Theorem dihatlat 37120

Description: The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.)

Hypotheses

Ref	Expression
dihatlat.a	⊢ 𝐴 = (Atoms‘𝐾)
dihatlat.h	⊢ 𝐻 = (LHyp‘𝐾)
dihatlat.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihatlat.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihatlat.l	⊢ 𝐿 = (LSAtoms‘𝑈)

Assertion

Ref	Expression
dihatlat	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → (𝐼‘𝑄) ∈ 𝐿)

Proof of Theorem dihatlat

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2817	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2817	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
3		dihatlat.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4		dihatlat.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2817	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
6		eqid 2817	. . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))
7		dihatlat.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		dihatlat.i	. . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
9		eqid 2817	. . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	dih1dimb2 37027	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄(le‘𝐾)𝑊)) → ∃𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))
11	10	anassrs 455	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → ∃𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))
12		simp3rr 1321	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩}))
13		simp1l 1247	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
14	4, 7, 13	dvhlmod 36896	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → 𝑈 ∈ LMod)
15		simp3l 1251	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))
16		eqid 2817	. . . . . . . . 9 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
17	1, 4, 5, 16, 6	tendo0cl 36576	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ ((TEndo‘𝐾)‘𝑊))
18	13, 17	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ ((TEndo‘𝐾)‘𝑊))
19		eqid 2817	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
20	4, 5, 16, 7, 19	dvhelvbasei 36874	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ∈ (Base‘𝑈))
21	13, 15, 18, 20	syl12anc 856	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ∈ (Base‘𝑈))
22		simp3rl 1320	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → 𝑔 ≠ ( I ↾ (Base‘𝐾)))
23	22	neneqd 2994	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → ¬ 𝑔 = ( I ↾ (Base‘𝐾)))
24	23	intnanrd 479	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → ¬ (𝑔 = ( I ↾ (Base‘𝐾)) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))
25		vex 3405	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
26		fvex 6428	. . . . . . . . . . 11 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
27	26	mptex 6718	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ V
28	25, 27	opth 5145	. . . . . . . . 9 ⊢ (⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ↔ (𝑔 = ( I ↾ (Base‘𝐾)) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))
29	28	necon3abii 3035	. . . . . . . 8 ⊢ (⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ≠ ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ↔ ¬ (𝑔 = ( I ↾ (Base‘𝐾)) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))
30	24, 29	sylibr 225	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ≠ ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
31		eqid 2817	. . . . . . . . 9 ⊢ (0g‘𝑈) = (0g‘𝑈)
32	1, 4, 5, 7, 31, 6	dvh0g 36897	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0g‘𝑈) = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
33	13, 32	syl 17	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → (0g‘𝑈) = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
34	30, 33	neeqtrrd 3063	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ≠ (0g‘𝑈))
35		dihatlat.l	. . . . . . 7 ⊢ 𝐿 = (LSAtoms‘𝑈)
36	19, 9, 31, 35	lsatlspsn2 34778	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ∈ (Base‘𝑈) ∧ ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ≠ (0g‘𝑈)) → ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩}) ∈ 𝐿)
37	14, 21, 34, 36	syl3anc 1483	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩}) ∈ 𝐿)
38	12, 37	eqeltrd 2896	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊 ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → (𝐼‘𝑄) ∈ 𝐿)
39	38	3expa 1140	. . 3 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑔 ≠ ( I ↾ (Base‘𝐾)) ∧ (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩})))) → (𝐼‘𝑄) ∈ 𝐿)
40	11, 39	rexlimddv 3234	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) ∈ 𝐿)
41		eqid 2817	. . . . 5 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
42		eqid 2817	. . . . 5 ⊢ (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) = (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄)
43	2, 3, 4, 41, 5, 8, 7, 9, 42	dih1dimc 37028	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄(le‘𝐾)𝑊)) → (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}))
44	43	anassrs 455	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) = ((LSpan‘𝑈)‘{⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}))
45		simpll 774	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46	4, 7, 45	dvhlmod 36896	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑈 ∈ LMod)
47		eqid 2817	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
48	2, 47, 3, 4	lhpocnel 35804	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
49	48	ad2antrr 708	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
50		simplr 776	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑄 ∈ 𝐴)
51		simpr 473	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ¬ 𝑄(le‘𝐾)𝑊)
52	2, 3, 4, 5, 42	ltrniotacl 36365	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄(le‘𝐾)𝑊)) → (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
53	45, 49, 50, 51, 52	syl112anc 1486	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
54	4, 5, 16	tendoidcl 36555	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))
55	54	ad2antrr 708	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))
56	4, 5, 16, 7, 19	dvhelvbasei 36874	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊) ∧ ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (Base‘𝑈))
57	45, 53, 55, 56	syl12anc 856	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (Base‘𝑈))
58	1, 4, 5, 16, 6	tendo1ne0 36614	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ≠ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))
59	58	ad2antrr 708	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ≠ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))
60	59	neneqd 2994	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ¬ ( I ↾ ((LTrn‘𝐾)‘𝑊)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))
61	60	intnand 478	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ¬ ((℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) = ( I ↾ (Base‘𝐾)) ∧ ( I ↾ ((LTrn‘𝐾)‘𝑊)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))
62		riotaex 6846	. . . . . . . 8 ⊢ (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ V
63		resiexg 7339	. . . . . . . . 9 ⊢ (((LTrn‘𝐾)‘𝑊) ∈ V → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ V)
64	26, 63	ax-mp 5	. . . . . . . 8 ⊢ ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ V
65	62, 64	opth 5145	. . . . . . 7 ⊢ (⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ↔ ((℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) = ( I ↾ (Base‘𝐾)) ∧ ( I ↾ ((LTrn‘𝐾)‘𝑊)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))
66	65	necon3abii 3035	. . . . . 6 ⊢ (⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ≠ ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ↔ ¬ ((℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) = ( I ↾ (Base‘𝐾)) ∧ ( I ↾ ((LTrn‘𝐾)‘𝑊)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))
67	61, 66	sylibr 225	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ≠ ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
68	32	ad2antrr 708	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (0g‘𝑈) = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
69	67, 68	neeqtrrd 3063	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ≠ (0g‘𝑈))
70	19, 9, 31, 35	lsatlspsn2 34778	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (Base‘𝑈) ∧ ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ≠ (0g‘𝑈)) → ((LSpan‘𝑈)‘{⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∈ 𝐿)
71	46, 57, 69, 70	syl3anc 1483	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ((LSpan‘𝑈)‘{⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∈ 𝐿)
72	44, 71	eqeltrd 2896	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) ∈ 𝐿)
73	40, 72	pm2.61dan 838	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → (𝐼‘𝑄) ∈ 𝐿)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1100   = wceq 1637   ∈ wcel 2157   ≠ wne 2989  ∃wrex 3108  Vcvv 3402  {csn 4381  ⟨cop 4387   class class class wbr 4855   ↦ cmpt 4934   I cid 5229   ↾ cres 5324  ‘cfv 6108  ℩crio 6841  Basecbs 16075  lecple 16167  occoc 16168  0_gc0g 16312  LModclmod 19074  LSpanclspn 19185  LSAtomsclsa 34760  Atomscatm 35049  HLchlt 35136  LHypclh 35770  LTrncltrn 35887  TEndoctendo 36538  DVecHcdvh 36864  DIsoHcdih 37014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4975  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186  ax-cnex 10284  ax-resscn 10285  ax-1cn 10286  ax-icn 10287  ax-addcl 10288  ax-addrcl 10289  ax-mulcl 10290  ax-mulrcl 10291  ax-mulcom 10292  ax-addass 10293  ax-mulass 10294  ax-distr 10295  ax-i2m1 10296  ax-1ne0 10297  ax-1rid 10298  ax-rnegex 10299  ax-rrecex 10300  ax-cnre 10301  ax-pre-lttri 10302  ax-pre-lttrn 10303  ax-pre-ltadd 10304  ax-pre-mulgt0 10305  ax-riotaBAD 34738

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-iin 4726  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5230  df-eprel 5235  df-po 5243  df-so 5244  df-fr 5281  df-we 5283  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-pred 5904  df-ord 5950  df-on 5951  df-lim 5952  df-suc 5953  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-riota 6842  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-om 7303  df-1st 7405  df-2nd 7406  df-tpos 7594  df-undef 7641  df-wrecs 7649  df-recs 7711  df-rdg 7749  df-1o 7803  df-oadd 7807  df-er 7986  df-map 8101  df-en 8200  df-dom 8201  df-sdom 8202  df-fin 8203  df-pnf 10368  df-mnf 10369  df-xr 10370  df-ltxr 10371  df-le 10372  df-sub 10560  df-neg 10561  df-nn 11313  df-2 11371  df-3 11372  df-4 11373  df-5 11374  df-6 11375  df-n0 11567  df-z 11651  df-uz 11912  df-fz 12557  df-struct 16077  df-ndx 16078  df-slot 16079  df-base 16081  df-sets 16082  df-ress 16083  df-plusg 16173  df-mulr 16174  df-sca 16176  df-vsca 16177  df-0g 16314  df-proset 17140  df-poset 17158  df-plt 17170  df-lub 17186  df-glb 17187  df-join 17188  df-meet 17189  df-p0 17251  df-p1 17252  df-lat 17258  df-clat 17320  df-mgm 17454  df-sgrp 17496  df-mnd 17507  df-submnd 17548  df-grp 17637  df-minusg 17638  df-sbg 17639  df-subg 17800  df-cntz 17958  df-lsm 18259  df-cmn 18403  df-abl 18404  df-mgp 18699  df-ur 18711  df-ring 18758  df-oppr 18832  df-dvdsr 18850  df-unit 18851  df-invr 18881  df-dvr 18892  df-drng 18960  df-lmod 19076  df-lss 19144  df-lsp 19186  df-lvec 19317  df-lsatoms 34762  df-oposet 34962  df-ol 34964  df-oml 34965  df-covers 35052  df-ats 35053  df-atl 35084  df-cvlat 35108  df-hlat 35137  df-llines 35284  df-lplanes 35285  df-lvols 35286  df-lines 35287  df-psubsp 35289  df-pmap 35290  df-padd 35582  df-lhyp 35774  df-laut 35775  df-ldil 35890  df-ltrn 35891  df-trl 35945  df-tendo 36541  df-edring 36543  df-disoa 36815  df-dvech 36865  df-dib 36925  df-dic 36959  df-dih 37015

This theorem is referenced by:  dihat  37121  dihjat3  37218  dihjat5N  37223  dvh4dimat  37224