Theorem mapd0 37476

Description: Projectivity map of the zero subspace. Part of property (f) in [Baer] p. 40. TODO: does proof need to be this long for this simple fact? (Contributed by NM, 15-Mar-2015.)

Hypotheses

Ref	Expression
mapd0.h	⊢ 𝐻 = (LHyp‘𝐾)
mapd0.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapd0.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapd0.o	⊢ 𝑂 = (0g‘𝑈)
mapd0.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapd0.z	⊢ 0 = (0g‘𝐶)
mapd0.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

Assertion

Ref	Expression
mapd0	⊢ (𝜑 → (𝑀‘{𝑂}) = { 0 })

Proof of Theorem mapd0

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapd0.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapd0.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		eqid 2771	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
4		eqid 2771	. . 3 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
5		eqid 2771	. . 3 ⊢ (LKer‘𝑈) = (LKer‘𝑈)
6		eqid 2771	. . 3 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
7		mapd0.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
8		mapd0.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	1, 2, 8	dvhlmod 36921	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		mapd0.o	. . . . 5 ⊢ 𝑂 = (0g‘𝑈)
11	10, 3	lsssn0 19159	. . . 4 ⊢ (𝑈 ∈ LMod → {𝑂} ∈ (LSubSp‘𝑈))
12	9, 11	syl 17	. . 3 ⊢ (𝜑 → {𝑂} ∈ (LSubSp‘𝑈))
13	1, 2, 3, 4, 5, 6, 7, 8, 12	mapdval 37439	. 2 ⊢ (𝜑 → (𝑀‘{𝑂}) = {𝑓 ∈ (LFnl‘𝑈) ∣ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂})})
14		simprrr 761	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})
15	9	adantr 466	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 𝑈 ∈ LMod)
16	8	adantr 466	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		eqid 2771	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑈) = (Base‘𝑈)
18		simprl 748	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 𝑔 ∈ (LFnl‘𝑈))
19	17, 4, 5, 15, 18	lkrssv 34906	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → ((LKer‘𝑈)‘𝑔) ⊆ (Base‘𝑈))
20	1, 2, 17, 3, 6	dochlss 37165	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((LKer‘𝑈)‘𝑔) ⊆ (Base‘𝑈)) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈))
21	16, 19, 20	syl2anc 567	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈))
22	10, 3	lssle0 19161	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ LMod ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈)) → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = {𝑂}))
23	15, 21, 22	syl2anc 567	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = {𝑂}))
24	14, 23	mpbid 222	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = {𝑂})
25	24	fveq2d 6337	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = (((ocH‘𝐾)‘𝑊)‘{𝑂}))
26		simprrl 760	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔))
27	1, 2, 6, 17, 10	doch0 37169	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈))
28	8, 27	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈))
29	28	adantr 466	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈))
30	25, 26, 29	3eqtr3d 2813	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → ((LKer‘𝑈)‘𝑔) = (Base‘𝑈))
31		eqid 2771	. . . . . . . . . 10 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
32		eqid 2771	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈))
33	31, 32, 17, 4, 5	lkr0f 34904	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑔 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘𝑔) = (Base‘𝑈) ↔ 𝑔 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
34	15, 18, 33	syl2anc 567	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → (((LKer‘𝑈)‘𝑔) = (Base‘𝑈) ↔ 𝑔 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
35	30, 34	mpbid 222	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 𝑔 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))
36		mapd0.c	. . . . . . . . 9 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
37		mapd0.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐶)
38	1, 2, 17, 31, 32, 36, 37, 8	lcd0v 37422	. . . . . . . 8 ⊢ (𝜑 → 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))
39	38	adantr 466	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))
40	35, 39	eqtr4d 2808	. . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))) → 𝑔 = 0 )
41	40	ex 397	. . . . 5 ⊢ (𝜑 → ((𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})) → 𝑔 = 0 ))
42		eqid 2771	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
43	1, 36, 42, 37, 8	lcd0vcl 37425	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (Base‘𝐶))
44	1, 36, 42, 2, 4, 8, 43	lcdvbaselfl 37406	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (LFnl‘𝑈))
45	31, 32, 17, 4, 5	lkr0f 34904	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ LMod ∧ 0 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
46	9, 44, 45	syl2anc 567	. . . . . . . . . . . 12 ⊢ (𝜑 → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
47	38, 46	mpbird 247	. . . . . . . . . . 11 ⊢ (𝜑 → ((LKer‘𝑈)‘ 0 ) = (Base‘𝑈))
48	47	fveq2d 6337	. . . . . . . . . 10 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)))
49	48	fveq2d 6337	. . . . . . . . 9 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))))
50	1, 2, 6, 17, 8	dochoc1 37172	. . . . . . . . 9 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))) = (Base‘𝑈))
51	49, 50	eqtrd 2805	. . . . . . . 8 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (Base‘𝑈))
52	51, 47	eqtr4d 2808	. . . . . . 7 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ))
53	1, 2, 6, 17, 10	doch1 37170	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂})
54	8, 53	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂})
55	48, 54	eqtrd 2805	. . . . . . . 8 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂})
56		eqimss 3807	. . . . . . . 8 ⊢ ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂} → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})
57	55, 56	syl 17	. . . . . . 7 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})
58	44, 52, 57	jca32 501	. . . . . 6 ⊢ (𝜑 → ( 0 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})))
59		eleq1 2838	. . . . . . 7 ⊢ (𝑔 = 0 → (𝑔 ∈ (LFnl‘𝑈) ↔ 0 ∈ (LFnl‘𝑈)))
60		fveq2 6333	. . . . . . . . . . 11 ⊢ (𝑔 = 0 → ((LKer‘𝑈)‘𝑔) = ((LKer‘𝑈)‘ 0 ))
61	60	fveq2d 6337	. . . . . . . . . 10 ⊢ (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )))
62	61	fveq2d 6337	. . . . . . . . 9 ⊢ (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))))
63	62, 60	eqeq12d 2786	. . . . . . . 8 ⊢ (𝑔 = 0 → ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ↔ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 )))
64	61	sseq1d 3782	. . . . . . . 8 ⊢ (𝑔 = 0 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}))
65	63, 64	anbi12d 610	. . . . . . 7 ⊢ (𝑔 = 0 → (((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}) ↔ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})))
66	59, 65	anbi12d 610	. . . . . 6 ⊢ (𝑔 = 0 → ((𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})) ↔ ( 0 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}))))
67	58, 66	syl5ibrcom 237	. . . . 5 ⊢ (𝜑 → (𝑔 = 0 → (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))))
68	41, 67	impbid 202	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})) ↔ 𝑔 = 0 ))
69		fveq2 6333	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((LKer‘𝑈)‘𝑓) = ((LKer‘𝑈)‘𝑔))
70	69	fveq2d 6337	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)))
71	70	fveq2d 6337	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))))
72	71, 69	eqeq12d 2786	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ↔ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔)))
73	70	sseq1d 3782	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))
74	72, 73	anbi12d 610	. . . . 5 ⊢ (𝑓 = 𝑔 → (((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂}) ↔ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})))
75	74	elrab 3516	. . . 4 ⊢ (𝑔 ∈ {𝑓 ∈ (LFnl‘𝑈) ∣ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂})} ↔ (𝑔 ∈ (LFnl‘𝑈) ∧ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂})))
76		velsn 4333	. . . 4 ⊢ (𝑔 ∈ { 0 } ↔ 𝑔 = 0 )
77	68, 75, 76	3bitr4g 303	. . 3 ⊢ (𝜑 → (𝑔 ∈ {𝑓 ∈ (LFnl‘𝑈) ∣ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂})} ↔ 𝑔 ∈ { 0 }))
78	77	eqrdv 2769	. 2 ⊢ (𝜑 → {𝑓 ∈ (LFnl‘𝑈) ∣ ((((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = ((LKer‘𝑈)‘𝑓) ∧ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂})} = { 0 })
79	13, 78	eqtrd 2805	1 ⊢ (𝜑 → (𝑀‘{𝑂}) = { 0 })

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {crab 3065   ⊆ wss 3724  {csn 4317   × cxp 5248  ‘cfv 6032  Basecbs 16065  Scalarcsca 16153  0_gc0g 16309  LModclmod 19074  LSubSpclss 19143  LFnlclfn 34867  LKerclk 34895  HLchlt 35160  LHypclh 35793  DVecHcdvh 36889  ocHcoch 37158  LCDualclcd 37397  mapdcmpd 37435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097  ax-cnex 10195  ax-resscn 10196  ax-1cn 10197  ax-icn 10198  ax-addcl 10199  ax-addrcl 10200  ax-mulcl 10201  ax-mulrcl 10202  ax-mulcom 10203  ax-addass 10204  ax-mulass 10205  ax-distr 10206  ax-i2m1 10207  ax-1ne0 10208  ax-1rid 10209  ax-rnegex 10210  ax-rrecex 10211  ax-cnre 10212  ax-pre-lttri 10213  ax-pre-lttrn 10214  ax-pre-ltadd 10215  ax-pre-mulgt0 10216  ax-riotaBAD 34762

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-iin 4658  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5824  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040  df-riota 6755  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-of 7045  df-om 7214  df-1st 7316  df-2nd 7317  df-tpos 7505  df-undef 7552  df-wrecs 7560  df-recs 7622  df-rdg 7660  df-1o 7714  df-oadd 7718  df-er 7897  df-map 8012  df-en 8111  df-dom 8112  df-sdom 8113  df-fin 8114  df-pnf 10279  df-mnf 10280  df-xr 10281  df-ltxr 10282  df-le 10283  df-sub 10471  df-neg 10472  df-nn 11224  df-2 11282  df-3 11283  df-4 11284  df-5 11285  df-6 11286  df-n0 11496  df-z 11581  df-uz 11890  df-fz 12535  df-struct 16067  df-ndx 16068  df-slot 16069  df-base 16071  df-sets 16072  df-ress 16073  df-plusg 16163  df-mulr 16164  df-sca 16166  df-vsca 16167  df-0g 16311  df-mre 16455  df-mrc 16456  df-acs 16458  df-preset 17137  df-poset 17155  df-plt 17167  df-lub 17183  df-glb 17184  df-join 17185  df-meet 17186  df-p0 17248  df-p1 17249  df-lat 17255  df-clat 17317  df-mgm 17451  df-sgrp 17493  df-mnd 17504  df-submnd 17545  df-grp 17634  df-minusg 17635  df-sbg 17636  df-subg 17800  df-cntz 17958  df-oppg 17984  df-lsm 18259  df-cmn 18403  df-abl 18404  df-mgp 18699  df-ur 18711  df-ring 18758  df-oppr 18832  df-dvdsr 18850  df-unit 18851  df-invr 18881  df-dvr 18892  df-drng 18960  df-lmod 19076  df-lss 19144  df-lsp 19186  df-lvec 19317  df-lsatoms 34786  df-lshyp 34787  df-lcv 34829  df-lfl 34868  df-lkr 34896  df-ldual 34934  df-oposet 34986  df-ol 34988  df-oml 34989  df-covers 35076  df-ats 35077  df-atl 35108  df-cvlat 35132  df-hlat 35161  df-llines 35307  df-lplanes 35308  df-lvols 35309  df-lines 35310  df-psubsp 35312  df-pmap 35313  df-padd 35605  df-lhyp 35797  df-laut 35798  df-ldil 35913  df-ltrn 35914  df-trl 35969  df-tgrp 36553  df-tendo 36565  df-edring 36567  df-dveca 36813  df-disoa 36840  df-dvech 36890  df-dib 36950  df-dic 36984  df-dih 37040  df-doch 37159  df-djh 37206  df-lcdual 37398  df-mapd 37436

This theorem is referenced by:  mapdcnvatN  37477  mapdat  37478  mapdspex  37479  mapdn0  37480  hdmap10  37651  hdmapeq0  37655