eqlkr - Mathbox for Norm Megill

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Theorem eqlkr 38285

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 1190	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)})) → 𝑊 ∈ LVec)
2		lveclmod 20865	. . . . . 6 ⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod)
3		eqlkr.d	. . . . . . 7 ⊢ 𝐷 = (Scalar‘𝑊)
4	3	lmodring 20626	. . . . . 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 × {(0_g‘𝐷)})) → 𝐷 ∈ Ring)
7		eqlkr.k	. . . . 5 ⊢ 𝐾 = (Base‘𝐷)
8		eqid 2731	. . . . 5 ⊢ (1_r‘𝐷) = (1_r‘𝐷)
9	7, 8	ringidcl 20158	. . . 4 ⊢ (𝐷 ∈ Ring → (1_r‘𝐷) ∈ 𝐾)
10	6, 9	syl 17	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)})) → (1_r‘𝐷) ∈ 𝐾)
11		simp11 1202	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ LVec)
12	11, 5	syl 17	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐷 ∈ Ring)
13		simp12l 1285	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐺 ∈ 𝐹)
14		simp3 1137	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘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 38250	. . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
18	11, 13, 14, 17	syl3anc 1370	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
19		eqlkr.t	. . . . . . . 8 ⊢ · = (._r‘𝐷)
20	7, 19, 8	ringridm 20162	. . . . . . 7 ⊢ ((𝐷 ∈ Ring ∧ (𝐺‘𝑥) ∈ 𝐾) → ((𝐺‘𝑥) · (1_r‘𝐷)) = (𝐺‘𝑥))
21	12, 18, 20	syl2anc 583	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥) · (1_r‘𝐷)) = (𝐺‘𝑥))
22		simp2 1136	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐺 = (𝑉 × {(0_g‘𝐷)}))
23		simp13 1204	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘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 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ LMod)
25		eqid 2731	. . . . . . . . . . . . 13 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
26		eqlkr.l	. . . . . . . . . . . . 13 ⊢ 𝐿 = (LKer‘𝑊)
27	3, 25, 15, 16, 26	lkr0f 38280	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹) → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0_g‘𝐷)})))
28	24, 13, 27	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = (𝑉 × {(0_g‘𝐷)})))
29	22, 28	mpbird 257	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝐺) = 𝑉)
30	23, 29	eqtr3d 2773	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝐻) = 𝑉)
31		simp12r 1286	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐻 ∈ 𝐹)
32	3, 25, 15, 16, 26	lkr0f 38280	. . . . . . . . . 10 ⊢ ((𝑊 ∈ LMod ∧ 𝐻 ∈ 𝐹) → ((𝐿‘𝐻) = 𝑉 ↔ 𝐻 = (𝑉 × {(0_g‘𝐷)})))
33	24, 31, 32	syl2anc 583	. . . . . . . . 9 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐻) = 𝑉 ↔ 𝐻 = (𝑉 × {(0_g‘𝐷)})))
34	30, 33	mpbid 231	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐻 = (𝑉 × {(0_g‘𝐷)}))
35	22, 34	eqtr4d 2774	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → 𝐺 = 𝐻)
36	35	fveq1d 6893	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) = (𝐻‘𝑥))
37	21, 36	eqtr2d 2772	. . . . 5 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)}) ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑥) = ((𝐺‘𝑥) · (1_r‘𝐷)))
38	37	3expia 1120	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)})) → (𝑥 ∈ 𝑉 → (𝐻‘𝑥) = ((𝐺‘𝑥) · (1_r‘𝐷))))
39	38	ralrimiv 3144	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)})) → ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (1_r‘𝐷)))
40		oveq2 7420	. . . . . 6 ⊢ (𝑟 = (1_r‘𝐷) → ((𝐺‘𝑥) · 𝑟) = ((𝐺‘𝑥) · (1_r‘𝐷)))
41	40	eqeq2d 2742	. . . . 5 ⊢ (𝑟 = (1_r‘𝐷) → ((𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟) ↔ (𝐻‘𝑥) = ((𝐺‘𝑥) · (1_r‘𝐷))))
42	41	ralbidv 3176	. . . 4 ⊢ (𝑟 = (1_r‘𝐷) → (∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟) ↔ ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (1_r‘𝐷))))
43	42	rspcev 3612	. . 3 ⊢ (((1_r‘𝐷) ∈ 𝐾 ∧ ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (1_r‘𝐷))) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
44	10, 39, 43	syl2anc 583	. 2 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 = (𝑉 × {(0_g‘𝐷)})) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
45		simpl1 1190	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0_g‘𝐷)})) → 𝑊 ∈ LVec)
46		simpl2l 1225	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0_g‘𝐷)})) → 𝐺 ∈ 𝐹)
47		simpr 484	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0_g‘𝐷)})) → 𝐺 ≠ (𝑉 × {(0_g‘𝐷)}))
48	3, 25, 8, 15, 16	lfl1 38256	. . . 4 ⊢ ((𝑊 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝐺 ≠ (𝑉 × {(0_g‘𝐷)})) → ∃𝑧 ∈ 𝑉 (𝐺‘𝑧) = (1_r‘𝐷))
49	45, 46, 47, 48	syl3anc 1370	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0_g‘𝐷)})) → ∃𝑧 ∈ 𝑉 (𝐺‘𝑧) = (1_r‘𝐷))
50		simpl1 1190	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷))) → 𝑊 ∈ LVec)
51		simpl2r 1226	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷))) → 𝐻 ∈ 𝐹)
52		simpr2 1194	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷))) → 𝑧 ∈ 𝑉)
53	3, 7, 15, 16	lflcl 38250	. . . . . . . 8 ⊢ ((𝑊 ∈ LVec ∧ 𝐻 ∈ 𝐹 ∧ 𝑧 ∈ 𝑉) → (𝐻‘𝑧) ∈ 𝐾)
54	50, 51, 52, 53	syl3anc 1370	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷))) → (𝐻‘𝑧) ∈ 𝐾)
55		simp11 1202	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ LVec)
56	55, 2	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝑊 ∈ LMod)
57		simp12r 1286	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝐻 ∈ 𝐹)
58		simp12l 1285	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝐺 ∈ 𝐹)
59		simp3 1137	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝑥 ∈ 𝑉)
60	3, 7, 15, 16	lflcl 38250	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
61	56, 58, 59, 60	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
62		simp22 1206	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝑧 ∈ 𝑉)
63		eqid 2731	. . . . . . . . . . . . . 14 ⊢ ( ·_𝑠 ‘𝑊) = ( ·_𝑠 ‘𝑊)
64	3, 7, 19, 15, 63, 16	lflmul 38254	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧)) = ((𝐺‘𝑥) · (𝐻‘𝑧)))
66	65	oveq2d 7428	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘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 20636	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ LMod ∧ (𝐺‘𝑥) ∈ 𝐾 ∧ 𝑧 ∈ 𝑉) → ((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
68	56, 61, 62, 67	syl3anc 1370	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧) ∈ 𝑉)
69		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝐷) = (-_g‘𝐷)
70		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (-_g‘𝑊) = (-_g‘𝑊)
71	3, 69, 15, 70, 16	lflsub 38253	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘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 20665	. . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝑥(-_g‘𝑊)((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧)) ∈ 𝑉)
75	3, 69, 15, 70, 16	lflsub 38253	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘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 1370	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑥) ∈ 𝐾)
78	3, 7, 19, 15, 63, 16	lflmul 38254	. . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧)) = ((𝐺‘𝑥) · (𝐺‘𝑧)))
80		simp23 1207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘𝑧) = (1_r‘𝐷))
81	80	oveq2d 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥) · (𝐺‘𝑧)) = ((𝐺‘𝑥) · (1_r‘𝐷)))
82	55, 5	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝐷 ∈ Ring)
83	82, 77, 20	syl2anc 583	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥) · (1_r‘𝐷)) = (𝐺‘𝑥))
84	79, 81, 83	3eqtrd 2775	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧)) = (𝐺‘𝑥))
85	84	oveq2d 7428	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥)(-_g‘𝐷)(𝐺‘((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧))) = ((𝐺‘𝑥)(-_g‘𝐷)(𝐺‘𝑥)))
86	3	lmodfgrp 20627	. . . . . . . . . . . . . . . . . . . 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 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → 𝐷 ∈ Grp)
89	7, 25, 69	grpsubid 18947	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 ∈ Grp ∧ (𝐺‘𝑥) ∈ 𝐾) → ((𝐺‘𝑥)(-_g‘𝐷)(𝐺‘𝑥)) = (0_g‘𝐷))
90	88, 77, 89	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥)(-_g‘𝐷)(𝐺‘𝑥)) = (0_g‘𝐷))
91	76, 85, 90	3eqtrd 2775	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐺‘(𝑥(-_g‘𝑊)((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧))) = (0_g‘𝐷))
92	15, 3, 25, 16, 26	ellkr 38275	. . . . . . . . . . . . . . . . 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))) = (0_g‘𝐷))))
93	55, 58, 92	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘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))) = (0_g‘𝐷))))
94	74, 91, 93	mpbir2and 710	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝑥(-_g‘𝑊)((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧)) ∈ (𝐿‘𝐺))
95		simp13 1204	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐿‘𝐺) = (𝐿‘𝐻))
96	94, 95	eleqtrd 2834	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝑥(-_g‘𝑊)((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧)) ∈ (𝐿‘𝐻))
97	15, 3, 25, 16, 26	ellkr 38275	. . . . . . . . . . . . . . 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))) = (0_g‘𝐷))))
98	55, 57, 97	syl2anc 583	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘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))) = (0_g‘𝐷))))
99	96, 98	mpbid 231	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝑥(-_g‘𝑊)((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧)) ∈ 𝑉 ∧ (𝐻‘(𝑥(-_g‘𝑊)((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧))) = (0_g‘𝐷)))
100	99	simprd 495	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘(𝑥(-_g‘𝑊)((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧))) = (0_g‘𝐷))
101	72, 100	eqtr3d 2773	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐻‘𝑥)(-_g‘𝐷)(𝐻‘((𝐺‘𝑥)( ·_𝑠 ‘𝑊)𝑧))) = (0_g‘𝐷))
102	66, 101	eqtr3d 2773	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐻‘𝑥)(-_g‘𝐷)((𝐺‘𝑥) · (𝐻‘𝑧))) = (0_g‘𝐷))
103	3, 7, 15, 16	lflcl 38250	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LVec ∧ 𝐻 ∈ 𝐹 ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑥) ∈ 𝐾)
104	55, 57, 59, 103	syl3anc 1370	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑥) ∈ 𝐾)
105	54	3adant3 1131	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑧) ∈ 𝐾)
106	3, 7, 19	lmodmcl 20631	. . . . . . . . . . . 12 ⊢ ((𝑊 ∈ LMod ∧ (𝐺‘𝑥) ∈ 𝐾 ∧ (𝐻‘𝑧) ∈ 𝐾) → ((𝐺‘𝑥) · (𝐻‘𝑧)) ∈ 𝐾)
107	56, 77, 105, 106	syl3anc 1370	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → ((𝐺‘𝑥) · (𝐻‘𝑧)) ∈ 𝐾)
108	7, 25, 69	grpsubeq0 18949	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ Grp ∧ (𝐻‘𝑥) ∈ 𝐾 ∧ ((𝐺‘𝑥) · (𝐻‘𝑧)) ∈ 𝐾) → (((𝐻‘𝑥)(-_g‘𝐷)((𝐺‘𝑥) · (𝐻‘𝑧))) = (0_g‘𝐷) ↔ (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))))
109	88, 104, 107, 108	syl3anc 1370	. . . . . . . . . 10 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (((𝐻‘𝑥)(-_g‘𝐷)((𝐺‘𝑥) · (𝐻‘𝑧))) = (0_g‘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 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷)) ∧ 𝑥 ∈ 𝑉) → (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧)))
111	110	3expia 1120	. . . . . . . 8 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷))) → (𝑥 ∈ 𝑉 → (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))))
112	111	ralrimiv 3144	. . . . . . 7 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷))) → ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧)))
113		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑟 = (𝐻‘𝑧) → ((𝐺‘𝑥) · 𝑟) = ((𝐺‘𝑥) · (𝐻‘𝑧)))
114	113	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑟 = (𝐻‘𝑧) → ((𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟) ↔ (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))))
115	114	ralbidv 3176	. . . . . . . 8 ⊢ (𝑟 = (𝐻‘𝑧) → (∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟) ↔ ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · (𝐻‘𝑧))))
116	115	rspcev 3612	. . . . . . 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 583	. . . . . 6 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) ∧ 𝑧 ∈ 𝑉 ∧ (𝐺‘𝑧) = (1_r‘𝐷))) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
118	117	3exp2 1353	. . . . 5 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → (𝐺 ≠ (𝑉 × {(0_g‘𝐷)}) → (𝑧 ∈ 𝑉 → ((𝐺‘𝑧) = (1_r‘𝐷) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟)))))
119	118	imp 406	. . . 4 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0_g‘𝐷)})) → (𝑧 ∈ 𝑉 → ((𝐺‘𝑧) = (1_r‘𝐷) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))))
120	119	rexlimdv 3152	. . 3 ⊢ (((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) ∧ 𝐺 ≠ (𝑉 × {(0_g‘𝐷)})) → (∃𝑧 ∈ 𝑉 (𝐺‘𝑧) = (1_r‘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 × {(0_g‘𝐷)})) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))
122	44, 121	pm2.61dane 3028	1 ⊢ ((𝑊 ∈ LVec ∧ (𝐺 ∈ 𝐹 ∧ 𝐻 ∈ 𝐹) ∧ (𝐿‘𝐺) = (𝐿‘𝐻)) → ∃𝑟 ∈ 𝐾 ∀𝑥 ∈ 𝑉 (𝐻‘𝑥) = ((𝐺‘𝑥) · 𝑟))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {csn 4628 × cxp 5674 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 ._rcmulr 17205 Scalarcsca 17207 ·_𝑠 cvsca 17208 0_gc0g 17392 Grpcgrp 18858 -_gcsg 18860 1_rcur 20079 Ringcrg 20131 LModclmod 20618 LVecclvec 20861 LFnlclfn 38243 LKerclk 38271

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-tpos 8217 df-frecs 8272 df-wrecs 8303 df-recs 8377 df-rdg 8416 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-nn 12220 df-2 12282 df-3 12283 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-0g 17394 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-grp 18861 df-minusg 18862 df-sbg 18863 df-cmn 19695 df-abl 19696 df-mgp 20033 df-rng 20051 df-ur 20080 df-ring 20133 df-oppr 20229 df-dvdsr 20252 df-unit 20253 df-invr 20283 df-drng 20506 df-lmod 20620 df-lvec 20862 df-lfl 38244 df-lkr 38272

This theorem is referenced by: eqlkr2 38286 eqlkr3 38287

W3C validator