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Theorem mapd0 41377
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 = (0g𝑈)
mapd0.c 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapd0.z 0 = (0g𝐶)
mapd0.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
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.
StepHypRef Expression
1 mapd0.h . . 3 𝐻 = (LHyp‘𝐾)
2 mapd0.u . . 3 𝑈 = ((DVecH‘𝐾)‘𝑊)
3 eqid 2726 . . 3 (LSubSp‘𝑈) = (LSubSp‘𝑈)
4 eqid 2726 . . 3 (LFnl‘𝑈) = (LFnl‘𝑈)
5 eqid 2726 . . 3 (LKer‘𝑈) = (LKer‘𝑈)
6 eqid 2726 . . 3 ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
7 mapd0.m . . 3 𝑀 = ((mapd‘𝐾)‘𝑊)
8 mapd0.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
91, 2, 8dvhlmod 40822 . . . 4 (𝜑𝑈 ∈ LMod)
10 mapd0.o . . . . 5 𝑂 = (0g𝑈)
1110, 3lsssn0 20921 . . . 4 (𝑈 ∈ LMod → {𝑂} ∈ (LSubSp‘𝑈))
129, 11syl 17 . . 3 (𝜑 → {𝑂} ∈ (LSubSp‘𝑈))
131, 2, 3, 4, 5, 6, 7, 8, 12mapdval 41340 . 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 780 . . . . . . . . . . 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})
159adantr 479 . . . . . . . . . . . 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)
168adantr 479 . . . . . . . . . . . . 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 ∧ 𝑊𝐻))
17 eqid 2726 . . . . . . . . . . . . . 14 (Base‘𝑈) = (Base‘𝑈)
18 simprl 769 . . . . . . . . . . . . . 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))
1917, 4, 5, 15, 18lkrssv 38807 . . . . . . . . . . . . 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))
201, 2, 17, 3, 6dochlss 41066 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((LKer‘𝑈)‘𝑔) ⊆ (Base‘𝑈)) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈))
2116, 19, 20syl2anc 582 . . . . . . . . . . . 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))
2210, 3lssle0 20923 . . . . . . . . . . . 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}))
2315, 21, 22syl2anc 582 . . . . . . . . . . 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}))
2414, 23mpbid 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})
2524fveq2d 6897 . . . . . . . . 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 779 . . . . . . . . 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))
271, 2, 6, 17, 10doch0 41070 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈))
288, 27syl 17 . . . . . . . . . 10 (𝜑 → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈))
2928adantr 479 . . . . . . . . 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))
3025, 26, 293eqtr3d 2774 . . . . . . . 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 2726 . . . . . . . . . 10 (Scalar‘𝑈) = (Scalar‘𝑈)
32 eqid 2726 . . . . . . . . . 10 (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈))
3331, 32, 17, 4, 5lkr0f 38805 . . . . . . . . 9 ((𝑈 ∈ LMod ∧ 𝑔 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘𝑔) = (Base‘𝑈) ↔ 𝑔 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
3415, 18, 33syl2anc 582 . . . . . . . 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))})))
3530, 34mpbid 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 = (0g𝐶)
381, 2, 17, 31, 32, 36, 37, 8lcd0v 41323 . . . . . . . 8 (𝜑0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))
3938adantr 479 . . . . . . 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))}))
4035, 39eqtr4d 2769 . . . . . 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 )
4140ex 411 . . . . 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 2726 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
431, 36, 42, 37, 8lcd0vcl 41326 . . . . . . . 8 (𝜑0 ∈ (Base‘𝐶))
441, 36, 42, 2, 4, 8, 43lcdvbaselfl 41307 . . . . . . 7 (𝜑0 ∈ (LFnl‘𝑈))
4531, 32, 17, 4, 5lkr0f 38805 . . . . . . . . . . . . 13 ((𝑈 ∈ LMod ∧ 0 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
469, 44, 45syl2anc 582 . . . . . . . . . . . 12 (𝜑 → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
4738, 46mpbird 256 . . . . . . . . . . 11 (𝜑 → ((LKer‘𝑈)‘ 0 ) = (Base‘𝑈))
4847fveq2d 6897 . . . . . . . . . 10 (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)))
4948fveq2d 6897 . . . . . . . . 9 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))))
501, 2, 6, 17, 8dochoc1 41073 . . . . . . . . 9 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))) = (Base‘𝑈))
5149, 50eqtrd 2766 . . . . . . . 8 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (Base‘𝑈))
5251, 47eqtr4d 2769 . . . . . . 7 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ))
531, 2, 6, 17, 10doch1 41071 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂})
548, 53syl 17 . . . . . . . . 9 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂})
5548, 54eqtrd 2766 . . . . . . . 8 (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂})
56 eqimss 4037 . . . . . . . 8 ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂} → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})
5755, 56syl 17 . . . . . . 7 (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂})
5844, 52, 57jca32 514 . . . . . 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 2814 . . . . . . 7 (𝑔 = 0 → (𝑔 ∈ (LFnl‘𝑈) ↔ 0 ∈ (LFnl‘𝑈)))
60 2fveq3 6898 . . . . . . . . . 10 (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )))
6160fveq2d 6897 . . . . . . . . 9 (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))))
62 fveq2 6893 . . . . . . . . 9 (𝑔 = 0 → ((LKer‘𝑈)‘𝑔) = ((LKer‘𝑈)‘ 0 ))
6361, 62eqeq12d 2742 . . . . . . . 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 )))
6460sseq1d 4010 . . . . . . . 8 (𝑔 = 0 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}))
6563, 64anbi12d 630 . . . . . . 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})))
6659, 65anbi12d 630 . . . . . 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}))))
6758, 66syl5ibrcom 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}))))
6841, 67impbid 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 6898 . . . . . . . 8 (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)))
7069fveq2d 6897 . . . . . . 7 (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))))
71 fveq2 6893 . . . . . . 7 (𝑓 = 𝑔 → ((LKer‘𝑈)‘𝑓) = ((LKer‘𝑈)‘𝑔))
7270, 71eqeq12d 2742 . . . . . 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)))
7369sseq1d 4010 . . . . . 6 (𝑓 = 𝑔 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))
7472, 73anbi12d 630 . . . . 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})))
7574elrab 3680 . . . 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 4639 . . . 4 (𝑔 ∈ { 0 } ↔ 𝑔 = 0 )
7768, 75, 763bitr4g 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 }))
7877eqrdv 2724 . 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 })
7913, 78eqtrd 2766 1 (𝜑 → (𝑀‘{𝑂}) = { 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1534  wcel 2099  {crab 3419  wss 3946  {csn 4623   × cxp 5672  cfv 6546  Basecbs 17208  Scalarcsca 17264  0gc0g 17449  LModclmod 20832  LSubSpclss 20904  LFnlclfn 38768  LKerclk 38796  HLchlt 39061  LHypclh 39696  DVecHcdvh 40790  ocHcoch 41059  LCDualclcd 41298  mapdcmpd 41336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-cnex 11205  ax-resscn 11206  ax-1cn 11207  ax-icn 11208  ax-addcl 11209  ax-addrcl 11210  ax-mulcl 11211  ax-mulrcl 11212  ax-mulcom 11213  ax-addass 11214  ax-mulass 11215  ax-distr 11216  ax-i2m1 11217  ax-1ne0 11218  ax-1rid 11219  ax-rnegex 11220  ax-rrecex 11221  ax-cnre 11222  ax-pre-lttri 11223  ax-pre-lttrn 11224  ax-pre-ltadd 11225  ax-pre-mulgt0 11226  ax-riotaBAD 38664
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-of 7682  df-om 7869  df-1st 7995  df-2nd 7996  df-tpos 8233  df-undef 8280  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-1o 8488  df-2o 8489  df-er 8726  df-map 8849  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-pnf 11291  df-mnf 11292  df-xr 11293  df-ltxr 11294  df-le 11295  df-sub 11487  df-neg 11488  df-nn 12259  df-2 12321  df-3 12322  df-4 12323  df-5 12324  df-6 12325  df-n0 12519  df-z 12605  df-uz 12869  df-fz 13533  df-struct 17144  df-sets 17161  df-slot 17179  df-ndx 17191  df-base 17209  df-ress 17238  df-plusg 17274  df-mulr 17275  df-sca 17277  df-vsca 17278  df-0g 17451  df-mre 17594  df-mrc 17595  df-acs 17597  df-proset 18315  df-poset 18333  df-plt 18350  df-lub 18366  df-glb 18367  df-join 18368  df-meet 18369  df-p0 18445  df-p1 18446  df-lat 18452  df-clat 18519  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-submnd 18769  df-grp 18926  df-minusg 18927  df-sbg 18928  df-subg 19113  df-cntz 19307  df-oppg 19336  df-lsm 19630  df-cmn 19776  df-abl 19777  df-mgp 20114  df-rng 20132  df-ur 20161  df-ring 20214  df-oppr 20312  df-dvdsr 20335  df-unit 20336  df-invr 20366  df-dvr 20379  df-nzr 20491  df-rlreg 20668  df-domn 20669  df-drng 20705  df-lmod 20834  df-lss 20905  df-lsp 20945  df-lvec 21077  df-lsatoms 38687  df-lshyp 38688  df-lcv 38730  df-lfl 38769  df-lkr 38797  df-ldual 38835  df-oposet 38887  df-ol 38889  df-oml 38890  df-covers 38977  df-ats 38978  df-atl 39009  df-cvlat 39033  df-hlat 39062  df-llines 39210  df-lplanes 39211  df-lvols 39212  df-lines 39213  df-psubsp 39215  df-pmap 39216  df-padd 39508  df-lhyp 39700  df-laut 39701  df-ldil 39816  df-ltrn 39817  df-trl 39871  df-tgrp 40455  df-tendo 40467  df-edring 40469  df-dveca 40715  df-disoa 40741  df-dvech 40791  df-dib 40851  df-dic 40885  df-dih 40941  df-doch 41060  df-djh 41107  df-lcdual 41299  df-mapd 41337
This theorem is referenced by:  mapdcnvatN  41378  mapdat  41379  mapdspex  41380  mapdn0  41381  hdmap10  41552  hdmapeq0  41556
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