Theorem mapd0 42546

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 2762	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
4		eqid 2762	. . 3 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
5		eqid 2762	. . 3 ⊢ (LKer‘𝑈) = (LKer‘𝑈)
6		eqid 2762	. . 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 41991	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		mapd0.o	. . . . 5 ⊢ 𝑂 = (0g‘𝑈)
11	10, 3	lsssn0 21138	. . . 4 ⊢ (𝑈 ∈ LMod → {𝑂} ∈ (LSubSp‘𝑈))
12	9, 11	syl 18	. . 3 ⊢ (𝜑 → {𝑂} ∈ (LSubSp‘𝑈))
13	1, 2, 3, 4, 5, 6, 7, 8, 12	mapdval 42509	. 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 794	. . . . . . . . . . 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 486	. . . . . . . . . . . 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 486	. . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑈) = (Base‘𝑈)
18		simprl 783	. . . . . . . . . . . . . 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 39977	. . . . . . . . . . . . 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 42235	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((LKer‘𝑈)‘𝑔) ⊆ (Base‘𝑈)) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈))
21	16, 19, 20	syl2anc 596	. . . . . . . . . . . 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 21140	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 235	. . . . . . . . . 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 6886	. . . . . . . . 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 793	. . . . . . . . 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 42239	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈))
28	8, 27	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈))
29	28	adantr 486	. . . . . . . . 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 2805	. . . . . . . 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 2762	. . . . . . . . . 10 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
32		eqid 2762	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈))
33	31, 32, 17, 4, 5	lkr0f 39975	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑔 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘𝑔) = (Base‘𝑈) ↔ 𝑔 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
34	15, 18, 33	syl2anc 596	. . . . . . . 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 235	. . . . . . 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 42492	. . . . . . . 8 ⊢ (𝜑 → 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))
39	38	adantr 486	. . . . . . 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 2800	. . . . . 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 418	. . . . 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 2762	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
43	1, 36, 42, 37, 8	lcd0vcl 42495	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (Base‘𝐶))
44	1, 36, 42, 2, 4, 8, 43	lcdvbaselfl 42476	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (LFnl‘𝑈))
45	31, 32, 17, 4, 5	lkr0f 39975	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ LMod ∧ 0 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
46	9, 44, 45	syl2anc 596	. . . . . . . . . . . 12 ⊢ (𝜑 → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
47	38, 46	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → ((LKer‘𝑈)‘ 0 ) = (Base‘𝑈))
48	47	fveq2d 6886	. . . . . . . . . 10 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)))
49	48	fveq2d 6886	. . . . . . . . 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 42242	. . . . . . . . 9 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))) = (Base‘𝑈))
51	49, 50	eqtrd 2797	. . . . . . . 8 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (Base‘𝑈))
52	51, 47	eqtr4d 2800	. . . . . . 7 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ))
53	1, 2, 6, 17, 10	doch1 42240	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂})
54	8, 53	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂})
55	48, 54	eqtrd 2797	. . . . . . . 8 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂})
56		eqimss 3992	. . . . . . . 8 ⊢ ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂} → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})
57	55, 56	syl 18	. . . . . . 7 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})
58	44, 52, 57	jca32 525	. . . . . 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 2850	. . . . . . 7 ⊢ (𝑔 = 0 → (𝑔 ∈ (LFnl‘𝑈) ↔ 0 ∈ (LFnl‘𝑈)))
60		2fveq3 6887	. . . . . . . . . 10 ⊢ (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )))
61	60	fveq2d 6886	. . . . . . . . 9 ⊢ (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))))
62		fveq2 6882	. . . . . . . . 9 ⊢ (𝑔 = 0 → ((LKer‘𝑈)‘𝑔) = ((LKer‘𝑈)‘ 0 ))
63	61, 62	eqeq12d 2778	. . . . . . . 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	60	sseq1d 3965	. . . . . . . 8 ⊢ (𝑔 = 0 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}))
65	63, 64	anbi12d 644	. . . . . . 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 644	. . . . . 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 250	. . . . 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 215	. . . 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		2fveq3 6887	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)))
70	69	fveq2d 6886	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))))
71		fveq2 6882	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((LKer‘𝑈)‘𝑓) = ((LKer‘𝑈)‘𝑔))
72	70, 71	eqeq12d 2778	. . . . . 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	69	sseq1d 3965	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))
74	72, 73	anbi12d 644	. . . . 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 3648	. . . 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 4603	. . . 4 ⊢ (𝑔 ∈ { 0 } ↔ 𝑔 = 0 )
77	68, 75, 76	3bitr4g 317	. . 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 2760	. 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 2797	1 ⊢ (𝜑 → (𝑀‘{𝑂}) = { 0 })

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3414   ⊆ wss 3902  {csn 4587   × cxp 5657  ‘cfv 6537  Basecbs 17307  Scalarcsca 17351  0_gc0g 17530  LModclmod 21050  LSubSpclss 21121  LFnlclfn 39938  LKerclk 39966  HLchlt 40231  LHypclh 40865  DVecHcdvh 41959  ocHcoch 42228  LCDualclcd 42467  mapdcmpd 42505

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-riotaBAD 39834

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-of 7682  df-om 7867  df-1st 7990  df-2nd 7991  df-tpos 8228  df-undef 8275  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-struct 17245  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-plusg 17361  df-mulr 17362  df-sca 17364  df-vsca 17365  df-0g 17532  df-mre 17676  df-mrc 17677  df-acs 17679  df-proset 18388  df-poset 18407  df-plt 18422  df-lub 18438  df-glb 18439  df-join 18440  df-meet 18441  df-p0 18517  df-p1 18518  df-lat 18526  df-clat 18593  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-submnd 18898  df-grp 19066  df-minusg 19067  df-sbg 19068  df-subg 19252  df-cntz 19450  df-oppg 19479  df-lsm 19769  df-cmn 19915  df-abl 19916  df-mgp 20280  df-rng 20294  df-ur 20327  df-ring 20380  df-oppr 20484  df-dvdsr 20504  df-unit 20505  df-invr 20535  df-dvr 20548  df-nzr 20679  df-rlreg 20862  df-domn 20863  df-drng 20898  df-lmod 21052  df-lss 21122  df-lsp 21162  df-lvec 21293  df-lsatoms 39857  df-lshyp 39858  df-lcv 39900  df-lfl 39939  df-lkr 39967  df-ldual 40005  df-oposet 40057  df-ol 40059  df-oml 40060  df-covers 40147  df-ats 40148  df-atl 40179  df-cvlat 40203  df-hlat 40232  df-llines 40379  df-lplanes 40380  df-lvols 40381  df-lines 40382  df-psubsp 40384  df-pmap 40385  df-padd 40677  df-lhyp 40869  df-laut 40870  df-ldil 40985  df-ltrn 40986  df-trl 41040  df-tgrp 41624  df-tendo 41636  df-edring 41638  df-dveca 41884  df-disoa 41910  df-dvech 41960  df-dib 42020  df-dic 42054  df-dih 42110  df-doch 42229  df-djh 42276  df-lcdual 42468  df-mapd 42506

This theorem is used by:  mapdcnvatN  42547  mapdat  42548  mapdspex  42549  mapdn0  42550  hdmap10  42721  hdmapeq0  42725