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Theorem clsocv 24630
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 24553 . . . . . . . 8 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ NrmGrp)
2 ngptps 23974 . . . . . . . 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 22299 . . . . . 6 (π‘Š ∈ TopSp ↔ 𝐽 ∈ (TopOnβ€˜π‘‰))
84, 7sylib 217 . . . . 5 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝐽 ∈ (TopOnβ€˜π‘‰))
9 topontop 22278 . . . . 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 22279 . . . . . 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 3985 . . . 4 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† βˆͺ 𝐽)
15 eqid 2733 . . . . 5 βˆͺ 𝐽 = βˆͺ 𝐽
1615sscls 22423 . . . 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 21093 . . 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 22426 . . . . . 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 3986 . . . 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 21090 . . . . 5 (π‘‚β€˜π‘†) βŠ† 𝑉
2625a1i 11 . . . 4 ((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (π‘‚β€˜π‘†) βŠ† 𝑉)
2726sselda 3945 . . 3 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ π‘₯ ∈ 𝑉)
28 df-ss 3928 . . . . . . . . . 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 4172 . . . . . . . 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 4270 . . . . . . . . . 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 4170 . . . . . . . . 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 4180 . . . . . . . . 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 4270 . . . . . . . 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 22414 . . . . . . . . . 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 6856 . . . . . . . . . 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 6192 . . . . . . . . . 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 23029 . . . . . . . . . . 11 (((π‘Š ∈ β„‚PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ (π‘‚β€˜π‘†)) β†’ (𝑦 ∈ 𝑉 ↦ π‘₯) ∈ (𝐽 Cn 𝐽))
4947cnmptid 23028 . . . . . . . . . . 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 24627 . . . . . . . . . 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 24164 . . . . . . . . . . 11 (TopOpenβ€˜β„‚fld) ∈ Haus
52 cphclm 24569 . . . . . . . . . . . . . 14 (π‘Š ∈ β„‚PreHil β†’ π‘Š ∈ β„‚Mod)
53 eqid 2733 . . . . . . . . . . . . . . 15 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
5453clm0 24451 . . . . . . . . . . . . . 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 11152 . . . . . . . . . . . 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 24165 . . . . . . . . . . . 12 β„‚ = βˆͺ (TopOpenβ€˜β„‚fld)
6059sncld 22738 . . . . . . . . . . 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 22635 . . . . . . . . . 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 22410 . . . . . . . 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 21089 . . . . . . . . 9 ((π‘₯ ∈ (π‘‚β€˜π‘†) ∧ 𝑦 ∈ 𝑆) β†’ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
7170ralrimiva 3140 . . . . . . . 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 4031 . . . . . . 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 22439 . . . . . 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 4038 . . . . . 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 3962 . . . 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 3435 . . . 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 21088 . . 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 3966 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 3061  {crab 3406  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  {csn 4587  βˆͺ cuni 4866   ↦ cmpt 5189  β—‘ccnv 5633   β€œ cima 5637  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11054  0cc0 11056  Basecbs 17088  Scalarcsca 17141  Β·π‘–cip 17143  TopOpenctopn 17308  0gc0g 17326  β„‚fldccnfld 20812  ocvcocv 21080  Topctop 22258  TopOnctopon 22275  TopSpctps 22297  Clsdccld 22383  clsccl 22385   Cn ccn 22591  Hauscha 22675  NrmGrpcngp 23949  β„‚Modcclm 24441  β„‚PreHilccph 24546
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134  ax-addf 11135  ax-mulf 11136
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7618  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-tpos 8158  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-2o 8414  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-fsupp 9309  df-fi 9352  df-sup 9383  df-inf 9384  df-oi 9451  df-card 9880  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-q 12879  df-rp 12921  df-xneg 13038  df-xadd 13039  df-xmul 13040  df-ico 13276  df-icc 13277  df-fz 13431  df-fzo 13574  df-seq 13913  df-exp 13974  df-hash 14237  df-cj 14990  df-re 14991  df-im 14992  df-sqrt 15126  df-abs 15127  df-struct 17024  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-mulr 17152  df-starv 17153  df-sca 17154  df-vsca 17155  df-ip 17156  df-tset 17157  df-ple 17158  df-ds 17160  df-unif 17161  df-hom 17162  df-cco 17163  df-rest 17309  df-topn 17310  df-0g 17328  df-gsum 17329  df-topgen 17330  df-pt 17331  df-prds 17334  df-xrs 17389  df-qtop 17394  df-imas 17395  df-xps 17397  df-mre 17471  df-mrc 17472  df-acs 17474  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-submnd 18607  df-grp 18756  df-minusg 18757  df-sbg 18758  df-mulg 18878  df-subg 18930  df-ghm 19011  df-cntz 19102  df-cmn 19569  df-abl 19570  df-mgp 19902  df-ur 19919  df-ring 19971  df-cring 19972  df-oppr 20054  df-dvdsr 20075  df-unit 20076  df-invr 20106  df-dvr 20117  df-rnghom 20153  df-drng 20199  df-subrg 20234  df-staf 20318  df-srng 20319  df-lmod 20338  df-lmhm 20498  df-lvec 20579  df-sra 20649  df-rgmod 20650  df-psmet 20804  df-xmet 20805  df-met 20806  df-bl 20807  df-mopn 20808  df-cnfld 20813  df-phl 21046  df-ipf 21047  df-ocv 21083  df-top 22259  df-topon 22276  df-topsp 22298  df-bases 22312  df-cld 22386  df-cls 22388  df-cn 22594  df-cnp 22595  df-t1 22681  df-haus 22682  df-tx 22929  df-hmeo 23122  df-xms 23689  df-ms 23690  df-tms 23691  df-nm 23954  df-ngp 23955  df-tng 23956  df-nlm 23958  df-clm 24442  df-cph 24548  df-tcph 24549
This theorem is referenced by: (None)
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