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Theorem dochocss 40895
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 4964 . 2 𝑋 βŠ† ∩ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧}
2 dochss.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
3 eqid 2725 . . . . 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 40882 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘‹) ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
8 eqid 2725 . . . . 5 (ocβ€˜πΎ) = (ocβ€˜πΎ)
98, 2, 3, 6dochvalr 40886 . . . 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 589 . . 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 40881 . . . . . . . 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 6896 . . . . . . 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 2725 . . . . . . . . . . 11 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
14 eqid 2725 . . . . . . . . . . 11 (LSubSpβ€˜π‘ˆ) = (LSubSpβ€˜π‘ˆ)
1513, 2, 3, 4, 14dihf11 40796 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((DIsoHβ€˜πΎ)β€˜π‘Š):(Baseβ€˜πΎ)–1-1β†’(LSubSpβ€˜π‘ˆ))
1615adantr 479 . . . . . . . . 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 6845 . . . . . . . . 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 38890 . . . . . . . . . 10 (𝐾 ∈ HL β†’ 𝐾 ∈ OP)
2019ad2antrr 724 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝐾 ∈ OP)
21 simpl 481 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
22 ssrab2 4069 . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . 16 (1.β€˜πΎ) = (1.β€˜πΎ)
2524, 2, 3, 4, 5dih1 40815 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(1.β€˜πΎ)) = 𝑉)
2625adantr 479 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ (((DIsoHβ€˜πΎ)β€˜π‘Š)β€˜(1.β€˜πΎ)) = 𝑉)
27 f1fn 6789 . . . . . . . . . . . . . . . 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 38713 . . . . . . . . . . . . . . . 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 7085 . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . 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 2826 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑉 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š))
34 simpr 483 . . . . . . . . . . . . 13 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑋 βŠ† 𝑉)
35 sseq2 3999 . . . . . . . . . . . . . 14 (𝑧 = 𝑉 β†’ (𝑋 βŠ† 𝑧 ↔ 𝑋 βŠ† 𝑉))
3635elrab 3674 . . . . . . . . . . . . 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 581 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 βŠ† 𝑉) β†’ 𝑉 ∈ {𝑧 ∈ ran ((DIsoHβ€˜πΎ)β€˜π‘Š) ∣ 𝑋 βŠ† 𝑧})
3837ne0d 4331 . . . . . . . . . . 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 40873 . . . . . . . . . . 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 7290 . . . . . . . . . 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 582 . . . . . . . . 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 38722 . . . . . . . . 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 582 . . . . . . . 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 7281 . . . . . . . 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 582 . . . . . . 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 2765 . . . . . 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 6896 . . . . 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 38723 . . . . . 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 582 . . . . 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 2765 . . . 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 6896 . . 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 7282 . . . 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 582 . . 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 2770 . 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 4025 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 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  {crab 3419   βŠ† wss 3939  βˆ…c0 4318  βˆ© cint 4944  β—‘ccnv 5671  ran crn 5673   Fn wfn 6538  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  Basecbs 17179  occoc 17240  1.cp1 18415  LSubSpclss 20819  OPcops 38700  HLchlt 38878  LHypclh 39513  DVecHcdvh 40607  DIsoHcdih 40757  ocHcoch 40876
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 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-riotaBAD 38481
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-tpos 8230  df-undef 8277  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-struct 17115  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-mulr 17246  df-sca 17248  df-vsca 17249  df-0g 17422  df-proset 18286  df-poset 18304  df-plt 18321  df-lub 18337  df-glb 18338  df-join 18339  df-meet 18340  df-p0 18416  df-p1 18417  df-lat 18423  df-clat 18490  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18740  df-grp 18897  df-minusg 18898  df-sbg 18899  df-subg 19082  df-cntz 19272  df-lsm 19595  df-cmn 19741  df-abl 19742  df-mgp 20079  df-rng 20097  df-ur 20126  df-ring 20179  df-oppr 20277  df-dvdsr 20300  df-unit 20301  df-invr 20331  df-dvr 20344  df-drng 20630  df-lmod 20749  df-lss 20820  df-lsp 20860  df-lvec 20992  df-lsatoms 38504  df-oposet 38704  df-ol 38706  df-oml 38707  df-covers 38794  df-ats 38795  df-atl 38826  df-cvlat 38850  df-hlat 38879  df-llines 39027  df-lplanes 39028  df-lvols 39029  df-lines 39030  df-psubsp 39032  df-pmap 39033  df-padd 39325  df-lhyp 39517  df-laut 39518  df-ldil 39633  df-ltrn 39634  df-trl 39688  df-tendo 40284  df-edring 40286  df-disoa 40558  df-dvech 40608  df-dib 40668  df-dic 40702  df-dih 40758  df-doch 40877
This theorem is referenced by:  dochsscl  40897  dochsat  40912  dochshpncl  40913  dochlkr  40914  dochdmj1  40919  dochnoncon  40920
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