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Theorem lsmcss 21112
Description: A subset of a pre-Hilbert space whose double orthocomplement has a projection decomposition is a closed subspace. This is the core of the proof that a topologically closed subspace is algebraically closed in a Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
lsmcss.c 𝐢 = (ClSubSpβ€˜π‘Š)
lsmcss.j 𝑉 = (Baseβ€˜π‘Š)
lsmcss.o βŠ₯ = (ocvβ€˜π‘Š)
lsmcss.p βŠ• = (LSSumβ€˜π‘Š)
lsmcss.1 (πœ‘ β†’ π‘Š ∈ PreHil)
lsmcss.2 (πœ‘ β†’ 𝑆 βŠ† 𝑉)
lsmcss.3 (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) βŠ† (𝑆 βŠ• ( βŠ₯ β€˜π‘†)))
Assertion
Ref Expression
lsmcss (πœ‘ β†’ 𝑆 ∈ 𝐢)

Proof of Theorem lsmcss
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lsmcss.3 . . . . . . 7 (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) βŠ† (𝑆 βŠ• ( βŠ₯ β€˜π‘†)))
21sseld 3944 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ π‘₯ ∈ (𝑆 βŠ• ( βŠ₯ β€˜π‘†))))
3 lsmcss.1 . . . . . . . 8 (πœ‘ β†’ π‘Š ∈ PreHil)
4 phllmod 21050 . . . . . . . 8 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
53, 4syl 17 . . . . . . 7 (πœ‘ β†’ π‘Š ∈ LMod)
6 lsmcss.2 . . . . . . 7 (πœ‘ β†’ 𝑆 βŠ† 𝑉)
7 lsmcss.j . . . . . . . . 9 𝑉 = (Baseβ€˜π‘Š)
8 lsmcss.o . . . . . . . . 9 βŠ₯ = (ocvβ€˜π‘Š)
97, 8ocvss 21090 . . . . . . . 8 ( βŠ₯ β€˜π‘†) βŠ† 𝑉
109a1i 11 . . . . . . 7 (πœ‘ β†’ ( βŠ₯ β€˜π‘†) βŠ† 𝑉)
11 eqid 2733 . . . . . . . 8 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
12 lsmcss.p . . . . . . . 8 βŠ• = (LSSumβ€˜π‘Š)
137, 11, 12lsmelvalx 19427 . . . . . . 7 ((π‘Š ∈ LMod ∧ 𝑆 βŠ† 𝑉 ∧ ( βŠ₯ β€˜π‘†) βŠ† 𝑉) β†’ (π‘₯ ∈ (𝑆 βŠ• ( βŠ₯ β€˜π‘†)) ↔ βˆƒπ‘¦ ∈ 𝑆 βˆƒπ‘§ ∈ ( βŠ₯ β€˜π‘†)π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧)))
145, 6, 10, 13syl3anc 1372 . . . . . 6 (πœ‘ β†’ (π‘₯ ∈ (𝑆 βŠ• ( βŠ₯ β€˜π‘†)) ↔ βˆƒπ‘¦ ∈ 𝑆 βˆƒπ‘§ ∈ ( βŠ₯ β€˜π‘†)π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧)))
152, 14sylibd 238 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ βˆƒπ‘¦ ∈ 𝑆 βˆƒπ‘§ ∈ ( βŠ₯ β€˜π‘†)π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧)))
163ad2antrr 725 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ π‘Š ∈ PreHil)
176ad2antrr 725 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ 𝑆 βŠ† 𝑉)
18 simplrl 776 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ 𝑦 ∈ 𝑆)
1917, 18sseldd 3946 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ 𝑦 ∈ 𝑉)
20 simplrr 777 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ 𝑧 ∈ ( βŠ₯ β€˜π‘†))
219, 20sselid 3943 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ 𝑧 ∈ 𝑉)
22 eqid 2733 . . . . . . . . . . . . . . . 16 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
23 eqid 2733 . . . . . . . . . . . . . . . 16 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
24 eqid 2733 . . . . . . . . . . . . . . . 16 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
2522, 23, 7, 11, 24ipdir 21059 . . . . . . . . . . . . . . 15 ((π‘Š ∈ PreHil ∧ (𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉)) β†’ ((𝑦(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)𝑧) = ((𝑦(Β·π‘–β€˜π‘Š)𝑧)(+gβ€˜(Scalarβ€˜π‘Š))(𝑧(Β·π‘–β€˜π‘Š)𝑧)))
2616, 19, 21, 21, 25syl13anc 1373 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ ((𝑦(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)𝑧) = ((𝑦(Β·π‘–β€˜π‘Š)𝑧)(+gβ€˜(Scalarβ€˜π‘Š))(𝑧(Β·π‘–β€˜π‘Š)𝑧)))
27 eqid 2733 . . . . . . . . . . . . . . . . . 18 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
287, 23, 22, 27, 8ocvi 21089 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑦 ∈ 𝑆) β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
2920, 18, 28syl2anc 585 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
3022, 23, 7, 27iporthcom 21055 . . . . . . . . . . . . . . . . 17 ((π‘Š ∈ PreHil ∧ 𝑧 ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) β†’ ((𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ (𝑦(Β·π‘–β€˜π‘Š)𝑧) = (0gβ€˜(Scalarβ€˜π‘Š))))
3116, 21, 19, 30syl3anc 1372 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ ((𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ (𝑦(Β·π‘–β€˜π‘Š)𝑧) = (0gβ€˜(Scalarβ€˜π‘Š))))
3229, 31mpbid 231 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (𝑦(Β·π‘–β€˜π‘Š)𝑧) = (0gβ€˜(Scalarβ€˜π‘Š)))
3332oveq1d 7373 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ ((𝑦(Β·π‘–β€˜π‘Š)𝑧)(+gβ€˜(Scalarβ€˜π‘Š))(𝑧(Β·π‘–β€˜π‘Š)𝑧)) = ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(𝑧(Β·π‘–β€˜π‘Š)𝑧)))
3416, 4syl 17 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ π‘Š ∈ LMod)
3522lmodfgrp 20345 . . . . . . . . . . . . . . . 16 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Grp)
3634, 35syl 17 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (Scalarβ€˜π‘Š) ∈ Grp)
37 eqid 2733 . . . . . . . . . . . . . . . . 17 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
3822, 23, 7, 37ipcl 21053 . . . . . . . . . . . . . . . 16 ((π‘Š ∈ PreHil ∧ 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑧) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
3916, 21, 21, 38syl3anc 1372 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑧) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
4037, 24, 27grplid 18785 . . . . . . . . . . . . . . 15 (((Scalarβ€˜π‘Š) ∈ Grp ∧ (𝑧(Β·π‘–β€˜π‘Š)𝑧) ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(𝑧(Β·π‘–β€˜π‘Š)𝑧)) = (𝑧(Β·π‘–β€˜π‘Š)𝑧))
4136, 39, 40syl2anc 585 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(𝑧(Β·π‘–β€˜π‘Š)𝑧)) = (𝑧(Β·π‘–β€˜π‘Š)𝑧))
4226, 33, 413eqtrd 2777 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ ((𝑦(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)𝑧) = (𝑧(Β·π‘–β€˜π‘Š)𝑧))
43 simpr 486 . . . . . . . . . . . . . 14 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)))
447, 23, 22, 27, 8ocvi 21089 . . . . . . . . . . . . . 14 (((𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†)) β†’ ((𝑦(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)𝑧) = (0gβ€˜(Scalarβ€˜π‘Š)))
4543, 20, 44syl2anc 585 . . . . . . . . . . . . 13 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ ((𝑦(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)𝑧) = (0gβ€˜(Scalarβ€˜π‘Š)))
4642, 45eqtr3d 2775 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑧) = (0gβ€˜(Scalarβ€˜π‘Š)))
47 eqid 2733 . . . . . . . . . . . . . 14 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
4822, 23, 7, 27, 47ipeq0 21058 . . . . . . . . . . . . 13 ((π‘Š ∈ PreHil ∧ 𝑧 ∈ 𝑉) β†’ ((𝑧(Β·π‘–β€˜π‘Š)𝑧) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ 𝑧 = (0gβ€˜π‘Š)))
4916, 21, 48syl2anc 585 . . . . . . . . . . . 12 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ ((𝑧(Β·π‘–β€˜π‘Š)𝑧) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ 𝑧 = (0gβ€˜π‘Š)))
5046, 49mpbid 231 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ 𝑧 = (0gβ€˜π‘Š))
5150oveq2d 7374 . . . . . . . . . 10 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (𝑦(+gβ€˜π‘Š)𝑧) = (𝑦(+gβ€˜π‘Š)(0gβ€˜π‘Š)))
52 lmodgrp 20343 . . . . . . . . . . . . 13 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
535, 52syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ π‘Š ∈ Grp)
5453ad2antrr 725 . . . . . . . . . . 11 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ π‘Š ∈ Grp)
557, 11, 47grprid 18786 . . . . . . . . . . 11 ((π‘Š ∈ Grp ∧ 𝑦 ∈ 𝑉) β†’ (𝑦(+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 𝑦)
5654, 19, 55syl2anc 585 . . . . . . . . . 10 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (𝑦(+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 𝑦)
5751, 56eqtrd 2773 . . . . . . . . 9 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (𝑦(+gβ€˜π‘Š)𝑧) = 𝑦)
5857, 18eqeltrd 2834 . . . . . . . 8 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑆)
5958ex 414 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ ((𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑆))
60 eleq1 2822 . . . . . . . 8 (π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧) β†’ (π‘₯ ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) ↔ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))))
61 eleq1 2822 . . . . . . . 8 (π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧) β†’ (π‘₯ ∈ 𝑆 ↔ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑆))
6260, 61imbi12d 345 . . . . . . 7 (π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧) β†’ ((π‘₯ ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ π‘₯ ∈ 𝑆) ↔ ((𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑆)))
6359, 62syl5ibrcom 247 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ (π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧) β†’ (π‘₯ ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ π‘₯ ∈ 𝑆)))
6463rexlimdvva 3202 . . . . 5 (πœ‘ β†’ (βˆƒπ‘¦ ∈ 𝑆 βˆƒπ‘§ ∈ ( βŠ₯ β€˜π‘†)π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧) β†’ (π‘₯ ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ π‘₯ ∈ 𝑆)))
6515, 64syld 47 . . . 4 (πœ‘ β†’ (π‘₯ ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ (π‘₯ ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ π‘₯ ∈ 𝑆)))
6665pm2.43d 53 . . 3 (πœ‘ β†’ (π‘₯ ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) β†’ π‘₯ ∈ 𝑆))
6766ssrdv 3951 . 2 (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) βŠ† 𝑆)
68 lsmcss.c . . . 4 𝐢 = (ClSubSpβ€˜π‘Š)
697, 68, 8iscss2 21106 . . 3 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (𝑆 ∈ 𝐢 ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) βŠ† 𝑆))
703, 6, 69syl2anc 585 . 2 (πœ‘ β†’ (𝑆 ∈ 𝐢 ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) βŠ† 𝑆))
7167, 70mpbird 257 1 (πœ‘ β†’ 𝑆 ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070   βŠ† wss 3911  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  Scalarcsca 17141  Β·π‘–cip 17143  0gc0g 17326  Grpcgrp 18753  LSSumclsm 19421  LModclmod 20336  PreHilcphl 21044  ocvcocv 21080  ClSubSpccss 21081
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
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-op 4594  df-uni 4867  df-iun 4957  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-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-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-tpos 8158  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  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-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-plusg 17151  df-mulr 17152  df-sca 17154  df-vsca 17155  df-ip 17156  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-mhm 18606  df-grp 18756  df-ghm 19011  df-lsm 19423  df-mgp 19902  df-ur 19919  df-ring 19971  df-oppr 20054  df-rnghom 20153  df-staf 20318  df-srng 20319  df-lmod 20338  df-lmhm 20498  df-lvec 20579  df-sra 20649  df-rgmod 20650  df-phl 21046  df-ocv 21083  df-css 21084
This theorem is referenced by:  pjcss  21138
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