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Theorem ocvlss 21092
Description: The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.)
Hypotheses
Ref Expression
ocvss.v 𝑉 = (Baseβ€˜π‘Š)
ocvss.o βŠ₯ = (ocvβ€˜π‘Š)
ocvlss.l 𝐿 = (LSubSpβ€˜π‘Š)
Assertion
Ref Expression
ocvlss ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘†) ∈ 𝐿)

Proof of Theorem ocvlss
Dummy variables π‘₯ π‘Ÿ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ocvss.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
2 ocvss.o . . . 4 βŠ₯ = (ocvβ€˜π‘Š)
31, 2ocvss 21090 . . 3 ( βŠ₯ β€˜π‘†) βŠ† 𝑉
43a1i 11 . 2 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘†) βŠ† 𝑉)
5 simpr 486 . . . 4 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
6 phllmod 21050 . . . . . 6 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
76adantr 482 . . . . 5 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ π‘Š ∈ LMod)
8 eqid 2733 . . . . . 6 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
91, 8lmod0vcl 20366 . . . . 5 (π‘Š ∈ LMod β†’ (0gβ€˜π‘Š) ∈ 𝑉)
107, 9syl 17 . . . 4 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (0gβ€˜π‘Š) ∈ 𝑉)
11 simpll 766 . . . . . 6 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ 𝑆) β†’ π‘Š ∈ PreHil)
125sselda 3945 . . . . . 6 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑉)
13 eqid 2733 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
14 eqid 2733 . . . . . . 7 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
15 eqid 2733 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
1613, 14, 1, 15, 8ip0l 21056 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘₯ ∈ 𝑉) β†’ ((0gβ€˜π‘Š)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
1711, 12, 16syl2anc 585 . . . . 5 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ 𝑆) β†’ ((0gβ€˜π‘Š)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
1817ralrimiva 3140 . . . 4 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ βˆ€π‘₯ ∈ 𝑆 ((0gβ€˜π‘Š)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
191, 14, 13, 15, 2elocv 21088 . . . 4 ((0gβ€˜π‘Š) ∈ ( βŠ₯ β€˜π‘†) ↔ (𝑆 βŠ† 𝑉 ∧ (0gβ€˜π‘Š) ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 ((0gβ€˜π‘Š)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
205, 10, 18, 19syl3anbrc 1344 . . 3 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (0gβ€˜π‘Š) ∈ ( βŠ₯ β€˜π‘†))
2120ne0d 4296 . 2 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘†) β‰  βˆ…)
225adantr 482 . . . 4 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ 𝑆 βŠ† 𝑉)
237adantr 482 . . . . 5 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ π‘Š ∈ LMod)
24 simpr1 1195 . . . . . 6 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
25 simpr2 1196 . . . . . . 7 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ 𝑦 ∈ ( βŠ₯ β€˜π‘†))
263, 25sselid 3943 . . . . . 6 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ 𝑦 ∈ 𝑉)
27 eqid 2733 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
28 eqid 2733 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
291, 13, 27, 28lmodvscl 20354 . . . . . 6 ((π‘Š ∈ LMod ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑉) β†’ (π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑉)
3023, 24, 26, 29syl3anc 1372 . . . . 5 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ (π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑉)
31 simpr3 1197 . . . . . 6 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ 𝑧 ∈ ( βŠ₯ β€˜π‘†))
323, 31sselid 3943 . . . . 5 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ 𝑧 ∈ 𝑉)
33 eqid 2733 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
341, 33lmodvacl 20351 . . . . 5 ((π‘Š ∈ LMod ∧ (π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) β†’ ((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧) ∈ 𝑉)
3523, 30, 32, 34syl3anc 1372 . . . 4 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ ((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧) ∈ 𝑉)
3611adantlr 714 . . . . . . 7 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ π‘Š ∈ PreHil)
3730adantr 482 . . . . . . 7 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ (π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑉)
3832adantr 482 . . . . . . 7 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ 𝑧 ∈ 𝑉)
3912adantlr 714 . . . . . . 7 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑉)
40 eqid 2733 . . . . . . . 8 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
4113, 14, 1, 33, 40ipdir 21059 . . . . . . 7 ((π‘Š ∈ PreHil ∧ ((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦) ∈ 𝑉 ∧ 𝑧 ∈ 𝑉 ∧ π‘₯ ∈ 𝑉)) β†’ (((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)π‘₯) = (((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)π‘₯)(+gβ€˜(Scalarβ€˜π‘Š))(𝑧(Β·π‘–β€˜π‘Š)π‘₯)))
4236, 37, 38, 39, 41syl13anc 1373 . . . . . 6 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ (((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)π‘₯) = (((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)π‘₯)(+gβ€˜(Scalarβ€˜π‘Š))(𝑧(Β·π‘–β€˜π‘Š)π‘₯)))
4324adantr 482 . . . . . . . . 9 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
4426adantr 482 . . . . . . . . 9 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ 𝑦 ∈ 𝑉)
45 eqid 2733 . . . . . . . . . 10 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
4613, 14, 1, 28, 27, 45ipass 21065 . . . . . . . . 9 ((π‘Š ∈ PreHil ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ 𝑉 ∧ π‘₯ ∈ 𝑉)) β†’ ((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)π‘₯) = (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)π‘₯)))
4736, 43, 44, 39, 46syl13anc 1373 . . . . . . . 8 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ ((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)π‘₯) = (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)π‘₯)))
481, 14, 13, 15, 2ocvi 21089 . . . . . . . . . 10 ((𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ π‘₯ ∈ 𝑆) β†’ (𝑦(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
4925, 48sylan 581 . . . . . . . . 9 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ (𝑦(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
5049oveq2d 7374 . . . . . . . 8 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)π‘₯)) = (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))))
5123adantr 482 . . . . . . . . . 10 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ π‘Š ∈ LMod)
5213lmodring 20344 . . . . . . . . . 10 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Ring)
5351, 52syl 17 . . . . . . . . 9 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ (Scalarβ€˜π‘Š) ∈ Ring)
5428, 45, 15ringrz 20017 . . . . . . . . 9 (((Scalarβ€˜π‘Š) ∈ Ring ∧ π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
5553, 43, 54syl2anc 585 . . . . . . . 8 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ (π‘Ÿ(.rβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
5647, 50, 553eqtrd 2777 . . . . . . 7 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ ((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
571, 14, 13, 15, 2ocvi 21089 . . . . . . . 8 ((𝑧 ∈ ( βŠ₯ β€˜π‘†) ∧ π‘₯ ∈ 𝑆) β†’ (𝑧(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
5831, 57sylan 581 . . . . . . 7 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ (𝑧(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
5956, 58oveq12d 7376 . . . . . 6 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ (((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)π‘₯)(+gβ€˜(Scalarβ€˜π‘Š))(𝑧(Β·π‘–β€˜π‘Š)π‘₯)) = ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))))
6013lmodfgrp 20345 . . . . . . 7 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Grp)
6128, 15grpidcl 18783 . . . . . . . 8 ((Scalarβ€˜π‘Š) ∈ Grp β†’ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
6228, 40, 15grplid 18785 . . . . . . . 8 (((Scalarβ€˜π‘Š) ∈ Grp ∧ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š))) β†’ ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
6361, 62mpdan 686 . . . . . . 7 ((Scalarβ€˜π‘Š) ∈ Grp β†’ ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
6451, 60, 633syl 18 . . . . . 6 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ ((0gβ€˜(Scalarβ€˜π‘Š))(+gβ€˜(Scalarβ€˜π‘Š))(0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
6542, 59, 643eqtrd 2777 . . . . 5 ((((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) ∧ π‘₯ ∈ 𝑆) β†’ (((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
6665ralrimiva 3140 . . . 4 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ βˆ€π‘₯ ∈ 𝑆 (((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
671, 14, 13, 15, 2elocv 21088 . . . 4 (((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜π‘†) ↔ (𝑆 βŠ† 𝑉 ∧ ((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧) ∈ 𝑉 ∧ βˆ€π‘₯ ∈ 𝑆 (((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
6822, 35, 66, 67syl3anbrc 1344 . . 3 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ (π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ 𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ 𝑧 ∈ ( βŠ₯ β€˜π‘†))) β†’ ((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜π‘†))
6968ralrimivvva 3197 . 2 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ ( βŠ₯ β€˜π‘†)βˆ€π‘§ ∈ ( βŠ₯ β€˜π‘†)((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜π‘†))
70 ocvlss.l . . 3 𝐿 = (LSubSpβ€˜π‘Š)
7113, 28, 1, 33, 27, 70islss 20410 . 2 (( βŠ₯ β€˜π‘†) ∈ 𝐿 ↔ (( βŠ₯ β€˜π‘†) βŠ† 𝑉 ∧ ( βŠ₯ β€˜π‘†) β‰  βˆ… ∧ βˆ€π‘Ÿ ∈ (Baseβ€˜(Scalarβ€˜π‘Š))βˆ€π‘¦ ∈ ( βŠ₯ β€˜π‘†)βˆ€π‘§ ∈ ( βŠ₯ β€˜π‘†)((π‘Ÿ( ·𝑠 β€˜π‘Š)𝑦)(+gβ€˜π‘Š)𝑧) ∈ ( βŠ₯ β€˜π‘†)))
724, 21, 69, 71syl3anbrc 1344 1 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘†) ∈ 𝐿)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061   βŠ† wss 3911  βˆ…c0 4283  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  .rcmulr 17139  Scalarcsca 17141   ·𝑠 cvsca 17142  Β·π‘–cip 17143  0gc0g 17326  Grpcgrp 18753  Ringcrg 19969  LModclmod 20336  LSubSpclss 20407  PreHilcphl 21044  ocvcocv 21080
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-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  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-sca 17154  df-vsca 17155  df-ip 17156  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-ghm 19011  df-mgp 19902  df-ring 19971  df-lmod 20338  df-lss 20408  df-lmhm 20498  df-lvec 20579  df-sra 20649  df-rgmod 20650  df-phl 21046  df-ocv 21083
This theorem is referenced by:  ocvin  21094  ocvlsp  21096  csslss  21111  pjdm2  21133  pjff  21134  pjf2  21136  pjfo  21137  ocvpj  21139  pjthlem2  24818  pjth  24819
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