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Theorem dochocss 40232
Description: Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)
Hypotheses
Ref Expression
dochss.h 𝐻 = (LHypβ€˜πΎ)
dochss.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dochss.v 𝑉 = (Baseβ€˜π‘ˆ)
dochss.o βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dochocss (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
Proof of Theorem dochocss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ssintub 4970 . 2 𝑋 βŠ† ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}
2 dochss.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
3 eqid 2732 . . . . 5 ((DIsoHβ€˜πΎ)β€˜π‘Š) = ((DIsoHβ€˜πΎ)β€˜π‘Š)
4 dochss.u . . . . 5 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
5 dochss.v . . . . 5 𝑉 = (Baseβ€˜π‘ˆ)
6 dochss.o . . . . 5 βŠ₯ = ((ocHβ€˜πΎ)β€˜π‘Š)
72, 3, 4, 5, 6dochcl 40219 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘‹) ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
8 eqid 2732 . . . . 5 (ocβ€˜πΎ) = (ocβ€˜πΎ)
98, 2, 3, 6dochvalr 40223 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ( βŠ₯ β€˜π‘‹) ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š)) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜( βŠ₯ β€˜π‘‹)))))
107, 9syldan 591 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜( βŠ₯ β€˜π‘‹)))))
118, 2, 3, 4, 5, 6dochval2 40218 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘‹) = (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}))))
1211fveq2d 6895 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜( βŠ₯ β€˜π‘‹)) = (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})))))
13 eqid 2732 . . . . . . . . . . 11 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
14 eqid 2732 . . . . . . . . . . 11 (LSubSpβ€˜π‘ˆ) = (LSubSpβ€˜π‘ˆ)
1513, 2, 3, 4, 14dihf11 40133 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1β†’(LSubSpβ€˜π‘ˆ))
1615adantr 481 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1β†’(LSubSpβ€˜π‘ˆ))
17 f1f1orn 6844 . . . . . . . . 9 (((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1β†’(LSubSpβ€˜π‘ˆ) β†’ ((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1-ontoβ†’ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
1816, 17syl 17 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1-ontoβ†’ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
19 hlop 38227 . . . . . . . . . 10 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
2019ad2antrr 724 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝐾 ∈ OP)
21 simpl 483 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
22 ssrab2 4077 . . . . . . . . . . . 12 {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} βŠ† ran ((DIsoHβ€˜πΎ)β€˜π‘Š)
2322a1i 11 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} βŠ† ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
24 eqid 2732 . . . . . . . . . . . . . . . 16 (1.β€˜πΎ) = (1.β€˜πΎ)
2524, 2, 3, 4, 5dih1 40152 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(1.β€˜πΎ)) = 𝑉)
2625adantr 481 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(1.β€˜πΎ)) = 𝑉)
27 f1fn 6788 . . . . . . . . . . . . . . . 16 (((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1β†’(LSubSpβ€˜π‘ˆ) β†’ ((DIsoHβ€˜πΎ)β€˜π‘Š) Fn (Baseβ€˜πΎ))
2816, 27syl 17 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ((DIsoHβ€˜πΎ)β€˜π‘Š) Fn (Baseβ€˜πΎ))
2913, 24op1cl 38050 . . . . . . . . . . . . . . . 16 (𝐾 ∈ OP β†’ (1.β€˜πΎ) ∈ (Baseβ€˜πΎ))
3020, 29syl 17 . . . . . . . . . . . . . . 15 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (1.β€˜πΎ) ∈ (Baseβ€˜πΎ))
31 fnfvelrn 7082 . . . . . . . . . . . . . . 15 ((((DIsoHβ€˜πΎ)β€˜π‘Š) Fn (Baseβ€˜πΎ) ∧ (1.β€˜πΎ) ∈ (Baseβ€˜πΎ)) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(1.β€˜πΎ)) ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
3228, 30, 31syl2anc 584 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(1.β€˜πΎ)) ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
3326, 32eqeltrrd 2834 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑉 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
34 simpr 485 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑋 βŠ† 𝑉)
35 sseq2 4008 . . . . . . . . . . . . . 14 (𝑧 = 𝑉 β†’ (𝑋 βŠ† 𝑧 ↔ 𝑋 βŠ† 𝑉))
3635elrab 3683 . . . . . . . . . . . . 13 (𝑉 ∈ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} ↔ (𝑉 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∧ 𝑋 βŠ† 𝑉))
3733, 34, 36sylanbrc 583 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑉 ∈ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})
3837ne0d 4335 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} β‰  βˆ…)
392, 3dihintcl 40210 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ({𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} βŠ† ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∧ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} β‰  βˆ…)) β†’ ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
4021, 23, 38, 39syl12anc 835 . . . . . . . . . 10 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
41 f1ocnvdm 7282 . . . . . . . . . 10 ((((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1-ontoβ†’ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∧ ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š)) β†’ (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}) ∈ (Baseβ€˜πΎ))
4218, 40, 41syl2anc 584 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}) ∈ (Baseβ€˜πΎ))
4313, 8opoccl 38059 . . . . . . . . 9 ((𝐾 ∈ OP ∧ (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}) ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})) ∈ (Baseβ€˜πΎ))
4420, 42, 43syl2anc 584 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})) ∈ (Baseβ€˜πΎ))
45 f1ocnvfv1 7273 . . . . . . . 8 ((((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1-ontoβ†’ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∧ ((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})) ∈ (Baseβ€˜πΎ)) β†’ (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})))) = ((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})))
4618, 44, 45syl2anc 584 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})))) = ((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})))
4712, 46eqtrd 2772 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜( βŠ₯ β€˜π‘‹)) = ((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})))
4847fveq2d 6895 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜( βŠ₯ β€˜π‘‹))) = ((ocβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}))))
4913, 8opococ 38060 . . . . . 6 ((𝐾 ∈ OP ∧ (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}) ∈ (Baseβ€˜πΎ)) β†’ ((ocβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}))) = (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}))
5020, 42, 49syl2anc 584 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ((ocβ€˜πΎ)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}))) = (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}))
5148, 50eqtrd 2772 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜( βŠ₯ β€˜π‘‹))) = (β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}))
5251fveq2d 6895 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜((ocβ€˜πΎ)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜( βŠ₯ β€˜π‘‹)))) = (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})))
53 f1ocnvfv2 7274 . . . 4 ((((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1-ontoβ†’ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∧ ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š)) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})) = ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})
5418, 40, 53syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(β—‘((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜βˆ© {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})) = ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})
5510, 52, 543eqtrrd 2777 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧} = ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
561, 55sseqtrid 4034 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432   βŠ† wss 3948  βˆ…c0 4322  βˆ© cint 4950  β—‘ccnv 5675  ran crn 5677   Fn wfn 6538  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  Basecbs 17143  occoc 17204  1.cp1 18376  LSubSpclss 20541  OPcops 38037  HLchlt 38215  LHypclh 38850  DVecHcdvh 39944  DIsoHcdih 40094  ocHcoch 40213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-riotaBAD 37818
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-tpos 8210  df-undef 8257  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-0g 17386  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-submnd 18671  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-cntz 19180  df-lsm 19503  df-cmn 19649  df-abl 19650  df-mgp 19987  df-ur 20004  df-ring 20057  df-oppr 20149  df-dvdsr 20170  df-unit 20171  df-invr 20201  df-dvr 20214  df-drng 20358  df-lmod 20472  df-lss 20542  df-lsp 20582  df-lvec 20713  df-lsatoms 37841  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-llines 38364  df-lplanes 38365  df-lvols 38366  df-lines 38367  df-psubsp 38369  df-pmap 38370  df-padd 38662  df-lhyp 38854  df-laut 38855  df-ldil 38970  df-ltrn 38971  df-trl 39025  df-tendo 39621  df-edring 39623  df-disoa 39895  df-dvech 39945  df-dib 40005  df-dic 40039  df-dih 40095  df-doch 40214
This theorem is referenced by:  dochsscl  40234  dochsat  40249  dochshpncl  40250  dochlkr  40251  dochdmj1  40256  dochnoncon  40257
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