dihatlat - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > dihatlat		Structured version Visualization version GIF version

Theorem dihatlat 39348

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 2738	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2738	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
3		dihatlat.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4		dihatlat.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2738	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
6		eqid 2738	. . . . 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 2738	. . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	dih1dimb2 39255	. . . 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 468	. . 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 1246	. . . . 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 1196	. . . . . . 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 39124	. . . . . 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 1200	. . . . . . 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 2738	. . . . . . . . 9 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
17	1, 4, 5, 16, 6	tendo0cl 38804	. . . . . . . 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 2738	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
20	4, 5, 16, 7, 19	dvhelvbasei 39102	. . . . . . 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 834	. . . . . 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 1245	. . . . . . . . . 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 2948	. . . . . . . . 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 490	. . . . . . . 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 3436	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
26		fvex 6787	. . . . . . . . . . 11 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
27	26	mptex 7099	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ V
28	25, 27	opth 5391	. . . . . . . . 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 2990	. . . . . . . 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 2738	. . . . . . . . 9 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
32	1, 4, 5, 7, 31, 6	dvh0g 39125	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0_g‘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)))⟩})))) → (0_g‘𝑈) = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
34	30, 33	neeqtrrd 3018	. . . . . 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)))⟩ ≠ (0_g‘𝑈))
35		dihatlat.l	. . . . . . 7 ⊢ 𝐿 = (LSAtoms‘𝑈)
36	19, 9, 31, 35	lsatlspsn2 37006	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ∈ (Base‘𝑈) ∧ ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩ ≠ (0_g‘𝑈)) → ((LSpan‘𝑈)‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩}) ∈ 𝐿)
37	14, 21, 34, 36	syl3anc 1370	. . . . 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 2839	. . . 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 1117	. . 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 3220	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) ∈ 𝐿)
41		eqid 2738	. . . . 5 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
42		eqid 2738	. . . . 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 39256	. . . 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 468	. . 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 764	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46	4, 7, 45	dvhlmod 39124	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑈 ∈ LMod)
47		eqid 2738	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
48	2, 47, 3, 4	lhpocnel 38032	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
49	48	ad2antrr 723	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
50		simplr 766	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑄 ∈ 𝐴)
51		simpr 485	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ¬ 𝑄(le‘𝐾)𝑊)
52	2, 3, 4, 5, 42	ltrniotacl 38593	. . . . . 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 1373	. . . . 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 38783	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))
55	54	ad2antrr 723	. . . . 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 39102	. . . . 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 834	. . . 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 38842	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ≠ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))
59	58	ad2antrr 723	. . . . . . . 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 2948	. . . . . . 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 489	. . . . . 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 7236	. . . . . . . 8 ⊢ (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ V
63		resiexg 7761	. . . . . . . . 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 5391	. . . . . . 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 2990	. . . . . 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 723	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (0_g‘𝑈) = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
69	67, 68	neeqtrrd 3018	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ≠ (0_g‘𝑈))
70	19, 9, 31, 35	lsatlspsn2 37006	. . . 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))⟩ ≠ (0_g‘𝑈)) → ((LSpan‘𝑈)‘{⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∈ 𝐿)
71	46, 57, 69, 70	syl3anc 1370	. . 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 2839	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) ∈ 𝐿)
73	40, 72	pm2.61dan 810	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → (𝐼‘𝑄) ∈ 𝐿)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3065 Vcvv 3432 {csn 4561 ⟨cop 4567 class class class wbr 5074 ↦ cmpt 5157 I cid 5488 ↾ cres 5591 ‘cfv 6433 ℩crio 7231 Basecbs 16912 lecple 16969 occoc 16970 0_gc0g 17150 LModclmod 20123 LSpanclspn 20233 LSAtomsclsa 36988 Atomscatm 37277 HLchlt 37364 LHypclh 37998 LTrncltrn 38115 TEndoctendo 38766 DVecHcdvh 39092 DIsoHcdih 39242

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tendo 38769 df-edring 38771 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243

This theorem is referenced by: dihat 39349 dihjat3 39446 dihjat5N 39451 dvh4dimat 39452

W3C validator