 7, 15, 16	lflcl 38762	. . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
18	11, 13, 14, 17	syl3anc 1368	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
19		eqlkr.t	. . . . . . . 8 ⊢ · = (.r‘𝐷)
20	7, 19, 8	ringridm 20249	. . . . . . 7 ⊢ ((𝐷 ∈ Ring ∧ (𝐺‘𝑥) ∈ 𝐾) → ((𝐺‘𝑥) · (1r‘𝐷)) = (𝐺‘𝑥))
21	12, 18, 20	syl2anc 582	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥) · (1r‘𝐷)) = (𝐺‘𝑥))
22		simp2 1134	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐺 = (𝑉 × {(0g‘𝐷)}))
23		simp13 1202	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝐺) = (𝐿‘𝐻))
24	11, 2	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ LMod)
25		eqid 2726	. . . . . . . . . . . . 13 ⊢ (0g‘𝐷) = (0g‘𝐷)
26		eqlkr.l	. . . . . . . . . . . . 13 ⊢ 𝐿 = (LKer‘𝑊)
27	3, 25, 15, 16, 26	lkr0f 38792	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘𝐷)})))
28	24, 13, 27	syl2anc 582	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0g‘𝐷)})))
29	22, 28	mpbird 256	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝐺) = 𝑉)
30	23, 29	eqtr3d 2768	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝐻) = 𝑉)
31		simp12r 1284	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐻 ∈ 𝐹)
32	3, 25, 15, 16, 26	lkr0f 38792	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → ((𝐿‘𝐻) = 𝑉 ↔ 𝐻 = (𝑉 × {(0g‘𝐷)})))
33	24, 31, 32	syl2anc 582	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐻) = 𝑉 ↔ 𝐻 = (𝑉 × {(0g‘𝐷)})))
34	30, 33	mpbid 231	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐻 = (𝑉 × {(0g‘𝐷)}))
35	22, 34	eqtr4d 2769	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐺 = 𝐻)
36	35	fveq1d 6903	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) = (𝐻‘𝑥))
37	21, 36	eqtr2d 2767	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑥) = ((𝐺‘𝑥) · (1r‘𝐷)))
38	37	3expia 1118	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → (𝑥 ∈ 𝑉 → (𝐻‘𝑥) = ((𝐺‘𝑥) · (1r‘𝐷))))
39	38	ralrimiv 3135	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (1r‘𝐷)))
40		oveq2 7432	. . . . . 6 ⊢ (𝑟 = (1r‘𝐷) → ((𝐺‘𝑥) · 𝑟) = ((𝐺‘𝑥) · (1r‘𝐷)))
41	40	eqeq2d 2737	. . . . 5 ⊢ (𝑟 = (1r‘𝐷) → ((𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟) ↔ (𝐻‘𝑥) = ((𝐺‘𝑥) · (1r‘𝐷))))
42	41	ralbidv 3168	. . . 4 ⊢ (𝑟 = (1r‘𝐷) → (∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟) ↔ ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (1r‘𝐷))))
43	42	rspcev 3608	. . 3 ⊢ (((1r‘𝐷) ∈ 𝐾 ∧ ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (1r‘𝐷))) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
44	10, 39, 43	syl2anc 582	. 2 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0g‘𝐷)})) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
45		simpl1 1188	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → 𝑊 ∈ LVec)
46		simpl2l 1223	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → 𝐺 ∈ 𝐹)
47		simpr 483	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → 𝐺 ≠ (𝑉 × {(0g‘𝐷)}))
48	3, 25, 8, 15, 16	lfl1 38768	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → ∃𝑧 ∈ 𝑉 (𝐺‘𝑧) = (1r‘𝐷))
49	45, 46, 47, 48	syl3anc 1368	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → ∃𝑧 ∈ 𝑉 (𝐺‘𝑧) = (1r‘𝐷))
50		simpl1 1188	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷))) → 𝑊 ∈ LVec)
51		simpl2r 1224	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷))) → 𝐻 ∈ 𝐹)
52		simpr2 1192	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷))) → 𝑧 ∈ 𝑉)
53	3, 7, 15, 16	lflcl 38762	. . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ 𝐻 ∈ 𝐹 ∧ 𝑧 ∈ 𝑉) → (𝐻‘𝑧) ∈ 𝐾)
54	50, 51, 52, 53	syl3anc 1368	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷))) → (𝐻‘𝑧) ∈ 𝐾)
55		simp11 1200	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ LVec)
56	55, 2	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ LMod)
57		simp12r 1284	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝐻 ∈ 𝐹)
58		simp12l 1283	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝐺 ∈ 𝐹)
59		simp3 1135	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
60	3, 7, 15, 16	lflcl 38762	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
61	56, 58, 59, 60	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
62		simp22 1204	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝑧 ∈ 𝑉)
63		eqid 2726	. . . . . . . . . . . . . 14 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊)
64	3, 7, 19, 15, 63, 16	lflmul 38766	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ ((𝐺‘𝑥) ∈ 𝐾 ∧ 𝑧 ∈ 𝑉)) → (𝐻‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) = ((𝐺‘𝑥) · (𝐻‘𝑧)))
65	56, 57, 61, 62, 64	syl112anc 1371	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) = ((𝐺‘𝑥) · (𝐻‘𝑧)))
66	65	oveq2d 7440	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐻‘𝑥)(-g‘𝐷)(𝐻‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = ((𝐻‘𝑥)(-g‘𝐷)((𝐺‘𝑥) · (𝐻‘𝑧))))
67	15, 3, 63, 7	lmodvscl 20854	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝐺‘𝑥) ∈ 𝐾 ∧ 𝑧 ∈ 𝑉) → ((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧) ∈ 𝑉)
68	56, 61, 62, 67	syl3anc 1368	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧) ∈ 𝑉)
69		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (-g‘𝐷) = (-g‘𝐷)
70		eqid 2726	. . . . . . . . . . . . . 14 ⊢ (-g‘𝑊) = (-g‘𝑊)
71	3, 69, 15, 70, 16	lflsub 38765	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹 ∧ (𝑥 ∈ 𝑉 ∧ ((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧) ∈ 𝑉)) → (𝐻‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = ((𝐻‘𝑥)(-g‘𝐷)(𝐻‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))))
72	56, 57, 59, 68, 71	syl112anc 1371	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = ((𝐻‘𝑥)(-g‘𝐷)(𝐻‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))))
73	15, 70	lmodvsubcl 20883	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ LMod ∧ 𝑥 ∈ 𝑉 ∧ ((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧) ∈ 𝑉) → (𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ 𝑉)
74	56, 59, 68, 73	syl3anc 1368	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ 𝑉)
75	3, 69, 15, 70, 16	lflsub 38765	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ (𝑥 ∈ 𝑉 ∧ ((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧) ∈ 𝑉)) → (𝐺‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = ((𝐺‘𝑥)(-g‘𝐷)(𝐺‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))))
76	56, 58, 59, 68, 75	syl112anc 1371	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = ((𝐺‘𝑥)(-g‘𝐷)(𝐺‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))))
77	55, 58, 59, 17	syl3anc 1368	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
78	3, 7, 19, 15, 63, 16	lflmul 38766	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ((𝐺‘𝑥) ∈ 𝐾 ∧ 𝑧 ∈ 𝑉)) → (𝐺‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) = ((𝐺‘𝑥) · (𝐺‘𝑧)))
79	56, 58, 77, 62, 78	syl112anc 1371	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) = ((𝐺‘𝑥) · (𝐺‘𝑧)))
80		simp23 1205	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑧) = (1r‘𝐷))
81	80	oveq2d 7440	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥) · (𝐺‘𝑧)) = ((𝐺‘𝑥) · (1r‘𝐷)))
82	55, 5	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝐷 ∈ Ring)
83	82, 77, 20	syl2anc 582	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥) · (1r‘𝐷)) = (𝐺‘𝑥))
84	79, 81, 83	3eqtrd 2770	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) = (𝐺‘𝑥))
85	84	oveq2d 7440	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥)(-g‘𝐷)(𝐺‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = ((𝐺‘𝑥)(-g‘𝐷)(𝐺‘𝑥)))
86	3	lmodfgrp 20845	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑊 ∈ LMod → 𝐷 ∈ Grp)
87	2, 86	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑊 ∈ LVec → 𝐷 ∈ Grp)
88	55, 87	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝐷 ∈ Grp)
89	7, 25, 69	grpsubid 19018	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ Grp ∧ (𝐺‘𝑥) ∈ 𝐾) → ((𝐺‘𝑥)(-g‘𝐷)(𝐺‘𝑥)) = (0g‘𝐷))
90	88, 77, 89	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥)(-g‘𝐷)(𝐺‘𝑥)) = (0g‘𝐷))
91	76, 85, 90	3eqtrd 2770	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = (0g‘𝐷))
92	15, 3, 25, 16, 26	ellkr 38787	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹) → ((𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ (𝐿‘𝐺) ↔ ((𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ 𝑉 ∧ (𝐺‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = (0g‘𝐷))))
93	55, 58, 92	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ (𝐿‘𝐺) ↔ ((𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ 𝑉 ∧ (𝐺‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = (0g‘𝐷))))
94	74, 91, 93	mpbir2and 711	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ (𝐿‘𝐺))
95		simp13 1202	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝐺) = (𝐿‘𝐻))
96	94, 95	eleqtrd 2828	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ (𝐿‘𝐻))
97	15, 3, 25, 16, 26	ellkr 38787	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ LVec ∧ 𝐻 ∈ 𝐹) → ((𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ (𝐿‘𝐻) ↔ ((𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ 𝑉 ∧ (𝐻‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = (0g‘𝐷))))
98	55, 57, 97	syl2anc 582	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ (𝐿‘𝐻) ↔ ((𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ 𝑉 ∧ (𝐻‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = (0g‘𝐷))))
99	96, 98	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧)) ∈ 𝑉 ∧ (𝐻‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = (0g‘𝐷)))
100	99	simprd 494	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘(𝑥(-g‘𝑊)((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = (0g‘𝐷))
101	72, 100	eqtr3d 2768	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐻‘𝑥)(-g‘𝐷)(𝐻‘((𝐺‘𝑥)( ·𝑠 ‘𝑊)𝑧))) = (0g‘𝐷))
102	66, 101	eqtr3d 2768	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐻‘𝑥)(-g‘𝐷)((𝐺‘𝑥) · (𝐻‘𝑧))) = (0g‘𝐷))
103	3, 7, 15, 16	lflcl 38762	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LVec ∧ 𝐻 ∈ 𝐹 ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑥) ∈ 𝐾)
104	55, 57, 59, 103	syl3anc 1368	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑥) ∈ 𝐾)
105	54	3adant3 1129	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑧) ∈ 𝐾)
106	3, 7, 19	lmodmcl 20849	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ (𝐺‘𝑥) ∈ 𝐾 ∧ (𝐻‘𝑧) ∈ 𝐾) → ((𝐺‘𝑥) · (𝐻‘𝑧)) ∈ 𝐾)
107	56, 77, 105, 106	syl3anc 1368	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥) · (𝐻‘𝑧)) ∈ 𝐾)
108	7, 25, 69	grpsubeq0 19020	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ Grp ∧ (𝐻‘𝑥) ∈ 𝐾 ∧ ((𝐺‘𝑥) · (𝐻‘𝑧)) ∈ 𝐾) → (((𝐻‘𝑥)(-g‘𝐷)((𝐺‘𝑥) · (𝐻‘𝑧))) = (0g‘𝐷) ↔ (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))))
109	88, 104, 107, 108	syl3anc 1368	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (((𝐻‘𝑥)(-g‘𝐷)((𝐺‘𝑥) · (𝐻‘𝑧))) = (0g‘𝐷) ↔ (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))))
110	102, 109	mpbid 231	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧)))
111	110	3expia 1118	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷))) → (𝑥 ∈ 𝑉 → (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))))
112	111	ralrimiv 3135	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷))) → ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧)))
113		oveq2 7432	. . . . . . . . . 10 ⊢ (𝑟 = (𝐻‘𝑧) → ((𝐺‘𝑥) · 𝑟) = ((𝐺‘𝑥) · (𝐻‘𝑧)))
114	113	eqeq2d 2737	. . . . . . . . 9 ⊢ (𝑟 = (𝐻‘𝑧) → ((𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟) ↔ (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))))
115	114	ralbidv 3168	. . . . . . . 8 ⊢ (𝑟 = (𝐻‘𝑧) → (∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟) ↔ ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))))
116	115	rspcev 3608	. . . . . . 7 ⊢ (((𝐻‘𝑧) ∈ 𝐾 ∧ ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
117	54, 112, 116	syl2anc 582	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1r‘𝐷))) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
118	117	3exp2 1351	. . . . 5 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → (𝐺 ≠ (𝑉 × {(0g‘𝐷)}) → (𝑧 ∈ 𝑉 → ((𝐺‘𝑧) = (1r‘𝐷) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟)))))
119	118	imp 405	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → (𝑧 ∈ 𝑉 → ((𝐺‘𝑧) = (1r‘𝐷) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))))
120	119	rexlimdv 3143	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → (∃𝑧 ∈ 𝑉 (𝐺‘𝑧) = (1r‘𝐷) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟)))
121	49, 120	mpd 15	. 2 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0g‘𝐷)})) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
122	44, 121	pm2.61dane 3019	1 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  {csn 4633   × cxp 5680  ‘cfv 6554  (class class class)co 7424  Basecbs 17213  ._rcmulr 17267  Scalarcsca 17269   ·_𝑠 cvsca 17270  0_gc0g 17454  Grpcgrp 18928  -_gcsg 18930  1_rcur 20164  Ringcrg 20216  LModclmod 20836  LVecclvec 21080  LFnlclfn 38755  LKerclk 38783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-tpos 8241  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-ress 17243  df-plusg 17279  df-mulr 17280  df-0g 17456  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-sbg 18933  df-cmn 19780  df-abl 19781  df-mgp 20118  df-rng 20136  df-ur 20165  df-ring 20218  df-oppr 20316  df-dvdsr 20339  df-unit 20340  df-invr 20370  df-drng 20709  df-lmod 20838  df-lvec 21081  df-lfl 38756  df-lkr 38784

This theorem is referenced by:  eqlkr2  38798  eqlkr3  38799