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Theorem mapd0 37456
 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 2760 . . 3 (LSubSp‘𝑈) = (LSubSp‘𝑈)
4 eqid 2760 . . 3 (LFnl‘𝑈) = (LFnl‘𝑈)
5 eqid 2760 . . 3 (LKer‘𝑈) = (LKer‘𝑈)
6 eqid 2760 . . 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 36901 . . . 4 (𝜑𝑈 ∈ LMod)
10 mapd0.o . . . . 5 𝑂 = (0g𝑈)
1110, 3lsssn0 19150 . . . 4 (𝑈 ∈ LMod → {𝑂} ∈ (LSubSp‘𝑈))
129, 11syl 17 . . 3 (𝜑 → {𝑂} ∈ (LSubSp‘𝑈))
131, 2, 3, 4, 5, 6, 7, 8, 12mapdval 37419 . 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 824 . . . . . . . . . . 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 472 . . . . . . . . . . . 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 472 . . . . . . . . . . . . 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 2760 . . . . . . . . . . . . . 14 (Base‘𝑈) = (Base‘𝑈)
18 simprl 811 . . . . . . . . . . . . . 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 34886 . . . . . . . . . . . . 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 37145 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((LKer‘𝑈)‘𝑔) ⊆ (Base‘𝑈)) → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ∈ (LSubSp‘𝑈))
2116, 19, 20syl2anc 696 . . . . . . . . . . . 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 19152 . . . . . . . . . . . 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 696 . . . . . . . . . . 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 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})
2524fveq2d 6356 . . . . . . . . 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 823 . . . . . . . . 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 37149 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈))
288, 27syl 17 . . . . . . . . . 10 (𝜑 → (((ocH‘𝐾)‘𝑊)‘{𝑂}) = (Base‘𝑈))
2928adantr 472 . . . . . . . . 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 2802 . . . . . . . 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 2760 . . . . . . . . . 10 (Scalar‘𝑈) = (Scalar‘𝑈)
32 eqid 2760 . . . . . . . . . 10 (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈))
3331, 32, 17, 4, 5lkr0f 34884 . . . . . . . . 9 ((𝑈 ∈ LMod ∧ 𝑔 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘𝑔) = (Base‘𝑈) ↔ 𝑔 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
3415, 18, 33syl2anc 696 . . . . . . . 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 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 = (0g𝐶)
381, 2, 17, 31, 32, 36, 37, 8lcd0v 37402 . . . . . . . 8 (𝜑0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))}))
3938adantr 472 . . . . . . 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 2797 . . . . . 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 449 . . . . 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 2760 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
431, 36, 42, 37, 8lcd0vcl 37405 . . . . . . . 8 (𝜑0 ∈ (Base‘𝐶))
441, 36, 42, 2, 4, 8, 43lcdvbaselfl 37386 . . . . . . 7 (𝜑0 ∈ (LFnl‘𝑈))
4531, 32, 17, 4, 5lkr0f 34884 . . . . . . . . . . . . 13 ((𝑈 ∈ LMod ∧ 0 ∈ (LFnl‘𝑈)) → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
469, 44, 45syl2anc 696 . . . . . . . . . . . 12 (𝜑 → (((LKer‘𝑈)‘ 0 ) = (Base‘𝑈) ↔ 0 = ((Base‘𝑈) × {(0g‘(Scalar‘𝑈))})))
4738, 46mpbird 247 . . . . . . . . . . 11 (𝜑 → ((LKer‘𝑈)‘ 0 ) = (Base‘𝑈))
4847fveq2d 6356 . . . . . . . . . 10 (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)))
4948fveq2d 6356 . . . . . . . . 9 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))))
501, 2, 6, 17, 8dochoc1 37152 . . . . . . . . 9 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘(Base‘𝑈))) = (Base‘𝑈))
5149, 50eqtrd 2794 . . . . . . . 8 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = (Base‘𝑈))
5251, 47eqtr4d 2797 . . . . . . 7 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))) = ((LKer‘𝑈)‘ 0 ))
531, 2, 6, 17, 10doch1 37150 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂})
548, 53syl 17 . . . . . . . . 9 (𝜑 → (((ocH‘𝐾)‘𝑊)‘(Base‘𝑈)) = {𝑂})
5548, 54eqtrd 2794 . . . . . . . 8 (𝜑 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) = {𝑂})
56 eqimss 3798 . . . . . . . 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 559 . . . . . 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 2827 . . . . . . 7 (𝑔 = 0 → (𝑔 ∈ (LFnl‘𝑈) ↔ 0 ∈ (LFnl‘𝑈)))
60 fveq2 6352 . . . . . . . . . . 11 (𝑔 = 0 → ((LKer‘𝑈)‘𝑔) = ((LKer‘𝑈)‘ 0 ))
6160fveq2d 6356 . . . . . . . . . 10 (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )))
6261fveq2d 6356 . . . . . . . . 9 (𝑔 = 0 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 ))))
6362, 60eqeq12d 2775 . . . . . . . 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 )))
6461sseq1d 3773 . . . . . . . 8 (𝑔 = 0 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘ 0 )) ⊆ {𝑂}))
6563, 64anbi12d 749 . . . . . . 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 749 . . . . . 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 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}))))
6841, 67impbid 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 6352 . . . . . . . . 9 (𝑓 = 𝑔 → ((LKer‘𝑈)‘𝑓) = ((LKer‘𝑈)‘𝑔))
7069fveq2d 6356 . . . . . . . 8 (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) = (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)))
7170fveq2d 6356 . . . . . . 7 (𝑓 = 𝑔 → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓))) = (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))))
7271, 69eqeq12d 2775 . . . . . 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)))
7370sseq1d 3773 . . . . . 6 (𝑓 = 𝑔 → ((((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑓)) ⊆ {𝑂} ↔ (((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔)) ⊆ {𝑂}))
7472, 73anbi12d 749 . . . . 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 3504 . . . 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 4337 . . . 4 (𝑔 ∈ { 0 } ↔ 𝑔 = 0 )
7768, 75, 763bitr4g 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 }))
7877eqrdv 2758 . 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 2794 1 (𝜑 → (𝑀‘{𝑂}) = { 0 })
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  {crab 3054   ⊆ wss 3715  {csn 4321   × cxp 5264  ‘cfv 6049  Basecbs 16059  Scalarcsca 16146  0gc0g 16302  LModclmod 19065  LSubSpclss 19134  LFnlclfn 34847  LKerclk 34875  HLchlt 35140  LHypclh 35773  DVecHcdvh 36869  ocHcoch 37138  LCDualclcd 37377  mapdcmpd 37415 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-riotaBAD 34742 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-tpos 7521  df-undef 7568  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-0g 16304  df-mre 16448  df-mrc 16449  df-acs 16451  df-preset 17129  df-poset 17147  df-plt 17159  df-lub 17175  df-glb 17176  df-join 17177  df-meet 17178  df-p0 17240  df-p1 17241  df-lat 17247  df-clat 17309  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-subg 17792  df-cntz 17950  df-oppg 17976  df-lsm 18251  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-ring 18749  df-oppr 18823  df-dvdsr 18841  df-unit 18842  df-invr 18872  df-dvr 18883  df-drng 18951  df-lmod 19067  df-lss 19135  df-lsp 19174  df-lvec 19305  df-lsatoms 34766  df-lshyp 34767  df-lcv 34809  df-lfl 34848  df-lkr 34876  df-ldual 34914  df-oposet 34966  df-ol 34968  df-oml 34969  df-covers 35056  df-ats 35057  df-atl 35088  df-cvlat 35112  df-hlat 35141  df-llines 35287  df-lplanes 35288  df-lvols 35289  df-lines 35290  df-psubsp 35292  df-pmap 35293  df-padd 35585  df-lhyp 35777  df-laut 35778  df-ldil 35893  df-ltrn 35894  df-trl 35949  df-tgrp 36533  df-tendo 36545  df-edring 36547  df-dveca 36793  df-disoa 36820  df-dvech 36870  df-dib 36930  df-dic 36964  df-dih 37020  df-doch 37139  df-djh 37186  df-lcdual 37378  df-mapd 37416 This theorem is referenced by:  mapdcnvatN  37457  mapdat  37458  mapdspex  37459  mapdn0  37460  hdmap10  37634  hdmapeq0  37638
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