Theorem dihatlat 42534

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 2763	. . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2763	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
3		dihatlat.a	. . . . 5 ⊢ 𝐴 = (Atoms‘𝐾)
4		dihatlat.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		eqid 2763	. . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
6		eqid 2763	. . . . 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 2763	. . . . 5 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	dih1dimb2 42441	. . . 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 473	. . 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 1266	. . . . 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 1216	. . . . . . 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 42310	. . . . . 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 1220	. . . . . . 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 2763	. . . . . . . . 9 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
17	1, 4, 5, 16, 6	tendo0cl 41990	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ ((TEndo‘𝐾)‘𝑊))
18	13, 17	syl 18	. . . . . . 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 2763	. . . . . . . 8 ⊢ (Base‘𝑈) = (Base‘𝑈)
20	4, 5, 16, 7, 19	dvhelvbasei 42288	. . . . . . 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 850	. . . . . 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 1265	. . . . . . . . . 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 2964	. . . . . . . . 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 495	. . . . . . . 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 3458	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
26		fvex 6909	. . . . . . . . . . 11 ⊢ ((LTrn‘𝐾)‘𝑊) ∈ V
27	26	mptex 7239	. . . . . . . . . 10 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) ∈ V
28	25, 27	opth 5449	. . . . . . . . 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 3005	. . . . . . . 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 237	. . . . . . 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 2763	. . . . . . . . 9 ⊢ (0g‘𝑈) = (0g‘𝑈)
32	1, 4, 5, 7, 31, 6	dvh0g 42311	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (0g‘𝑈) = ⟨( I ↾ (Base‘𝐾)), (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))⟩)
33	13, 32	syl 18	. . . . . . 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 3033	. . . . . 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 40192	. . . . . 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 1398	. . . . 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 2864	. . . 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 1136	. . 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 3173	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) ∈ 𝐿)
41		eqid 2763	. . . . 5 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
42		eqid 2763	. . . . 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 42442	. . . 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 473	. . 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 779	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
46	4, 7, 45	dvhlmod 42310	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑈 ∈ LMod)
47		eqid 2763	. . . . . . . 8 ⊢ (oc‘𝐾) = (oc‘𝐾)
48	2, 47, 3, 4	lhpocnel 41218	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
49	48	ad2antrr 739	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
50		simplr 781	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑄 ∈ 𝐴)
51		simpr 490	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ¬ 𝑄(le‘𝐾)𝑊)
52	2, 3, 4, 5, 42	ltrniotacl 41779	. . . . . 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 1401	. . . . 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 41969	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))
55	54	ad2antrr 739	. . . . 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 42288	. . . . 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 850	. . . 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 42028	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ≠ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))))
59	58	ad2antrr 739	. . . . . . . 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 2964	. . . . . . 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 494	. . . . . 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 7391	. . . . . . . 8 ⊢ (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ V
63		resiexg 7934	. . . . . . . . 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 5449	. . . . . . 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 3005	. . . . . 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 237	. . . . 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 739	. . . . 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 3033	. . . 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 40192	. . . 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 1398	. . 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 2864	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) ∈ 𝐿)
73	40, 72	pm2.61dan 825	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → (𝐼‘𝑄) ∈ 𝐿)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ≠ wne 2959  ∃wrex 3090  Vcvv 3454  {csn 4588  ⟨cop 4838   class class class wbr 5106   ↦ cmpt 5189   I cid 5549   ↾ cres 5657  ‘cfv 6547  ℩crio 7386  Basecbs 17402  lecple 17450  occoc 17451  0_gc0g 17625  LModclmod 21150  LSpanclspn 21261  LSAtomsclsa 40174  Atomscatm 40463  HLchlt 40550  LHypclh 41184  LTrncltrn 41301  TEndoctendo 41952  DVecHcdvh 42278  DIsoHcdih 42428

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761  ax-cnex 11269  ax-resscn 11270  ax-1cn 11271  ax-icn 11272  ax-addcl 11273  ax-addrcl 11274  ax-mulcl 11275  ax-mulrcl 11276  ax-mulcom 11277  ax-addass 11278  ax-mulass 11279  ax-distr 11280  ax-i2m1 11281  ax-1ne0 11282  ax-1rid 11283  ax-rnegex 11284  ax-rrecex 11285  ax-cnre 11286  ax-pre-lttri 11287  ax-pre-lttrn 11288  ax-pre-ltadd 11289  ax-pre-mulgt0 11290  ax-riotaBAD 40153

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-1st 8011  df-2nd 8012  df-tpos 8257  df-undef 8304  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-er 8731  df-map 8863  df-en 8988  df-dom 8989  df-sdom 8990  df-fin 8991  df-pnf 11358  df-mnf 11359  df-xr 11360  df-ltxr 11361  df-le 11362  df-sub 11556  df-neg 11557  df-nn 12349  df-2 12418  df-3 12419  df-4 12420  df-5 12421  df-6 12422  df-n0 12620  df-z 12707  df-uz 12979  df-fz 13653  df-struct 17340  df-sets 17357  df-slot 17375  df-ndx 17387  df-base 17403  df-ress 17424  df-plusg 17456  df-mulr 17457  df-sca 17459  df-vsca 17460  df-0g 17627  df-proset 18483  df-poset 18502  df-plt 18517  df-lub 18533  df-glb 18534  df-join 18535  df-meet 18536  df-p0 18612  df-p1 18613  df-lat 18621  df-clat 18688  df-mgm 18831  df-sgrp 18923  df-mnd 18939  df-submnd 18994  df-grp 19162  df-minusg 19163  df-sbg 19164  df-subg 19348  df-cntz 19546  df-lsm 19865  df-cmn 20011  df-abl 20012  df-mgp 20376  df-rng 20390  df-ur 20423  df-ring 20476  df-oppr 20582  df-dvdsr 20602  df-unit 20603  df-invr 20633  df-dvr 20646  df-drng 20997  df-lmod 21152  df-lss 21222  df-lsp 21262  df-lvec 21393  df-lsatoms 40176  df-oposet 40376  df-ol 40378  df-oml 40379  df-covers 40466  df-ats 40467  df-atl 40498  df-cvlat 40522  df-hlat 40551  df-llines 40698  df-lplanes 40699  df-lvols 40700  df-lines 40701  df-psubsp 40703  df-pmap 40704  df-padd 40996  df-lhyp 41188  df-laut 41189  df-ldil 41304  df-ltrn 41305  df-trl 41359  df-tendo 41955  df-edring 41957  df-disoa 42229  df-dvech 42279  df-dib 42339  df-dic 42373  df-dih 42429

This theorem is used by:  dihat  42535  dihjat3  42632  dihjat5N  42637  dvh4dimat  42638