dihatexv - Mathbox for Norm Megill

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Theorem dihatexv 41845

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 733	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		simplr 775	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → 𝑄 ∈ 𝐴)
4		simpr 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → 𝑄(le‘𝐾)𝑊)
5		dihatexv.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
6		eqid 2741	. . . . . . . . 9 ⊢ (le‘𝐾) = (le‘𝐾)
7		dihatexv.a	. . . . . . . . 9 ⊢ 𝐴 = (Atoms‘𝐾)
8		dihatexv.h	. . . . . . . . 9 ⊢ 𝐻 = (LHyp‘𝐾)
9		eqid 2741	. . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
10		eqid 2741	. . . . . . . . 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 41748	. . . . . . . 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 843	. . . . . . 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 737	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		simpr 486	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → 𝑔 ∈ ((LTrn‘𝐾)‘𝑊))
18		eqid 2741	. . . . . . . . . . . . . 14 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
19	5, 8, 9, 18, 10	tendo0cl 41297	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵)) ∈ ((TEndo‘𝐾)‘𝑊))
20	16, 19	syl 17	. . . . . . . . . . . 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 41595	. . . . . . . . . . . 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 843	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) ∧ 𝑔 ∈ ((LTrn‘𝐾)‘𝑊)) → ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩ ∈ 𝑉)
24		sneq 4568	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩ → {𝑥} = {⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩})
25	24	fveq2d 6835	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩ → (𝑁‘{𝑥}) = (𝑁‘{⟨𝑔, (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ 𝐵))⟩}))
26	25	rspceeqv 3585	. . . . . . . . . . 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 587	. . . . . . . . . 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 414	. . . . . . . . 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 492	. . . . . . . 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 3142	. . . . . . 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 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑄(le‘𝐾)𝑊) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}))
32	1	ad2antrr 733	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
33		eqid 2741	. . . . . . . . . . 11 ⊢ ((oc‘𝐾)‘𝑊) = ((oc‘𝐾)‘𝑊)
34	6, 7, 8, 33	lhpocnel2 40526	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
35	32, 34	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → (((oc‘𝐾)‘𝑊) ∈ 𝐴 ∧ ¬ ((oc‘𝐾)‘𝑊)(le‘𝐾)𝑊))
36		simplr 775	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → 𝑄 ∈ 𝐴)
37		simpr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ¬ 𝑄(le‘𝐾)𝑊)
38		eqid 2741	. . . . . . . . . 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 41086	. . . . . . . . 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 1383	. . . . . . . 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 41276	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))
42	32, 41	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ( I ↾ ((LTrn‘𝐾)‘𝑊)) ∈ ((TEndo‘𝐾)‘𝑊))
43	8, 9, 18, 11, 21	dvhelvbasei 41595	. . . . . . . 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 843	. . . . . . 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 41749	. . . . . . . 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 843	. . . . . . 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 4568	. . . . . . . . 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 6835	. . . . . . . 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 3585	. . . . . . 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 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄(le‘𝐾)𝑊) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}))
51	31, 50	pm2.61dan 819	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}))
52	1	simpld 496	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ HL)
53	52	ad3antrrr 737	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝐾 ∈ HL)
54		hlatl 39867	. . . . . . . . . . 11 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
55	53, 54	syl 17	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝐾 ∈ AtLat)
56		simpllr 782	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑄 ∈ 𝐴)
57		eqid 2741	. . . . . . . . . . 11 ⊢ (0.‘𝐾) = (0.‘𝐾)
58	57, 7	atn0 39815	. . . . . . . . . 10 ⊢ ((𝐾 ∈ AtLat ∧ 𝑄 ∈ 𝐴) → 𝑄 ≠ (0.‘𝐾))
59	55, 56, 58	syl2anc 591	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑄 ≠ (0.‘𝐾))
60		sneq 4568	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 0 → {𝑥} = { 0 })
61	60	fveq2d 6835	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 0 → (𝑁‘{𝑥}) = (𝑁‘{ 0 }))
62	61	3ad2ant3 1142	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝑁‘{𝑥}) = (𝑁‘{ 0 }))
63		simp1ll 1244	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → 𝜑)
64	8, 11, 1	dvhlmod 41617	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑈 ∈ LMod)
65		dihatexv.o	. . . . . . . . . . . . . . . 16 ⊢ 0 = (0_g‘𝑈)
66	65, 13	lspsn0 21002	. . . . . . . . . . . . . . 15 ⊢ (𝑈 ∈ LMod → (𝑁‘{ 0 }) = { 0 })
67	63, 64, 66	3syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝑁‘{ 0 }) = { 0 })
68	62, 67	eqtrd 2776	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝑁‘{𝑥}) = { 0 })
69		simp2 1144	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝐼‘𝑄) = (𝑁‘{𝑥}))
70	57, 8, 12, 11, 65	dih0 41787	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘(0.‘𝐾)) = { 0 })
71	63, 1, 70	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝐼‘(0.‘𝐾)) = { 0 })
72	68, 69, 71	3eqtr4d 2786	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (𝐼‘𝑄) = (𝐼‘(0.‘𝐾)))
73	63, 1	syl 17	. . . . . . . . . . . . 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 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → 𝑄 ∈ 𝐵)
76	63, 52	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → 𝐾 ∈ HL)
77		hlop 39869	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP)
78	5, 57	op0cl 39691	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ OP → (0.‘𝐾) ∈ 𝐵)
79	76, 77, 78	3syl 18	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → (0.‘𝐾) ∈ 𝐵)
80	5, 8, 12	dih11 41772	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵 ∧ (0.‘𝐾) ∈ 𝐵) → ((𝐼‘𝑄) = (𝐼‘(0.‘𝐾)) ↔ 𝑄 = (0.‘𝐾)))
81	73, 75, 79, 80	syl3anc 1380	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → ((𝐼‘𝑄) = (𝐼‘(0.‘𝐾)) ↔ 𝑄 = (0.‘𝐾)))
82	72, 81	mpbid 234	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}) ∧ 𝑥 = 0 ) → 𝑄 = (0.‘𝐾))
83	82	3expia 1128	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → (𝑥 = 0 → 𝑄 = (0.‘𝐾)))
84	83	necon3d 2957	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → (𝑄 ≠ (0.‘𝐾) → 𝑥 ≠ 0 ))
85	59, 84	mpd 15	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑥 ≠ 0 )
86	85	ex 414	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) → ((𝐼‘𝑄) = (𝑁‘{𝑥}) → 𝑥 ≠ 0 ))
87	86	ancrd 557	. . . . . 6 ⊢ (((𝜑 ∧ 𝑄 ∈ 𝐴) ∧ 𝑥 ∈ 𝑉) → ((𝐼‘𝑄) = (𝑁‘{𝑥}) → (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))))
88	87	reximdva 3154	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → (∃𝑥 ∈ 𝑉 (𝐼‘𝑄) = (𝑁‘{𝑥}) → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))))
89	51, 88	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})))
90	89	ex 414	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))))
91	1	ad2antrr 733	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
92	74	ad2antrr 733	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑄 ∈ 𝐵)
93	5, 8, 12	dihcnvid1 41779	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐵) → (^◡𝐼‘(𝐼‘𝑄)) = 𝑄)
94	91, 92, 93	syl2anc 591	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (^◡𝐼‘(𝐼‘𝑄)) = 𝑄)
95		fveq2 6831	. . . . . . . 8 ⊢ ((𝐼‘𝑄) = (𝑁‘{𝑥}) → (^◡𝐼‘(𝐼‘𝑄)) = (^◡𝐼‘(𝑁‘{𝑥})))
96	95	ad2antll 736	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (^◡𝐼‘(𝐼‘𝑄)) = (^◡𝐼‘(𝑁‘{𝑥})))
97	94, 96	eqtr3d 2778	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑄 = (^◡𝐼‘(𝑁‘{𝑥})))
98	64	ad2antrr 733	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑈 ∈ LMod)
99		simplr 775	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑥 ∈ 𝑉)
100		simprl 777	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑥 ≠ 0 )
101		eqid 2741	. . . . . . . . 9 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
102	21, 13, 65, 101	lsatlspsn2 39499	. . . . . . . 8 ⊢ ((𝑈 ∈ LMod ∧ 𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 ) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈))
103	98, 99, 100, 102	syl3anc 1380	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈))
104	7, 8, 11, 12, 101	dihlatat 41844	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈)) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ 𝐴)
105	91, 103, 104	syl2anc 591	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → (^◡𝐼‘(𝑁‘{𝑥})) ∈ 𝐴)
106	97, 105	eqeltrd 2841	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))) → 𝑄 ∈ 𝐴)
107	106	ex 414	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑄 ∈ 𝐴))
108	107	rexlimdva 3142	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})) → 𝑄 ∈ 𝐴))
109	90, 108	impbid 214	. 2 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥}))))
110		rexdifsn 4730	. 2 ⊢ (∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ (𝐼‘𝑄) = (𝑁‘{𝑥})))
111	109, 110	bitr4di 291	1 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥})))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 ∃wrex 3065 ∖ cdif 3882 {csn 4558 ⟨cop 4564 class class class wbr 5075 ↦ cmpt 5156 I cid 5515 ^◡ccnv 5620 ↾ cres 5623 ‘cfv 6489 ℩crio 7316 Basecbs 17174 lecple 17222 occoc 17223 0_gc0g 17397 0.cp0 18382 LModclmod 20854 LSpanclspn 20965 LSAtomsclsa 39481 OPcops 39679 Atomscatm 39770 AtLatcal 39771 HLchlt 39857 LHypclh 40491 LTrncltrn 40608 TEndoctendo 41259 DVecHcdvh 41585 DIsoHcdih 41735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-riotaBAD 39460

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-iin 4927 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-n0 12433 df-z 12520 df-uz 12784 df-fz 13457 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-sca 17231 df-vsca 17232 df-0g 17399 df-proset 18255 df-poset 18274 df-plt 18289 df-lub 18305 df-glb 18306 df-join 18307 df-meet 18308 df-p0 18384 df-p1 18385 df-lat 18393 df-clat 18460 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18747 df-grp 18907 df-minusg 18908 df-sbg 18909 df-subg 19094 df-cntz 19287 df-lsm 19606 df-cmn 19752 df-abl 19753 df-mgp 20117 df-rng 20129 df-ur 20158 df-ring 20211 df-oppr 20312 df-dvdsr 20332 df-unit 20333 df-invr 20363 df-dvr 20376 df-drng 20707 df-lmod 20856 df-lss 20926 df-lsp 20966 df-lvec 21097 df-lsatoms 39483 df-oposet 39683 df-ol 39685 df-oml 39686 df-covers 39773 df-ats 39774 df-atl 39805 df-cvlat 39829 df-hlat 39858 df-llines 40005 df-lplanes 40006 df-lvols 40007 df-lines 40008 df-psubsp 40010 df-pmap 40011 df-padd 40303 df-lhyp 40495 df-laut 40496 df-ldil 40611 df-ltrn 40612 df-trl 40666 df-tendo 41262 df-edring 41264 df-disoa 41536 df-dvech 41586 df-dib 41646 df-dic 41680 df-dih 41736

This theorem is referenced by: dihatexv2 41846

W3C validator