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Theorem clsocv 24767
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 24690 . . . . . . . 8 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ NrmGrp)
2 ngptps 24111 . . . . . . . 8 (π‘Š ∈ NrmGrp β†’ π‘Š ∈ TopSp)
31, 2syl 17 . . . . . . 7 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ TopSp)
43adantr 482 . . . . . 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 22436 . . . . . 6 (π‘Š ∈ TopSp ↔ 𝐽 ∈ (TopOnβ€˜π‘‰))
84, 7sylib 217 . . . . 5 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝐽 ∈ (TopOnβ€˜π‘‰))
9 topontop 22415 . . . . 5 (𝐽 ∈ (TopOnβ€˜π‘‰) β†’ 𝐽 ∈ Top)
108, 9syl 17 . . . 4 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝐽 ∈ Top)
11 simpr 486 . . . . 5 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
12 toponuni 22416 . . . . . 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 2733 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
1615sscls 22560 . . . 4 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
1710, 14, 16syl2anc 585 . . 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 21226 . . 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 22563 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† βˆͺ 𝐽)
2210, 14, 21syl2anc 585 . . . . 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 482 . . 3 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ ((clsβ€˜π½)β€˜π‘†) βŠ† 𝑉)
255, 18ocvss 21223 . . . . 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 2764 . . . . . . . 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 2798 . . . . . . 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 22551 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑆 βŠ† βˆͺ 𝐽) β†’ ((clsβ€˜π½)β€˜π‘†) ∈ (Clsdβ€˜π½))
3810, 14, 37syl2anc 585 . . . . . . . . 9 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ((clsβ€˜π½)β€˜π‘†) ∈ (Clsdβ€˜π½))
3938adantr 482 . . . . . . . 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 2733 . . . . . . . . . . 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 2733 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) = (TopOpenβ€˜β„‚fld)
45 eqid 2733 . . . . . . . . . . 11 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
46 simpll 766 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ π‘Š ∈ β„‚PreHil)
478adantr 482 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‰))
4847, 47, 27cnmptc 23166 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (𝑦 ∈ 𝑉 ↦ π‘₯) ∈ (𝐽 Cn 𝐽))
4947cnmptid 23165 . . . . . . . . . . 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 24764 . . . . . . . . . 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 24301 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) ∈ Haus
52 cphclm 24706 . . . . . . . . . . . . . 14 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ β„‚Mod)
53 eqid 2733 . . . . . . . . . . . . . . 15 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5453clm0 24588 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . 12 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ 0 = (0gβ€˜(Scalarβ€˜π‘Š)))
57 0cn 11206 . . . . . . . . . . . 12 0 ∈ β„‚
5856, 57eqeltrrdi 2843 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ β„‚)
59 unicntop 24302 . . . . . . . . . . . 12 β„‚ = βˆͺ (TopOpenβ€˜β„‚fld)
6059sncld 22875 . . . . . . . . . . 11 (((TopOpenβ€˜β„‚fld) ∈ Haus ∧ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ β„‚) β†’ {(0gβ€˜(Scalarβ€˜π‘Š))} ∈ (Clsdβ€˜(TopOpenβ€˜β„‚fld)))
6151, 58, 60sylancr 588 . . . . . . . . . 10 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ {(0gβ€˜(Scalarβ€˜π‘Š))} ∈ (Clsdβ€˜(TopOpenβ€˜β„‚fld)))
62 cnclima 22772 . . . . . . . . . 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 585 . . . . . . . . 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 2839 . . . . . . . 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 22547 . . . . . . . 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 585 . . . . . . 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 2835 . . . . . 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 482 . . . . . . 7 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ 𝑆 βŠ† ((clsβ€˜π½)β€˜π‘†))
69 eqid 2733 . . . . . . . . . 10 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
705, 45, 53, 69, 18ocvi 21222 . . . . . . . . 9 ((π‘₯ ∈ (π‘‚β€˜π‘†) ∧ 𝑦 ∈ 𝑆) β†’ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
7170ralrimiva 3147 . . . . . . . 8 (π‘₯ ∈ (π‘‚β€˜π‘†) β†’ βˆ€π‘¦ ∈ 𝑆 (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
7271adantl 483 . . . . . . 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 584 . . . . . 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 22576 . . . . . 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 585 . . . . 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 3465 . . . 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 21221 . . 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 1344 . 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 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  {crab 3433  Vcvv 3475   ∩ 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 7409  β„‚cc 11108  0cc0 11110  Basecbs 17144  Scalarcsca 17200  Β·π‘–cip 17202  TopOpenctopn 17367  0gc0g 17385  β„‚fldccnfld 20944  ocvcocv 21213  Topctop 22395  TopOnctopon 22412  TopSpctps 22434  Clsdccld 22520  clsccl 22522   Cn ccn 22728  Hauscha 22812  NrmGrpcngp 24086  β„‚Modcclm 24578  β„‚PreHilccph 24683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188  ax-addf 11189  ax-mulf 11190
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-om 7856  df-1st 7975  df-2nd 7976  df-supp 8147  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-fsupp 9362  df-fi 9406  df-sup 9437  df-inf 9438  df-oi 9505  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ico 13330  df-icc 13331  df-fz 13485  df-fzo 13628  df-seq 13967  df-exp 14028  df-hash 14291  df-cj 15046  df-re 15047  df-im 15048  df-sqrt 15182  df-abs 15183  df-struct 17080  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-ress 17174  df-plusg 17210  df-mulr 17211  df-starv 17212  df-sca 17213  df-vsca 17214  df-ip 17215  df-tset 17216  df-ple 17217  df-ds 17219  df-unif 17220  df-hom 17221  df-cco 17222  df-rest 17368  df-topn 17369  df-0g 17387  df-gsum 17388  df-topgen 17389  df-pt 17390  df-prds 17393  df-xrs 17448  df-qtop 17453  df-imas 17454  df-xps 17456  df-mre 17530  df-mrc 17531  df-acs 17533  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-submnd 18672  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mulg 18951  df-subg 19003  df-ghm 19090  df-cntz 19181  df-cmn 19650  df-abl 19651  df-mgp 19988  df-ur 20005  df-ring 20058  df-cring 20059  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-invr 20202  df-dvr 20215  df-rnghom 20251  df-subrg 20317  df-drng 20359  df-staf 20453  df-srng 20454  df-lmod 20473  df-lmhm 20633  df-lvec 20714  df-sra 20785  df-rgmod 20786  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-cnfld 20945  df-phl 21179  df-ipf 21180  df-ocv 21216  df-top 22396  df-topon 22413  df-topsp 22435  df-bases 22449  df-cld 22523  df-cls 22525  df-cn 22731  df-cnp 22732  df-t1 22818  df-haus 22819  df-tx 23066  df-hmeo 23259  df-xms 23826  df-ms 23827  df-tms 23828  df-nm 24091  df-ngp 24092  df-tng 24093  df-nlm 24095  df-clm 24579  df-cph 24685  df-tcph 24686
This theorem is referenced by: (None)
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