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Theorem dochocss 40750
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 4963 . 2 𝑋 βŠ† ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}
2 dochss.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
3 eqid 2726 . . . . 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 40737 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘‹) ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
8 eqid 2726 . . . . 5 (ocβ€˜πΎ) = (ocβ€˜πΎ)
98, 2, 3, 6dochvalr 40741 . . . 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 590 . . 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 40736 . . . . . . . 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 6889 . . . . . . 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 2726 . . . . . . . . . . 11 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
14 eqid 2726 . . . . . . . . . . 11 (LSubSpβ€˜π‘ˆ) = (LSubSpβ€˜π‘ˆ)
1513, 2, 3, 4, 14dihf11 40651 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1β†’(LSubSpβ€˜π‘ˆ))
1615adantr 480 . . . . . . . . 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 6838 . . . . . . . . 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 38745 . . . . . . . . . 10 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
2019ad2antrr 723 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝐾 ∈ OP)
21 simpl 482 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
22 ssrab2 4072 . . . . . . . . . . . 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 2726 . . . . . . . . . . . . . . . 16 (1.β€˜πΎ) = (1.β€˜πΎ)
2524, 2, 3, 4, 5dih1 40670 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(1.β€˜πΎ)) = 𝑉)
2625adantr 480 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(1.β€˜πΎ)) = 𝑉)
27 f1fn 6782 . . . . . . . . . . . . . . . 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 38568 . . . . . . . . . . . . . . . 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 7076 . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑉 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
34 simpr 484 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑋 βŠ† 𝑉)
35 sseq2 4003 . . . . . . . . . . . . . 14 (𝑧 = 𝑉 β†’ (𝑋 βŠ† 𝑧 ↔ 𝑋 βŠ† 𝑉))
3635elrab 3678 . . . . . . . . . . . . 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 582 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑉 ∈ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})
3837ne0d 4330 . . . . . . . . . . 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 40728 . . . . . . . . . . 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 834 . . . . . . . . . 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 7279 . . . . . . . . . 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 583 . . . . . . . . 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 38577 . . . . . . . . 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 583 . . . . . . . 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 7270 . . . . . . . 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 583 . . . . . . 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 2766 . . . . . 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 6889 . . . . 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 38578 . . . . . 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 583 . . . . 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 2766 . . . 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 6889 . . 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 7271 . . . 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 583 . . 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 2771 . 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 4029 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  {crab 3426   βŠ† wss 3943  βˆ…c0 4317  βˆ© cint 4943  β—‘ccnv 5668  ran crn 5670   Fn wfn 6532  β€“1-1β†’wf1 6534  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  Basecbs 17153  occoc 17214  1.cp1 18389  LSubSpclss 20778  OPcops 38555  HLchlt 38733  LHypclh 39368  DVecHcdvh 40462  DIsoHcdih 40612  ocHcoch 40731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-riotaBAD 38336
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-tpos 8212  df-undef 8259  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-mulr 17220  df-sca 17222  df-vsca 17223  df-0g 17396  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-p1 18391  df-lat 18397  df-clat 18464  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19050  df-cntz 19233  df-lsm 19556  df-cmn 19702  df-abl 19703  df-mgp 20040  df-rng 20058  df-ur 20087  df-ring 20140  df-oppr 20236  df-dvdsr 20259  df-unit 20260  df-invr 20290  df-dvr 20303  df-drng 20589  df-lmod 20708  df-lss 20779  df-lsp 20819  df-lvec 20951  df-lsatoms 38359  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734  df-llines 38882  df-lplanes 38883  df-lvols 38884  df-lines 38885  df-psubsp 38887  df-pmap 38888  df-padd 39180  df-lhyp 39372  df-laut 39373  df-ldil 39488  df-ltrn 39489  df-trl 39543  df-tendo 40139  df-edring 40141  df-disoa 40413  df-dvech 40463  df-dib 40523  df-dic 40557  df-dih 40613  df-doch 40732
This theorem is referenced by:  dochsscl  40752  dochsat  40767  dochshpncl  40768  dochlkr  40769  dochdmj1  40774  dochnoncon  40775
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