Theorem mapd0 39984

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 2736	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
4		eqid 2736	. . 3 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈)
5		eqid 2736	. . 3 ⊢ (LKer‘𝑈) = (LKer‘𝑈)
6		eqid 2736	. . 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 39429	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		mapd0.o	. . . . 5 ⊢ 𝑂 = (0g‘𝑈)
11	10, 3	lsssn0 20316	. . . 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 39947	. 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 779	. . . . . . . . . . 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 481	. . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . 14 ⊢ (Base‘𝑈) = (Base‘𝑈)
18		simprl 768	. . . . . . . . . . . . . 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 37414	. . . . . . . . . . . . 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 39673	. . . . . . . . . . . . 13 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((LKer‘𝑈)‘𝑔) ⊆ (Base‘𝑈)) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈))
21	16, 19, 20	syl2anc 584	. . . . . . . . . . . 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 20318	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 231	. . . . . . . . . 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 6830	. . . . . . . . 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 778	. . . . . . . . 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 39677	. . . . . . . . . . 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 481	. . . . . . . . 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 2784	. . . . . . . 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 2736	. . . . . . . . . 10 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
32		eqid 2736	. . . . . . . . . 10 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈))
33	31, 32, 17, 4, 5	lkr0f 37412	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑔 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘𝑔) = (Base‘𝑈) ↔ 𝑔 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
34	15, 18, 33	syl2anc 584	. . . . . . . 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 231	. . . . . . 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 39930	. . . . . . . 8 ⊢ (𝜑 → 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))
39	38	adantr 481	. . . . . . 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 2779	. . . . . 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 413	. . . . 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 2736	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
43	1, 36, 42, 37, 8	lcd0vcl 39933	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ (Base‘𝐶))
44	1, 36, 42, 2, 4, 8, 43	lcdvbaselfl 39914	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (LFnl‘𝑈))
45	31, 32, 17, 4, 5	lkr0f 37412	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ LMod ∧ 0 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
46	9, 44, 45	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
47	38, 46	mpbird 256	. . . . . . . . . . 11 ⊢ (𝜑 → ((LKer‘𝑈)‘ 0 ) = (Base‘𝑈))
48	47	fveq2d 6830	. . . . . . . . . 10 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)))
49	48	fveq2d 6830	. . . . . . . . 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 39680	. . . . . . . . 9 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))) = (Base‘𝑈))
51	49, 50	eqtrd 2776	. . . . . . . 8 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (Base‘𝑈))
52	51, 47	eqtr4d 2779	. . . . . . 7 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ))
53	1, 2, 6, 17, 10	doch1 39678	. . . . . . . . . 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 2776	. . . . . . . 8 ⊢ (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂})
56		eqimss 3988	. . . . . . . 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 516	. . . . . 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 2824	. . . . . . 7 ⊢ (𝑔 = 0 → (𝑔 ∈ (LFnl‘𝑈) ↔ 0 ∈ (LFnl‘𝑈)))
60		2fveq3 6831	. . . . . . . . . 10 ⊢ (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )))
61	60	fveq2d 6830	. . . . . . . . 9 ⊢ (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))))
62		fveq2 6826	. . . . . . . . 9 ⊢ (𝑔 = 0 → ((LKer‘𝑈)‘𝑔) = ((LKer‘𝑈)‘ 0 ))
63	61, 62	eqeq12d 2752	. . . . . . . 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 3963	. . . . . . . 8 ⊢ (𝑔 = 0 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}))
65	63, 64	anbi12d 631	. . . . . . 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 631	. . . . . 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 246	. . . . 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 211	. . . 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 6831	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)))
70	69	fveq2d 6830	. . . . . . 7 ⊢ (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))))
71		fveq2 6826	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((LKer‘𝑈)‘𝑓) = ((LKer‘𝑈)‘𝑔))
72	70, 71	eqeq12d 2752	. . . . . 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 3963	. . . . . 6 ⊢ (𝑓 = 𝑔 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))
74	72, 73	anbi12d 631	. . . . 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 3634	. . . 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 4590	. . . 4 ⊢ (𝑔 ∈ { 0 } ↔ 𝑔 = 0 )
77	68, 75, 76	3bitr4g 313	. . 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 2734	. 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 2776	1 ⊢ (𝜑 → (𝑀‘{𝑂}) = { 0 })

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  {crab 3403   ⊆ wss 3898  {csn 4574   × cxp 5619  ‘cfv 6480  Basecbs 17010  Scalarcsca 17063  0_gc0g 17248  LModclmod 20230  LSubSpclss 20300  LFnlclfn 37375  LKerclk 37403  HLchlt 37668  LHypclh 38303  DVecHcdvh 39397  ocHcoch 39666  LCDualclcd 39905  mapdcmpd 39943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5230  ax-sep 5244  ax-nul 5251  ax-pow 5309  ax-pr 5373  ax-un 7651  ax-cnex 11029  ax-resscn 11030  ax-1cn 11031  ax-icn 11032  ax-addcl 11033  ax-addrcl 11034  ax-mulcl 11035  ax-mulrcl 11036  ax-mulcom 11037  ax-addass 11038  ax-mulass 11039  ax-distr 11040  ax-i2m1 11041  ax-1ne0 11042  ax-1rid 11043  ax-rnegex 11044  ax-rrecex 11045  ax-cnre 11046  ax-pre-lttri 11047  ax-pre-lttrn 11048  ax-pre-ltadd 11049  ax-pre-mulgt0 11050  ax-riotaBAD 37271

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4271  df-if 4475  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4854  df-int 4896  df-iun 4944  df-iin 4945  df-br 5094  df-opab 5156  df-mpt 5177  df-tr 5211  df-id 5519  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5576  df-we 5578  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 6239  df-ord 6306  df-on 6307  df-lim 6308  df-suc 6309  df-iota 6432  df-fun 6482  df-fn 6483  df-f 6484  df-f1 6485  df-fo 6486  df-f1o 6487  df-fv 6488  df-riota 7294  df-ov 7341  df-oprab 7342  df-mpo 7343  df-of 7596  df-om 7782  df-1st 7900  df-2nd 7901  df-tpos 8113  df-undef 8160  df-frecs 8168  df-wrecs 8199  df-recs 8273  df-rdg 8312  df-1o 8368  df-er 8570  df-map 8689  df-en 8806  df-dom 8807  df-sdom 8808  df-fin 8809  df-pnf 11113  df-mnf 11114  df-xr 11115  df-ltxr 11116  df-le 11117  df-sub 11309  df-neg 11310  df-nn 12076  df-2 12138  df-3 12139  df-4 12140  df-5 12141  df-6 12142  df-n0 12336  df-z 12422  df-uz 12685  df-fz 13342  df-struct 16946  df-sets 16963  df-slot 16981  df-ndx 16993  df-base 17011  df-ress 17040  df-plusg 17073  df-mulr 17074  df-sca 17076  df-vsca 17077  df-0g 17250  df-mre 17393  df-mrc 17394  df-acs 17396  df-proset 18111  df-poset 18129  df-plt 18146  df-lub 18162  df-glb 18163  df-join 18164  df-meet 18165  df-p0 18241  df-p1 18242  df-lat 18248  df-clat 18315  df-mgm 18424  df-sgrp 18473  df-mnd 18484  df-submnd 18529  df-grp 18677  df-minusg 18678  df-sbg 18679  df-subg 18849  df-cntz 19020  df-oppg 19047  df-lsm 19338  df-cmn 19484  df-abl 19485  df-mgp 19817  df-ur 19834  df-ring 19881  df-oppr 19958  df-dvdsr 19979  df-unit 19980  df-invr 20010  df-dvr 20021  df-drng 20096  df-lmod 20232  df-lss 20301  df-lsp 20341  df-lvec 20472  df-lsatoms 37294  df-lshyp 37295  df-lcv 37337  df-lfl 37376  df-lkr 37404  df-ldual 37442  df-oposet 37494  df-ol 37496  df-oml 37497  df-covers 37584  df-ats 37585  df-atl 37616  df-cvlat 37640  df-hlat 37669  df-llines 37817  df-lplanes 37818  df-lvols 37819  df-lines 37820  df-psubsp 37822  df-pmap 37823  df-padd 38115  df-lhyp 38307  df-laut 38308  df-ldil 38423  df-ltrn 38424  df-trl 38478  df-tgrp 39062  df-tendo 39074  df-edring 39076  df-dveca 39322  df-disoa 39348  df-dvech 39398  df-dib 39458  df-dic 39492  df-dih 39548  df-doch 39667  df-djh 39714  df-lcdual 39906  df-mapd 39944

This theorem is referenced by:  mapdcnvatN  39985  mapdat  39986  mapdspex  39987  mapdn0  39988  hdmap10  40159  hdmapeq0  40163