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Theorem clsocv 25000
Description: The orthogonal complement of the closure of a subset is the same as the orthogonal complement of the subset itself. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
clsocv.v 𝑉 = (Baseβ€˜π‘Š)
clsocv.o 𝑂 = (ocvβ€˜π‘Š)
clsocv.j 𝐽 = (TopOpenβ€˜π‘Š)
Assertion
Ref Expression
clsocv ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (π‘‚β€˜((clsβ€˜π½)β€˜π‘†)) = (π‘‚β€˜π‘†))

Proof of Theorem clsocv
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cphngp 24923 . . . . . . . 8 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ NrmGrp)
2 ngptps 24333 . . . . . . . 8 (π‘Š ∈ NrmGrp β†’ π‘Š ∈ TopSp)
31, 2syl 17 . . . . . . 7 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ TopSp)
43adantr 479 . . . . . 6 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ π‘Š ∈ TopSp)
5 clsocv.v . . . . . . 7 𝑉 = (Baseβ€˜π‘Š)
6 clsocv.j . . . . . . 7 𝐽 = (TopOpenβ€˜π‘Š)
75, 6istps 22658 . . . . . 6 (π‘Š ∈ TopSp ↔ 𝐽 ∈ (TopOnβ€˜π‘‰))
84, 7sylib 217 . . . . 5 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝐽 ∈ (TopOnβ€˜π‘‰))
9 topontop 22637 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‰) β†’ 𝐽 ∈ Top)
108, 9syl 17 . . . 4 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝐽 ∈ Top)
11 simpr 483 . . . . 5 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
12 toponuni 22638 . . . . . 6 (𝐽 ∈ (TopOnβ€˜π‘‰) β†’ 𝑉 = βˆͺ 𝐽)
138, 12syl 17 . . . . 5 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝑉 = βˆͺ 𝐽)
1411, 13sseqtrd 4023 . . . 4 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† βˆͺ 𝐽)
15 eqid 2730 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
1615sscls 22782 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
1710, 14, 16syl2anc 582 . . 3 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
18 clsocv.o . . . 4 𝑂 = (ocvβ€˜π‘Š)
1918ocv2ss 21447 . . 3 (𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†) β†’ (π‘‚β€˜((clsβ€˜π½)β€˜π‘†)) βŠ† (π‘‚β€˜π‘†))
2017, 19syl 17 . 2 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (π‘‚β€˜((clsβ€˜π½)β€˜π‘†)) βŠ† (π‘‚β€˜π‘†))
2115clsss3 22785 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
2210, 14, 21syl2anc 582 . . . . 5 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
2322, 13sseqtrrd 4024 . . . 4 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑉)
2423adantr 479 . . 3 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑉)
255, 18ocvss 21444 . . . . 5 (π‘‚β€˜π‘†) βŠ† 𝑉
2625a1i 11 . . . 4 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (π‘‚β€˜π‘†) βŠ† 𝑉)
2726sselda 3983 . . 3 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ π‘₯ ∈ 𝑉)
28 df-ss 3966 . . . . . . . . . 10 (((clsβ€˜π½)β€˜π‘†) βŠ† 𝑉 ↔ (((clsβ€˜π½)β€˜π‘†) ∩ 𝑉) = ((clsβ€˜π½)β€˜π‘†))
2924, 28sylib 217 . . . . . . . . 9 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (((clsβ€˜π½)β€˜π‘†) ∩ 𝑉) = ((clsβ€˜π½)β€˜π‘†))
3029ineq1d 4212 . . . . . . . 8 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ ((((clsβ€˜π½)β€˜π‘†) ∩ 𝑉) ∩ {𝑦 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}) = (((clsβ€˜π½)β€˜π‘†) ∩ {𝑦 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}))
31 dfrab3 4310 . . . . . . . . . 10 {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} = (𝑉 ∩ {𝑦 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
3231ineq2i 4210 . . . . . . . . 9 (((clsβ€˜π½)β€˜π‘†) ∩ {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}) = (((clsβ€˜π½)β€˜π‘†) ∩ (𝑉 ∩ {𝑦 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}))
33 inass 4220 . . . . . . . . 9 ((((clsβ€˜π½)β€˜π‘†) ∩ 𝑉) ∩ {𝑦 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}) = (((clsβ€˜π½)β€˜π‘†) ∩ (𝑉 ∩ {𝑦 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}))
3432, 33eqtr4i 2761 . . . . . . . 8 (((clsβ€˜π½)β€˜π‘†) ∩ {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}) = ((((clsβ€˜π½)β€˜π‘†) ∩ 𝑉) ∩ {𝑦 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
35 dfrab3 4310 . . . . . . . 8 {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} = (((clsβ€˜π½)β€˜π‘†) ∩ {𝑦 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
3630, 34, 353eqtr4g 2795 . . . . . . 7 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (((clsβ€˜π½)β€˜π‘†) ∩ {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}) = {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
3715clscld 22773 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) ∈ (Clsdβ€˜π½))
3810, 14, 37syl2anc 582 . . . . . . . . 9 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ((clsβ€˜π½)β€˜π‘†) ∈ (Clsdβ€˜π½))
3938adantr 479 . . . . . . . 8 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) ∈ (Clsdβ€˜π½))
40 fvex 6905 . . . . . . . . . 10 (0gβ€˜(Scalarβ€˜π‘Š)) ∈ V
41 eqid 2730 . . . . . . . . . . 11 (𝑦 ∈ 𝑉 ↦ (π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = (𝑦 ∈ 𝑉 ↦ (π‘₯(Β·π‘–β€˜π‘Š)𝑦))
4241mptiniseg 6239 . . . . . . . . . 10 ((0gβ€˜(Scalarβ€˜π‘Š)) ∈ V β†’ (β—‘(𝑦 ∈ 𝑉 ↦ (π‘₯(Β·π‘–β€˜π‘Š)𝑦)) β€œ {(0gβ€˜(Scalarβ€˜π‘Š))}) = {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
4340, 42ax-mp 5 . . . . . . . . 9 (β—‘(𝑦 ∈ 𝑉 ↦ (π‘₯(Β·π‘–β€˜π‘Š)𝑦)) β€œ {(0gβ€˜(Scalarβ€˜π‘Š))}) = {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}
44 eqid 2730 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
45 eqid 2730 . . . . . . . . . . 11 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
46 simpll 763 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ π‘Š ∈ β„‚PreHil)
478adantr 479 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‰))
4847, 47, 27cnmptc 23388 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (𝑦 ∈ 𝑉 ↦ π‘₯) ∈ (𝐽 Cn 𝐽))
4947cnmptid 23387 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (𝑦 ∈ 𝑉 ↦ 𝑦) ∈ (𝐽 Cn 𝐽))
506, 44, 45, 46, 47, 48, 49cnmpt1ip 24997 . . . . . . . . . 10 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (𝑦 ∈ 𝑉 ↦ (π‘₯(Β·π‘–β€˜π‘Š)𝑦)) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)))
5144cnfldhaus 24523 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) ∈ Haus
52 cphclm 24939 . . . . . . . . . . . . . 14 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ β„‚Mod)
53 eqid 2730 . . . . . . . . . . . . . . 15 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5453clm0 24821 . . . . . . . . . . . . . 14 (π‘Š ∈ β„‚Mod β†’ 0 = (0gβ€˜(Scalarβ€˜π‘Š)))
5552, 54syl 17 . . . . . . . . . . . . 13 (π‘Š ∈ β„‚PreHil β†’ 0 = (0gβ€˜(Scalarβ€˜π‘Š)))
5655ad2antrr 722 . . . . . . . . . . . 12 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ 0 = (0gβ€˜(Scalarβ€˜π‘Š)))
57 0cn 11212 . . . . . . . . . . . 12 0 ∈ β„‚
5856, 57eqeltrrdi 2840 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ β„‚)
59 unicntop 24524 . . . . . . . . . . . 12 β„‚ = βˆͺ (TopOpenβ€˜β„‚fld)
6059sncld 23097 . . . . . . . . . . 11 (((TopOpenβ€˜β„‚fld) ∈ Haus ∧ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ β„‚) β†’ {(0gβ€˜(Scalarβ€˜π‘Š))} ∈ (Clsdβ€˜(TopOpenβ€˜β„‚fld)))
6151, 58, 60sylancr 585 . . . . . . . . . 10 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ {(0gβ€˜(Scalarβ€˜π‘Š))} ∈ (Clsdβ€˜(TopOpenβ€˜β„‚fld)))
62 cnclima 22994 . . . . . . . . . 10 (((𝑦 ∈ 𝑉 ↦ (π‘₯(Β·π‘–β€˜π‘Š)𝑦)) ∈ (𝐽 Cn (TopOpenβ€˜β„‚fld)) ∧ {(0gβ€˜(Scalarβ€˜π‘Š))} ∈ (Clsdβ€˜(TopOpenβ€˜β„‚fld))) β†’ (β—‘(𝑦 ∈ 𝑉 ↦ (π‘₯(Β·π‘–β€˜π‘Š)𝑦)) β€œ {(0gβ€˜(Scalarβ€˜π‘Š))}) ∈ (Clsdβ€˜π½))
6350, 61, 62syl2anc 582 . . . . . . . . 9 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (β—‘(𝑦 ∈ 𝑉 ↦ (π‘₯(Β·π‘–β€˜π‘Š)𝑦)) β€œ {(0gβ€˜(Scalarβ€˜π‘Š))}) ∈ (Clsdβ€˜π½))
6443, 63eqeltrrid 2836 . . . . . . . 8 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} ∈ (Clsdβ€˜π½))
65 incld 22769 . . . . . . . 8 ((((clsβ€˜π½)β€˜π‘†) ∈ (Clsdβ€˜π½) ∧ {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} ∈ (Clsdβ€˜π½)) β†’ (((clsβ€˜π½)β€˜π‘†) ∩ {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}) ∈ (Clsdβ€˜π½))
6639, 64, 65syl2anc 582 . . . . . . 7 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (((clsβ€˜π½)β€˜π‘†) ∩ {𝑦 ∈ 𝑉 ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}) ∈ (Clsdβ€˜π½))
6736, 66eqeltrrd 2832 . . . . . 6 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} ∈ (Clsdβ€˜π½))
6817adantr 479 . . . . . . 7 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
69 eqid 2730 . . . . . . . . . 10 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
705, 45, 53, 69, 18ocvi 21443 . . . . . . . . 9 ((π‘₯ ∈ (π‘‚β€˜π‘†) ∧ 𝑦 ∈ 𝑆) β†’ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
7170ralrimiva 3144 . . . . . . . 8 (π‘₯ ∈ (π‘‚β€˜π‘†) β†’ βˆ€π‘¦ ∈ 𝑆 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
7271adantl 480 . . . . . . 7 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ βˆ€π‘¦ ∈ 𝑆 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
73 ssrab 4071 . . . . . . 7 (𝑆 βŠ† {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} ↔ (𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†) ∧ βˆ€π‘¦ ∈ 𝑆 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
7468, 72, 73sylanbrc 581 . . . . . 6 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ 𝑆 βŠ† {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
7515clsss2 22798 . . . . . 6 (({𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} ∈ (Clsdβ€˜π½) ∧ 𝑆 βŠ† {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))}) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
7667, 74, 75syl2anc 582 . . . . 5 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
77 ssrab2 4078 . . . . . 6 {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} βŠ† ((clsβ€˜π½)β€˜π‘†)
7877a1i 11 . . . . 5 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} βŠ† ((clsβ€˜π½)β€˜π‘†))
7976, 78eqssd 4000 . . . 4 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) = {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))})
80 rabid2 3462 . . . 4 (((clsβ€˜π½)β€˜π‘†) = {𝑦 ∈ ((clsβ€˜π½)β€˜π‘†) ∣ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))} ↔ βˆ€π‘¦ ∈ ((clsβ€˜π½)β€˜π‘†)(π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
8179, 80sylib 217 . . 3 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ βˆ€π‘¦ ∈ ((clsβ€˜π½)β€˜π‘†)(π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
825, 45, 53, 69, 18elocv 21442 . . 3 (π‘₯ ∈ (π‘‚β€˜((clsβ€˜π½)β€˜π‘†)) ↔ (((clsβ€˜π½)β€˜π‘†) βŠ† 𝑉 ∧ π‘₯ ∈ 𝑉 ∧ βˆ€π‘¦ ∈ ((clsβ€˜π½)β€˜π‘†)(π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š))))
8324, 27, 81, 82syl3anbrc 1341 . 2 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ π‘₯ ∈ (π‘‚β€˜((clsβ€˜π½)β€˜π‘†)))
8420, 83eqelssd 4004 1 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (π‘‚β€˜((clsβ€˜π½)β€˜π‘†)) = (π‘‚β€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  {crab 3430  Vcvv 3472   ∩ cin 3948   βŠ† wss 3949  {csn 4629  βˆͺ cuni 4909   ↦ cmpt 5232  β—‘ccnv 5676   β€œ cima 5680  β€˜cfv 6544  (class class class)co 7413  β„‚cc 11112  0cc0 11114  Basecbs 17150  Scalarcsca 17206  Β·π‘–cip 17208  TopOpenctopn 17373  0gc0g 17391  β„‚fldccnfld 21146  ocvcocv 21434  Topctop 22617  TopOnctopon 22634  TopSpctps 22656  Clsdccld 22742  clsccl 22744   Cn ccn 22950  Hauscha 23034  NrmGrpcngp 24308  β„‚Modcclm 24811  β„‚PreHilccph 24916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192  ax-addf 11193  ax-mulf 11194
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7674  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8151  df-tpos 8215  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-er 8707  df-map 8826  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9366  df-fi 9410  df-sup 9441  df-inf 9442  df-oi 9509  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-div 11878  df-nn 12219  df-2 12281  df-3 12282  df-4 12283  df-5 12284  df-6 12285  df-7 12286  df-8 12287  df-9 12288  df-n0 12479  df-z 12565  df-dec 12684  df-uz 12829  df-q 12939  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-ico 13336  df-icc 13337  df-fz 13491  df-fzo 13634  df-seq 13973  df-exp 14034  df-hash 14297  df-cj 15052  df-re 15053  df-im 15054  df-sqrt 15188  df-abs 15189  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-mulr 17217  df-starv 17218  df-sca 17219  df-vsca 17220  df-ip 17221  df-tset 17222  df-ple 17223  df-ds 17225  df-unif 17226  df-hom 17227  df-cco 17228  df-rest 17374  df-topn 17375  df-0g 17393  df-gsum 17394  df-topgen 17395  df-pt 17396  df-prds 17399  df-xrs 17454  df-qtop 17459  df-imas 17460  df-xps 17462  df-mre 17536  df-mrc 17537  df-acs 17539  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18707  df-submnd 18708  df-grp 18860  df-minusg 18861  df-sbg 18862  df-mulg 18989  df-subg 19041  df-ghm 19130  df-cntz 19224  df-cmn 19693  df-abl 19694  df-mgp 20031  df-rng 20049  df-ur 20078  df-ring 20131  df-cring 20132  df-oppr 20227  df-dvdsr 20250  df-unit 20251  df-invr 20281  df-dvr 20294  df-rhm 20365  df-subrng 20436  df-subrg 20461  df-drng 20504  df-staf 20598  df-srng 20599  df-lmod 20618  df-lmhm 20779  df-lvec 20860  df-sra 20932  df-rgmod 20933  df-psmet 21138  df-xmet 21139  df-met 21140  df-bl 21141  df-mopn 21142  df-cnfld 21147  df-phl 21400  df-ipf 21401  df-ocv 21437  df-top 22618  df-topon 22635  df-topsp 22657  df-bases 22671  df-cld 22745  df-cls 22747  df-cn 22953  df-cnp 22954  df-t1 23040  df-haus 23041  df-tx 23288  df-hmeo 23481  df-xms 24048  df-ms 24049  df-tms 24050  df-nm 24313  df-ngp 24314  df-tng 24315  df-nlm 24317  df-clm 24812  df-cph 24918  df-tcph 24919
This theorem is referenced by: (None)
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