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Theorem lsmcss 21631
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 3981 . . . . . 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 21569 . . . . . . . 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 21609 . . . . . . . 8 ( βŠ₯ β€˜π‘†) βŠ† 𝑉
109a1i 11 . . . . . . 7 (πœ‘ β†’ ( βŠ₯ β€˜π‘†) βŠ† 𝑉)
11 eqid 2728 . . . . . . . 8 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
12 lsmcss.p . . . . . . . 8 βŠ• = (LSSumβ€˜π‘Š)
137, 11, 12lsmelvalx 19602 . . . . . . 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 1368 . . . . . 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 724 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ π‘Š ∈ PreHil)
176ad2antrr 724 . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . 16 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ 𝑦 ∈ 𝑆)
1917, 18sseldd 3983 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ 𝑦 ∈ 𝑉)
20 simplrr 776 . . . . . . . . . . . . . . . 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 3980 . . . . . . . . . . . . . . 15 (((πœ‘ ∧ (𝑦 ∈ 𝑆 ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) β†’ 𝑧 ∈ 𝑉)
22 eqid 2728 . . . . . . . . . . . . . . . 16 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
23 eqid 2728 . . . . . . . . . . . . . . . 16 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
24 eqid 2728 . . . . . . . . . . . . . . . 16 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
2522, 23, 7, 11, 24ipdir 21578 . . . . . . . . . . . . . . 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 1369 . . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . . . . . 18 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
287, 23, 22, 27, 8ocvi 21608 . . . . . . . . . . . . . . . . 17 ((𝑧 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑦 ∈ 𝑆) β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑦) = (0gβ€˜(Scalarβ€˜π‘Š)))
2920, 18, 28syl2anc 582 . . . . . . . . . . . . . . . 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 21574 . . . . . . . . . . . . . . . . 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 1368 . . . . . . . . . . . . . . . 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 7441 . . . . . . . . . . . . . 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 20759 . . . . . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . . . . 17 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
3822, 23, 7, 37ipcl 21572 . . . . . . . . . . . . . . . 16 ((π‘Š ∈ PreHil ∧ 𝑧 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) β†’ (𝑧(Β·π‘–β€˜π‘Š)𝑧) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
3916, 21, 21, 38syl3anc 1368 . . . . . . . . . . . . . . 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 18931 . . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . . 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 2772 . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . 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 21608 . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . 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 2770 . . . . . . . . . . . 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 2728 . . . . . . . . . . . . . 14 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
4822, 23, 7, 27, 47ipeq0 21577 . . . . . . . . . . . . 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 582 . . . . . . . . . . . 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 7442 . . . . . . . . . 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 20757 . . . . . . . . . . . . 13 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
535, 52syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ π‘Š ∈ Grp)
5453ad2antrr 724 . . . . . . . . . . 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 18932 . . . . . . . . . . 11 ((π‘Š ∈ Grp ∧ 𝑦 ∈ 𝑉) β†’ (𝑦(+gβ€˜π‘Š)(0gβ€˜π‘Š)) = 𝑦)
5654, 19, 55syl2anc 582 . . . . . . . . . 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 2768 . . . . . . . . 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 2829 . . . . . . . 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 411 . . . . . . 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 2817 . . . . . . . 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 2817 . . . . . . . 8 (π‘₯ = (𝑦(+gβ€˜π‘Š)𝑧) β†’ (π‘₯ ∈ 𝑆 ↔ (𝑦(+gβ€˜π‘Š)𝑧) ∈ 𝑆))
6260, 61imbi12d 343 . . . . . . 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 246 . . . . . 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 3209 . . . . 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 3988 . 2 (πœ‘ β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) βŠ† 𝑆)
68 lsmcss.c . . . 4 𝐢 = (ClSubSpβ€˜π‘Š)
697, 68, 8iscss2 21625 . . 3 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (𝑆 ∈ 𝐢 ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) βŠ† 𝑆))
703, 6, 69syl2anc 582 . 2 (πœ‘ β†’ (𝑆 ∈ 𝐢 ↔ ( βŠ₯ β€˜( βŠ₯ β€˜π‘†)) βŠ† 𝑆))
7167, 70mpbird 256 1 (πœ‘ β†’ 𝑆 ∈ 𝐢)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3067   βŠ† wss 3949  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  +gcplusg 17240  Scalarcsca 17243  Β·π‘–cip 17245  0gc0g 17428  Grpcgrp 18897  LSSumclsm 19596  LModclmod 20750  PreHilcphl 21563  ocvcocv 21599  ClSubSpccss 21600
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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-tpos 8238  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-map 8853  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-ip 17258  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-mhm 18747  df-grp 18900  df-ghm 19175  df-lsm 19598  df-mgp 20082  df-ur 20129  df-ring 20182  df-oppr 20280  df-rhm 20418  df-staf 20732  df-srng 20733  df-lmod 20752  df-lmhm 20914  df-lvec 20995  df-sra 21065  df-rgmod 21066  df-phl 21565  df-ocv 21602  df-css 21603
This theorem is referenced by:  pjcss  21657
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