Theorem dihatlat 40839

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 2728	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2728	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
3		dihatlat.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4		dihatlat.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2728	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
6		eqid 2728	. . . . 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 2728	. . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	dih1dimb2 40746	. . . 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 466	. . 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 1244	. . . . 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 1194	. . . . . . 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 40615	. . . . . 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 1198	. . . . . . 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 2728	. . . . . . . . 9 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
17	1, 4, 5, 16, 6	tendo0cl 40295	. . . . . . . 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 2728	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
20	4, 5, 16, 7, 19	dvhelvbasei 40593	. . . . . . 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 835	. . . . . 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 1243	. . . . . . . . . 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 2942	. . . . . . . . 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 488	. . . . . . . 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 3477	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
26		fvex 6915	. . . . . . . . . . 11 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
27	26	mptex 7241	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ V
28	25, 27	opth 5482	. . . . . . . . 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 2984	. . . . . . . 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 233	. . . . . . 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 2728	. . . . . . . . 9 ⊢ (0g‘𝑈) = (0g‘𝑈)
32	1, 4, 5, 7, 31, 6	dvh0g 40616	. . . . . . . 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 3012	. . . . . 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 38496	. . . . . 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 1368	. . . . 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 2829	. . . 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 1115	. . 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 3158	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) ∈ 𝐿)
41		eqid 2728	. . . . 5 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
42		eqid 2728	. . . . 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 40747	. . . 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 466	. . 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 765	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46	4, 7, 45	dvhlmod 40615	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑈 ∈ LMod)
47		eqid 2728	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
48	2, 47, 3, 4	lhpocnel 39523	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
49	48	ad2antrr 724	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
50		simplr 767	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑄 ∈ 𝐴)
51		simpr 483	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ¬ 𝑄(le‘𝐾)𝑊)
52	2, 3, 4, 5, 42	ltrniotacl 40084	. . . . . 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 1371	. . . . 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 40274	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))
55	54	ad2antrr 724	. . . . 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 40593	. . . . 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 835	. . . 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 40333	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ≠ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))
59	58	ad2antrr 724	. . . . . . . 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 2942	. . . . . . 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 487	. . . . . 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 7386	. . . . . . . 8 ⊢ (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ V
63		resiexg 7926	. . . . . . . . 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 5482	. . . . . . 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 2984	. . . . . 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 233	. . . . 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 724	. . . . 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 3012	. . . 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 38496	. . . 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 1368	. . 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 2829	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) ∈ 𝐿)
73	40, 72	pm2.61dan 811	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → (𝐼‘𝑄) ∈ 𝐿)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  ∃wrex 3067  Vcvv 3473  {csn 4632  ⟨cop 4638   class class class wbr 5152   ↦ cmpt 5235   I cid 5579   ↾ cres 5684  ‘cfv 6553  ℩crio 7381  Basecbs 17187  lecple 17247  occoc 17248  0_gc0g 17428  LModclmod 20750  LSpanclspn 20862  LSAtomsclsa 38478  Atomscatm 38767  HLchlt 38854  LHypclh 39489  LTrncltrn 39606  TEndoctendo 40257  DVecHcdvh 40583  DIsoHcdih 40733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-riotaBAD 38457

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  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 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-tpos 8238  df-undef 8285  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-n0 12511  df-z 12597  df-uz 12861  df-fz 13525  df-struct 17123  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-0g 17430  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-submnd 18748  df-grp 18900  df-minusg 18901  df-sbg 18902  df-subg 19085  df-cntz 19275  df-lsm 19598  df-cmn 19744  df-abl 19745  df-mgp 20082  df-rng 20100  df-ur 20129  df-ring 20182  df-oppr 20280  df-dvdsr 20303  df-unit 20304  df-invr 20334  df-dvr 20347  df-drng 20633  df-lmod 20752  df-lss 20823  df-lsp 20863  df-lvec 20995  df-lsatoms 38480  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005  df-lines 39006  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493  df-laut 39494  df-ldil 39609  df-ltrn 39610  df-trl 39664  df-tendo 40260  df-edring 40262  df-disoa 40534  df-dvech 40584  df-dib 40644  df-dic 40678  df-dih 40734

This theorem is referenced by:  dihat  40840  dihjat3  40937  dihjat5N  40942  dvh4dimat  40943