Theorem diclspsn 41830

Description: The value of isomorphism C is spanned by vector 𝐹. Part of proof of Lemma N of [Crawley] p. 121 line 29. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)

Hypotheses

Ref	Expression
diclspsn.l	⊢ ≤ = (le‘𝐾)
diclspsn.a	⊢ 𝐴 = (Atoms‘𝐾)
diclspsn.h	⊢ 𝐻 = (LHyp‘𝐾)
diclspsn.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
diclspsn.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
diclspsn.i	⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
diclspsn.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
diclspsn.n	⊢ 𝑁 = (LSpan‘𝑈)
diclspsn.f	⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)

Assertion

Ref	Expression
diclspsn	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}))

Distinct variable groups:   ≤ ,𝑓   𝑃,𝑓   𝐴,𝑓   𝑓,𝐻   𝑇,𝑓   𝑓,𝐾   𝑄,𝑓   𝑓,𝑊

Allowed substitution hints:   𝑈(𝑓)   𝐹(𝑓)   𝐼(𝑓)   𝑁(𝑓)

Proof of Theorem diclspsn

Dummy variables 𝑔 𝑠 𝑣 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3418	. . 3 ⊢ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} = {𝑣 ∣ (𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))}
2		relopabv 5799	. . . . 5 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ (𝑦 = (𝑧‘𝐹) ∧ 𝑧 ∈ ((TEndo‘𝐾)‘𝑊))}
3		diclspsn.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
4		diclspsn.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
5		diclspsn.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
6		diclspsn.p	. . . . . . 7 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
7		diclspsn.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		eqid 2765	. . . . . . 7 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
9		diclspsn.i	. . . . . . 7 ⊢ 𝐼 = ((DIsoC‘𝐾)‘𝑊)
10		diclspsn.f	. . . . . . 7 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑄)
11	3, 4, 5, 6, 7, 8, 9, 10	dicval2 41815	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 = (𝑧‘𝐹) ∧ 𝑧 ∈ ((TEndo‘𝐾)‘𝑊))})
12	11	releqd 5756	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (Rel (𝐼‘𝑄) ↔ Rel {⟨𝑦, 𝑧⟩ ∣ (𝑦 = (𝑧‘𝐹) ∧ 𝑧 ∈ ((TEndo‘𝐾)‘𝑊))}))
13	2, 12	mpbiri 261	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → Rel (𝐼‘𝑄))
14		ssrab2 4036	. . . . . 6 ⊢ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} ⊆ (𝑇 × ((TEndo‘𝐾)‘𝑊))
15		relxp 5670	. . . . . 6 ⊢ Rel (𝑇 × ((TEndo‘𝐾)‘𝑊))
16		relss 5759	. . . . . 6 ⊢ ({𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} ⊆ (𝑇 × ((TEndo‘𝐾)‘𝑊)) → (Rel (𝑇 × ((TEndo‘𝐾)‘𝑊)) → Rel {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}))
17	14, 15, 16	mp2 9	. . . . 5 ⊢ Rel {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}
18	17	a1i 11	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → Rel {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
19		id 23	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)))
20		vex 3461	. . . . . . 7 ⊢ 𝑔 ∈ V
21		vex 3461	. . . . . . 7 ⊢ 𝑠 ∈ V
22	3, 4, 5, 6, 7, 8, 9, 10, 20, 21	dicopelval2 41817	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑔, 𝑠⟩ ∈ (𝐼‘𝑄) ↔ (𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))))
23		simprl 782	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑔 = (𝑠‘𝐹))
24		simpll 778	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
25		simprr 784	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))
26		simpl 487	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
27	3, 4, 5, 6	lhpocnel2 40655	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
28	27	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
29		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
30	3, 4, 5, 7, 10	ltrniotacl 41215	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
31	26, 28, 29, 30	syl3anc 1394	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
32	31	adantr 485	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝐹 ∈ 𝑇)
33	5, 7, 8	tendocl 41403	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇)
34	24, 25, 32, 33	syl3anc 1394	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑠‘𝐹) ∈ 𝑇)
35	23, 34	eqeltrd 2865	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → 𝑔 ∈ 𝑇)
36	35, 25, 23	3jca 1144	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹)))
37		simpr3 1213	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹))) → 𝑔 = (𝑠‘𝐹))
38		simpr2 1212	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹))) → 𝑠 ∈ ((TEndo‘𝐾)‘𝑊))
39	37, 38	jca 520	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹))) → (𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)))
40	36, 39	impbida 812	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹))))
41		diclspsn.u	. . . . . . . . . . . . . 14 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
42		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
43		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (Base‘(Scalar‘𝑈)) = (Base‘(Scalar‘𝑈))
44	5, 8, 41, 42, 43	dvhbase 41719	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
45	44	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (Base‘(Scalar‘𝑈)) = ((TEndo‘𝐾)‘𝑊))
46	45	rexeqdv 3324	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
47		simpll 778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
48		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
49	31	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → 𝐹 ∈ 𝑇)
50	5, 7, 8	tendoidcl 41405	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
51	50	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
52		eqid 2765	. . . . . . . . . . . . . . . . . 18 ⊢ ( ·𝑠 ‘𝑈) = ( ·𝑠 ‘𝑈)
53	5, 7, 8, 41, 52	dvhopvsca 41738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝐹 ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) = ⟨(𝑥‘𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩)
54	47, 48, 49, 51, 53	syl13anc 1395	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) = ⟨(𝑥‘𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩)
55	54	eqeq2d 2776	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ⟨𝑔, 𝑠⟩ = ⟨(𝑥‘𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩))
56	20, 21	opth 5449	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑔, 𝑠⟩ = ⟨(𝑥‘𝐹), (𝑥 ∘ ( I ↾ 𝑇))⟩ ↔ (𝑔 = (𝑥‘𝐹) ∧ 𝑠 = (𝑥 ∘ ( I ↾ 𝑇))))
57	55, 56	bitrdi 290	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑔 = (𝑥‘𝐹) ∧ 𝑠 = (𝑥 ∘ ( I ↾ 𝑇)))))
58	5, 7, 8	tendo1mulr 41407	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( I ↾ 𝑇)) = 𝑥)
59	58	adantlr 727	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑥 ∘ ( I ↾ 𝑇)) = 𝑥)
60	59	eqeq2d 2776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑠 = (𝑥 ∘ ( I ↾ 𝑇)) ↔ 𝑠 = 𝑥))
61		equcom 2041	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑥 ↔ 𝑥 = 𝑠)
62	60, 61	bitrdi 290	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑠 = (𝑥 ∘ ( I ↾ 𝑇)) ↔ 𝑥 = 𝑠))
63	62	anbi2d 641	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → ((𝑔 = (𝑥‘𝐹) ∧ 𝑠 = (𝑥 ∘ ( I ↾ 𝑇))) ↔ (𝑔 = (𝑥‘𝐹) ∧ 𝑥 = 𝑠)))
64	57, 63	bitrd 282	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑔 = (𝑥‘𝐹) ∧ 𝑥 = 𝑠)))
65		ancom 465	. . . . . . . . . . . . 13 ⊢ ((𝑔 = (𝑥‘𝐹) ∧ 𝑥 = 𝑠) ↔ (𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹)))
66	64, 65	bitrdi 290	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)) → (⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹))))
67	66	rexbidva 3187	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹))))
68	46, 67	bitrd 282	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹))))
69	68	3anbi3d 1466	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)) ↔ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹)))))
70		fveq1 6870	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑠 → (𝑥‘𝐹) = (𝑠‘𝐹))
71	70	eqeq2d 2776	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑠 → (𝑔 = (𝑥‘𝐹) ↔ 𝑔 = (𝑠‘𝐹)))
72	71	ceqsrexv 3617	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) → (∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹)) ↔ 𝑔 = (𝑠‘𝐹)))
73	72	pm5.32i 584	. . . . . . . . . . 11 ⊢ ((𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹))) ↔ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹)))
74	73	anbi2i 634	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹)))) ↔ (𝑔 ∈ 𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹))))
75		3anass 1109	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹))) ↔ (𝑔 ∈ 𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹)))))
76		3anass 1109	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹)) ↔ (𝑔 ∈ 𝑇 ∧ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹))))
77	74, 75, 76	3bitr4i 306	. . . . . . . . 9 ⊢ ((𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ ((TEndo‘𝐾)‘𝑊)(𝑥 = 𝑠 ∧ 𝑔 = (𝑥‘𝐹))) ↔ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹)))
78	69, 77	bitr2di 291	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ 𝑔 = (𝑠‘𝐹)) ↔ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))))
79	40, 78	bitrd 282	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))))
80		eqeq1 2769	. . . . . . . . . . 11 ⊢ (𝑣 = ⟨𝑔, 𝑠⟩ → (𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
81	80	rexbidv 3189	. . . . . . . . . 10 ⊢ (𝑣 = ⟨𝑔, 𝑠⟩ → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
82	81	rabxp 5700	. . . . . . . . 9 ⊢ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} = {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))}
83	82	eleq2i 2857	. . . . . . . 8 ⊢ (⟨𝑔, 𝑠⟩ ∈ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} ↔ ⟨𝑔, 𝑠⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))})
84		opabidw 5499	. . . . . . . 8 ⊢ (⟨𝑔, 𝑠⟩ ∈ {⟨𝑔, 𝑠⟩ ∣ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))} ↔ (𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)))
85	83, 84	bitr2i 279	. . . . . . 7 ⊢ ((𝑔 ∈ 𝑇 ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))⟨𝑔, 𝑠⟩ = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)) ↔ ⟨𝑔, 𝑠⟩ ∈ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
86	79, 85	bitrdi 290	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝑔 = (𝑠‘𝐹) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑊)) ↔ ⟨𝑔, 𝑠⟩ ∈ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}))
87	22, 86	bitrd 282	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (⟨𝑔, 𝑠⟩ ∈ (𝐼‘𝑄) ↔ ⟨𝑔, 𝑠⟩ ∈ {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}))
88	87	eqrelrdv2 5772	. . . 4 ⊢ (((Rel (𝐼‘𝑄) ∧ Rel {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)}) ∧ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))) → (𝐼‘𝑄) = {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
89	13, 18, 19, 88	syl21anc 850	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
90		simpll 778	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
91	45	eleq2d 2851	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑥 ∈ (Base‘(Scalar‘𝑈)) ↔ 𝑥 ∈ ((TEndo‘𝐾)‘𝑊)))
92	91	biimpa 481	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → 𝑥 ∈ ((TEndo‘𝐾)‘𝑊))
93	50	adantr 485	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))
94		opelxpi 5689	. . . . . . . . . 10 ⊢ ((𝐹 ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
95	31, 93, 94	syl2anc 595	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
96	95	adantr 485	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
97	5, 7, 8, 41, 52	dvhvscacl 41739	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ ((TEndo‘𝐾)‘𝑊) ∧ ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))) → (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
98	90, 92, 96, 97	syl12anc 849	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)))
99		eleq1a 2860	. . . . . . 7 ⊢ ((𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) → (𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) → 𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊))))
100	98, 99	syl 18	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ 𝑥 ∈ (Base‘(Scalar‘𝑈))) → (𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) → 𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊))))
101	100	rexlimdva 3166	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) → 𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊))))
102	101	pm4.71rd 571	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩) ↔ (𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))))
103	102	abbidv 2831	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)} = {𝑣 ∣ (𝑣 ∈ (𝑇 × ((TEndo‘𝐾)‘𝑊)) ∧ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩))})
104	1, 89, 103	3eqtr4a 2826	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
105	5, 41, 26	dvhlmod 41746	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝑈 ∈ LMod)
106		eqid 2765	. . . . 5 ⊢ (Base‘𝑈) = (Base‘𝑈)
107	5, 7, 8, 41, 106	dvhelvbasei 41724	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
108	26, 31, 93, 107	syl12anc 849	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈))
109		diclspsn.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
110	42, 43, 106, 52, 109	lspsn 21092	. . 3 ⊢ ((𝑈 ∈ LMod ∧ ⟨𝐹, ( I ↾ 𝑇)⟩ ∈ (Base‘𝑈)) → (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
111	105, 108, 110	syl2anc 595	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) = {𝑣 ∣ ∃𝑥 ∈ (Base‘(Scalar‘𝑈))𝑣 = (𝑥( ·𝑠 ‘𝑈)⟨𝐹, ( I ↾ 𝑇)⟩)})
112	104, 111	eqtr4d 2803	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145  {cab 2743  ∃wrex 3089  {crab 3417   ⊆ wss 3907  {csn 4585  ⟨cop 4591   class class class wbr 5105  {copab 5167   I cid 5546   × cxp 5650   ↾ cres 5654   ∘ ccom 5656  Rel wrel 5657  ‘cfv 6525  ℩crio 7356  (class class class)co 7400  Basecbs 17259  Scalarcsca 17303   ·_𝑠 cvsca 17304  lecple 17307  occoc 17308  LModclmod 20950  LSpanclspn 21061  Atomscatm 39899  HLchlt 39986  LHypclh 40620  LTrncltrn 40737  TEndoctendo 41388  DVecHcdvh 41714  DIsoCcdic 41808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-riotaBAD 39589

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-tpos 8210  df-undef 8257  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-er 8682  df-map 8814  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12225  df-2 12294  df-3 12295  df-4 12296  df-5 12297  df-6 12298  df-n0 12496  df-z 12583  df-uz 12854  df-fz 13527  df-struct 17197  df-sets 17214  df-slot 17232  df-ndx 17244  df-base 17260  df-ress 17281  df-plusg 17313  df-mulr 17314  df-sca 17316  df-vsca 17317  df-0g 17484  df-proset 18340  df-poset 18359  df-plt 18374  df-lub 18390  df-glb 18391  df-join 18392  df-meet 18393  df-p0 18469  df-p1 18470  df-lat 18478  df-clat 18545  df-mgm 18688  df-sgrp 18767  df-mnd 18783  df-grp 18993  df-minusg 18994  df-sbg 18995  df-cmn 19843  df-abl 19844  df-mgp 20208  df-rng 20222  df-ur 20255  df-ring 20308  df-oppr 20410  df-dvdsr 20430  df-unit 20431  df-invr 20461  df-dvr 20474  df-drng 20806  df-lmod 20952  df-lss 21022  df-lsp 21062  df-lvec 21193  df-oposet 39812  df-ol 39814  df-oml 39815  df-covers 39902  df-ats 39903  df-atl 39934  df-cvlat 39958  df-hlat 39987  df-llines 40134  df-lplanes 40135  df-lvols 40136  df-lines 40137  df-psubsp 40139  df-pmap 40140  df-padd 40432  df-lhyp 40624  df-laut 40625  df-ldil 40740  df-ltrn 40741  df-trl 40795  df-tendo 41391  df-edring 41393  df-dvech 41715  df-dic 41809

This theorem is referenced by:  cdlemn5pre  41836  dih1dimc  41878