dihatexv - Mathbox for Norm Megill

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Theorem dihatexv 42315

Description: There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)

**Hypotheses**
Ref	Expression
dihatexv.b	⊢ 𝐵 = (Base‘𝐾)
dihatexv.a	⊢ 𝐴 = (Atoms‘𝐾)
dihatexv.h	⊢ 𝐻 = (LHyp‘𝐾)
dihatexv.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihatexv.v	⊢ 𝑉 = (Base‘𝑈)
dihatexv.o	⊢ 0 = (0_g‘𝑈)
dihatexv.n	⊢ 𝑁 = (LSpan‘𝑈)
dihatexv.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihatexv.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dihatexv.q	⊢ (𝜑 → 𝑄 ∈ 𝐵)

**Assertion**
Ref	Expression
dihatexv	⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥})))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐼 𝑥,𝐾 𝑥,𝑁 𝑥,𝑄 𝑥,𝑉 𝑥,𝑊 𝜑,𝑥 Allowed substitution hints: 𝑈(𝑥) 𝐻(𝑥) 0 (𝑥)

Proof of Theorem dihatexv Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihatexv.k	. . . . . . . . 9 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2	1	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → 𝑄 ∈ 𝐴)
4		simpr 490	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → 𝑄(le‘𝐾)𝑊)
5		dihatexv.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
6		eqid 2760	. . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾)
7		dihatexv.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
8		dihatexv.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
9		eqid 2760	. . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10		eqid 2760	. . . . . . . . 9 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))
11		dihatexv.u	. . . . . . . . 9 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
12		dihatexv.i	. . . . . . . . 9 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
13		dihatexv.n	. . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈)
14	5, 6, 7, 8, 9, 10, 11, 12, 13	dih1dimb2 42218	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄(le‘𝐾)𝑊)) → ∃𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩})))
15	2, 3, 4, 14	syl12anc 850	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → ∃𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩})))
16	1	ad3antrrr 743	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		simpr 490	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))
18		eqid 2760	. . . . . . . . . . . . . 14 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
19	5, 8, 9, 18, 10	tendo0cl 41767	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
20	16, 19	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
21		dihatexv.v	. . . . . . . . . . . . 13 ⊢ 𝑉 = (Base‘𝑈)
22	8, 9, 18, 11, 21	dvhelvbasei 42065	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑔 ∈ ((LTrn‘𝐾)‘𝑊) ∧ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))) → ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩ ∈ 𝑉)
23	16, 17, 20, 22	syl12anc 850	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩ ∈ 𝑉)
24		sneq 4593	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩ → {𝑥} = {⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩})
25	24	fveq2d 6877	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩ → (𝑁‘{𝑥}) = (𝑁‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩}))
26	25	rspceeqv 3598	. . . . . . . . . . 11 ⊢ ((⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩ ∈ 𝑉 ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩})) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}))
27	23, 26	sylan 592	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩})) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}))
28	27	ex 418	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝐼‘𝑄) = (𝑁‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩}) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥})))
29	28	adantld 496	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ((𝑔 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩})) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥})))
30	29	rexlimdva 3163	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → (∃𝑔 ∈ ((LTrn‘𝐾)‘𝑊)(𝑔 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩})) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥})))
31	15, 30	mpd 16	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}))
32	1	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		eqid 2760	. . . . . . . . . . 11 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
34	6, 7, 8, 33	lhpocnel2 40996	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
35	32, 34	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
36		simplr 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑄 ∈ 𝐴)
37		simpr 490	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ¬ 𝑄(le‘𝐾)𝑊)
38		eqid 2760	. . . . . . . . . 10 ⊢ (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) = (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄)
39	6, 7, 8, 9, 38	ltrniotacl 41556	. . . . . . . . 9 ⊢ (((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))
40	32, 35, 36, 37, 39	syl112anc 1401	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄) ∈ ((LTrn‘𝐾)‘𝑊))
41	8, 9, 18	tendoidcl 41746	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))
42	32, 41	syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))
43	8, 9, 18, 11, 21	dvhelvbasei 42065	. . . . . . . 8 ⊢ (((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))⟩ ∈ 𝑉)
44	32, 40, 42, 43	syl12anc 850	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ 𝑉)
45	6, 7, 8, 33, 9, 12, 11, 13, 38	dih1dimc 42219	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄(le‘𝐾)𝑊)) → (𝐼‘𝑄) = (𝑁‘{⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}))
46	32, 36, 37, 45	syl12anc 850	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐼‘𝑄) = (𝑁‘{⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}))
47		sneq 4593	. . . . . . . . 9 ⊢ (𝑥 = ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ → {𝑥} = {⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩})
48	47	fveq2d 6877	. . . . . . . 8 ⊢ (𝑥 = ⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ → (𝑁‘{𝑥}) = (𝑁‘{⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}))
49	48	rspceeqv 3598	. . . . . . 7 ⊢ ((⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ 𝑉 ∧ (𝐼‘𝑄) = (𝑁‘{⟨(℩𝑓 ∈ ((LTrn‘𝐾)‘𝑊)(𝑓‘((oc‘𝐾)‘𝑊)) = 𝑄), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩})) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}))
50	44, 46, 49	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}))
51	31, 50	pm2.61dan 825	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}))
52	1	simpld 500	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ HL)
53	52	ad3antrrr 743	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝐾 ∈ HL)
54		hlatl 40337	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
55	53, 54	syl 18	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝐾 ∈ AtLat)
56		simpllr 788	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑄 ∈ 𝐴)
57		eqid 2760	. . . . . . . . . . 11 ⊢ (0.‘𝐾) = (0.‘𝐾)
58	57, 7	atn0 40285	. . . . . . . . . 10 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴) → 𝑄 ≠ (0.‘𝐾))
59	55, 56, 58	syl2anc 596	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑄 ≠ (0.‘𝐾))
60		sneq 4593	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0 → {𝑥} = { 0 })
61	60	fveq2d 6877	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0 → (𝑁‘{𝑥}) = (𝑁‘{ 0 }))
62	61	3ad2ant3 1153	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝑁‘{𝑥}) = (𝑁‘{ 0 }))
63		simp1ll 1255	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → 𝜑)
64	8, 11, 1	dvhlmod 42087	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ LMod)
65		dihatexv.o	. . . . . . . . . . . . . . . 16 ⊢ 0 = (0_g‘𝑈)
66	65, 13	lspsn0 21244	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ LMod → (𝑁‘{ 0 }) = { 0 })
67	63, 64, 66	3syl 19	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝑁‘{ 0 }) = { 0 })
68	62, 67	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝑁‘{𝑥}) = { 0 })
69		simp2 1155	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝐼‘𝑄) = (𝑁‘{𝑥}))
70	57, 8, 12, 11, 65	dih0 42257	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘(0.‘𝐾)) = { 0 })
71	63, 1, 70	3syl 19	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝐼‘(0.‘𝐾)) = { 0 })
72	68, 69, 71	3eqtr4d 2805	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝐼‘𝑄) = (𝐼‘(0.‘𝐾)))
73	63, 1	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
74		dihatexv.q	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑄 ∈ 𝐵)
75	63, 74	syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → 𝑄 ∈ 𝐵)
76	63, 52	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → 𝐾 ∈ HL)
77		hlop 40339	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
78	5, 57	op0cl 40161	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ 𝐵)
79	76, 77, 78	3syl 19	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (0.‘𝐾) ∈ 𝐵)
80	5, 8, 12	dih11 42242	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵 ∧ (0.‘𝐾) ∈ 𝐵) → ((𝐼‘𝑄) = (𝐼‘(0.‘𝐾)) ↔ 𝑄 = (0.‘𝐾)))
81	73, 75, 79, 80	syl3anc 1398	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → ((𝐼‘𝑄) = (𝐼‘(0.‘𝐾)) ↔ 𝑄 = (0.‘𝐾)))
82	72, 81	mpbid 235	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → 𝑄 = (0.‘𝐾))
83	82	3expia 1139	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → (𝑥 = 0 → 𝑄 = (0.‘𝐾)))
84	83	necon3d 2976	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → (𝑄 ≠ (0.‘𝐾) → 𝑥 ≠ 0 ))
85	59, 84	mpd 16	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑥 ≠ 0 )
86	85	ex 418	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) → ((𝐼‘𝑄) = (𝑁‘{𝑥}) → 𝑥 ≠ 0 ))
87	86	ancrd 561	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) → ((𝐼‘𝑄) = (𝑁‘{𝑥}) → (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))))
88	87	reximdva 3175	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → (∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}) → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))))
89	51, 88	mpd 16	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})))
90	89	ex 418	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))))
91	1	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
92	74	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑄 ∈ 𝐵)
93	5, 8, 12	dihcnvid1 42249	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (^◡𝐼‘(𝐼‘𝑄)) = 𝑄)
94	91, 92, 93	syl2anc 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (^◡𝐼‘(𝐼‘𝑄)) = 𝑄)
95		fveq2 6873	. . . . . . . 8 ⊢ ((𝐼‘𝑄) = (𝑁‘{𝑥}) → (^◡𝐼‘(𝐼‘𝑄)) = (^◡𝐼‘(𝑁‘{𝑥})))
96	95	ad2antll 742	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (^◡𝐼‘(𝐼‘𝑄)) = (^◡𝐼‘(𝑁‘{𝑥})))
97	94, 96	eqtr3d 2797	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑄 = (^◡𝐼‘(𝑁‘{𝑥})))
98	64	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑈 ∈ LMod)
99		simplr 781	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑥 ∈ 𝑉)
100		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑥 ≠ 0 )
101		eqid 2760	. . . . . . . . 9 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
102	21, 13, 65, 101	lsatlspsn2 39969	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 ) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈))
103	98, 99, 100, 102	syl3anc 1398	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈))
104	7, 8, 11, 12, 101	dihlatat 42314	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈)) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ 𝐴)
105	91, 103, 104	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ 𝐴)
106	97, 105	eqeltrd 2860	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑄 ∈ 𝐴)
107	106	ex 418	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑄 ∈ 𝐴))
108	107	rexlimdva 3163	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑄 ∈ 𝐴))
109	90, 108	impbid 215	. 2 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))))
110		rexdifsn 4756	. 2 ⊢ (∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})))
111	109, 110	bitr4di 292	1 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∃wrex 3086 ∖ cdif 3895 {csn 4583 ⟨cop 4589 class class class wbr 5102 ↦ cmpt 5185 I cid 5541 ^◡ccnv 5646 ↾ cres 5649 ‘cfv 6527 ℩crio 7364 Basecbs 17348 lecple 17396 occoc 17397 0_gc0g 17571 0.cp0 18556 LModclmod 21096 LSpanclspn 21207 LSAtomsclsa 39951 OPcops 40149 Atomscatm 40240 AtLatcal 40241 HLchlt 40327 LHypclh 40961 LTrncltrn 41078 TEndoctendo 41729 DVecHcdvh 42055 DIsoHcdih 42205

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248 ax-riotaBAD 39930

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-tpos 8221 df-undef 8268 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-er 8695 df-map 8827 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-nn 12305 df-2 12374 df-3 12375 df-4 12376 df-5 12377 df-6 12378 df-n0 12576 df-z 12663 df-uz 12935 df-fz 13609 df-struct 17286 df-sets 17303 df-slot 17321 df-ndx 17333 df-base 17349 df-ress 17370 df-plusg 17402 df-mulr 17403 df-sca 17405 df-vsca 17406 df-0g 17573 df-proset 18429 df-poset 18448 df-plt 18463 df-lub 18479 df-glb 18480 df-join 18481 df-meet 18482 df-p0 18558 df-p1 18559 df-lat 18567 df-clat 18634 df-mgm 18777 df-sgrp 18869 df-mnd 18885 df-submnd 18940 df-grp 19108 df-minusg 19109 df-sbg 19110 df-subg 19294 df-cntz 19492 df-lsm 19811 df-cmn 19957 df-abl 19958 df-mgp 20322 df-rng 20336 df-ur 20369 df-ring 20422 df-oppr 20528 df-dvdsr 20548 df-unit 20549 df-invr 20579 df-dvr 20592 df-drng 20943 df-lmod 21098 df-lss 21168 df-lsp 21208 df-lvec 21339 df-lsatoms 39953 df-oposet 40153 df-ol 40155 df-oml 40156 df-covers 40243 df-ats 40244 df-atl 40275 df-cvlat 40299 df-hlat 40328 df-llines 40475 df-lplanes 40476 df-lvols 40477 df-lines 40478 df-psubsp 40480 df-pmap 40481 df-padd 40773 df-lhyp 40965 df-laut 40966 df-ldil 41081 df-ltrn 41082 df-trl 41136 df-tendo 41732 df-edring 41734 df-disoa 42006 df-dvech 42056 df-dib 42116 df-dic 42150 df-dih 42206

This theorem is used by: dihatexv2 42316

W3C validator