dochsatshpb - Mathbox for Norm Megill

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Theorem dochsatshpb 40311

Description: The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.)

**Hypotheses**
Ref	Expression
dochsatshpb.h	⊢ 𝐻 = (LHyp‘𝐾)
dochsatshpb.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochsatshpb.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochsatshpb.s	⊢ 𝑆 = (LSubSp‘𝑈)
dochsatshpb.a	⊢ 𝐴 = (LSAtoms‘𝑈)
dochsatshpb.y	⊢ 𝑌 = (LSHyp‘𝑈)
dochsatshpb.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochsatshpb.q	⊢ (𝜑 → 𝑄 ∈ 𝑆)

**Assertion**
Ref	Expression
dochsatshpb	⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ( ⊥ ‘𝑄) ∈ 𝑌))

Proof of Theorem dochsatshpb Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dochsatshpb.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		dochsatshpb.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		dochsatshpb.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
4		dochsatshpb.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑈)
5		dochsatshpb.y	. . 3 ⊢ 𝑌 = (LSHyp‘𝑈)
6		dochsatshpb.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8		simpr 485	. . 3 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → 𝑄 ∈ 𝐴)
9	1, 2, 3, 4, 5, 7, 8	dochsatshp 40310	. 2 ⊢ ((𝜑 ∧ 𝑄 ∈ 𝐴) → ( ⊥ ‘𝑄) ∈ 𝑌)
10		dochsatshpb.q	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ 𝑆)
11		eqid 2732	. . . . . . . . . . . 12 ⊢ (Base‘𝑈) = (Base‘𝑈)
12		dochsatshpb.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (LSubSp‘𝑈)
13	11, 12	lssss 20539	. . . . . . . . . . 11 ⊢ (𝑄 ∈ 𝑆 → 𝑄 ⊆ (Base‘𝑈))
14	10, 13	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ⊆ (Base‘𝑈))
15		eqid 2732	. . . . . . . . . . 11 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
16	1, 15, 2, 11, 3	dochcl 40212	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
17	6, 14, 16	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
18	1, 15, 3	dochoc 40226	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑄))) = ( ⊥ ‘𝑄))
19	6, 17, 18	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑄))) = ( ⊥ ‘𝑄))
20	19	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑄))) = ( ⊥ ‘𝑄))
21	1, 2, 6	dvhlmod 39969	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
22	21	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → 𝑈 ∈ LMod)
23		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ( ⊥ ‘𝑄) ∈ 𝑌)
24	11, 5, 22, 23	lshpne 37840	. . . . . . 7 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ( ⊥ ‘𝑄) ≠ (Base‘𝑈))
25	20, 24	eqnetrd 3008	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑄))) ≠ (Base‘𝑈))
26		eqid 2732	. . . . . . . 8 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
27	1, 2, 11, 3	dochssv 40214	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
28	6, 14, 27	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
29	1, 3, 2, 11, 26, 6, 28	dochn0nv 40234	. . . . . . 7 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑄)) ≠ {(0_g‘𝑈)} ↔ ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑄))) ≠ (Base‘𝑈)))
30	29	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → (( ⊥ ‘( ⊥ ‘𝑄)) ≠ {(0_g‘𝑈)} ↔ ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑄))) ≠ (Base‘𝑈)))
31	25, 30	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘𝑄)) ≠ {(0_g‘𝑈)})
32	1, 2, 11, 12, 3	dochlss 40213	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ⊆ (Base‘𝑈)) → ( ⊥ ‘𝑄) ∈ 𝑆)
33	6, 14, 32	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝑆)
34	11, 12	lssss 20539	. . . . . . . . 9 ⊢ (( ⊥ ‘𝑄) ∈ 𝑆 → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
35	33, 34	syl 17	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘𝑄) ⊆ (Base‘𝑈))
36	1, 2, 11, 12, 3	dochlss 40213	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑄) ⊆ (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝑆)
37	6, 35, 36	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝑆)
38	37	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝑆)
39	26, 12	lssne0 20553	. . . . . 6 ⊢ (( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝑆 → (( ⊥ ‘( ⊥ ‘𝑄)) ≠ {(0_g‘𝑈)} ↔ ∃𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄))𝑣 ≠ (0_g‘𝑈)))
40	38, 39	syl 17	. . . . 5 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → (( ⊥ ‘( ⊥ ‘𝑄)) ≠ {(0_g‘𝑈)} ↔ ∃𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄))𝑣 ≠ (0_g‘𝑈)))
41	31, 40	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ∃𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄))𝑣 ≠ (0_g‘𝑈))
42	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
43	42	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
44	17	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
45	44	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘𝑄) ∈ ran ((DIsoH‘𝐾)‘𝑊))
46	43, 45, 18	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑄))) = ( ⊥ ‘𝑄))
47		eqid 2732	. . . . . . . . . . . 12 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
48	22	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → 𝑈 ∈ LMod)
49	38	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝑆)
50		simp2 1137	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)))
51	12, 47, 48, 49, 50	lspsnel5a 20599	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ((LSpan‘𝑈)‘{𝑣}) ⊆ ( ⊥ ‘( ⊥ ‘𝑄)))
52	11, 12	lssel 20540	. . . . . . . . . . . . . 14 ⊢ ((( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝑆 ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄))) → 𝑣 ∈ (Base‘𝑈))
53	49, 50, 52	syl2anc 584	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → 𝑣 ∈ (Base‘𝑈))
54	1, 2, 11, 47, 15	dihlsprn 40190	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑣 ∈ (Base‘𝑈)) → ((LSpan‘𝑈)‘{𝑣}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
55	43, 53, 54	syl2anc 584	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ((LSpan‘𝑈)‘{𝑣}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
56	1, 15, 2, 11, 3	dochcl 40212	. . . . . . . . . . . . . . 15 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑄) ⊆ (Base‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
57	6, 35, 56	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
58	57	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
59	58	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ ran ((DIsoH‘𝐾)‘𝑊))
60	1, 15, 3, 43, 55, 59	dochord 40229	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → (((LSpan‘𝑈)‘{𝑣}) ⊆ ( ⊥ ‘( ⊥ ‘𝑄)) ↔ ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑄))) ⊆ ( ⊥ ‘((LSpan‘𝑈)‘{𝑣}))))
61	51, 60	mpbid 231	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑄))) ⊆ ( ⊥ ‘((LSpan‘𝑈)‘{𝑣})))
62	46, 61	eqsstrrd 4020	. . . . . . . . 9 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘𝑄) ⊆ ( ⊥ ‘((LSpan‘𝑈)‘{𝑣})))
63	1, 2, 6	dvhlvec 39968	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ LVec)
64	63	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → 𝑈 ∈ LVec)
65	64	3ad2ant1 1133	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → 𝑈 ∈ LVec)
66		simp1r 1198	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘𝑄) ∈ 𝑌)
67		simp3 1138	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → 𝑣 ≠ (0_g‘𝑈))
68	11, 47, 26, 4	lsatlspsn2 37850	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ 𝑣 ∈ (Base‘𝑈) ∧ 𝑣 ≠ (0_g‘𝑈)) → ((LSpan‘𝑈)‘{𝑣}) ∈ 𝐴)
69	48, 53, 67, 68	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ((LSpan‘𝑈)‘{𝑣}) ∈ 𝐴)
70	1, 2, 3, 4, 5, 43, 69	dochsatshp 40310	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘((LSpan‘𝑈)‘{𝑣})) ∈ 𝑌)
71	5, 65, 66, 70	lshpcmp 37846	. . . . . . . . 9 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → (( ⊥ ‘𝑄) ⊆ ( ⊥ ‘((LSpan‘𝑈)‘{𝑣})) ↔ ( ⊥ ‘𝑄) = ( ⊥ ‘((LSpan‘𝑈)‘{𝑣}))))
72	62, 71	mpbid 231	. . . . . . . 8 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘𝑄) = ( ⊥ ‘((LSpan‘𝑈)‘{𝑣})))
73	72	fveq2d 6892	. . . . . . 7 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑄)) = ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑣}))))
74	1, 15, 3	dochoc 40226	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((LSpan‘𝑈)‘{𝑣}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘{𝑣}))
75	43, 55, 74	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘((LSpan‘𝑈)‘{𝑣}))) = ((LSpan‘𝑈)‘{𝑣}))
76	73, 75	eqtrd 2772	. . . . . 6 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑄)) = ((LSpan‘𝑈)‘{𝑣}))
77	76, 69	eqeltrd 2833	. . . . 5 ⊢ (((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) ∧ 𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄)) ∧ 𝑣 ≠ (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝐴)
78	77	rexlimdv3a 3159	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → (∃𝑣 ∈ ( ⊥ ‘( ⊥ ‘𝑄))𝑣 ≠ (0_g‘𝑈) → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝐴))
79	41, 78	mpd 15	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → ( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝐴)
80	10	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → 𝑄 ∈ 𝑆)
81	1, 3, 2, 12, 4, 42, 80	dochsat 40242	. . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → (( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝐴 ↔ 𝑄 ∈ 𝐴))
82	79, 81	mpbid 231	. 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑄) ∈ 𝑌) → 𝑄 ∈ 𝐴)
83	9, 82	impbida 799	1 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ( ⊥ ‘𝑄) ∈ 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∃wrex 3070 ⊆ wss 3947 {csn 4627 ran crn 5676 ‘cfv 6540 Basecbs 17140 0_gc0g 17381 LModclmod 20463 LSubSpclss 20534 LSpanclspn 20574 LVecclvec 20705 LSAtomsclsa 37832 LSHypclsh 37833 HLchlt 38208 LHypclh 38843 DVecHcdvh 39937 DIsoHcdih 40087 ocHcoch 40206

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37811

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-lsm 19498 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lvec 20706 df-lsatoms 37834 df-lshyp 37835 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 df-tgrp 39602 df-tendo 39614 df-edring 39616 df-dveca 39862 df-disoa 39888 df-dvech 39938 df-dib 39998 df-dic 40032 df-dih 40088 df-doch 40207 df-djh 40254

This theorem is referenced by: dochshpsat 40313 dochkrsat 40314 lcfl4N 40354

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