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Theorem ocvlss 21561
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 21559 . . 3 ( βŠ₯ β€˜π‘†) βŠ† 𝑉
43a1i 11 . 2 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘†) βŠ† 𝑉)
5 simpr 484 . . . 4 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
6 phllmod 21519 . . . . . 6 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
76adantr 480 . . . . 5 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ π‘Š ∈ LMod)
8 eqid 2726 . . . . . 6 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
91, 8lmod0vcl 20735 . . . . 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 764 . . . . . 6 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ 𝑆) β†’ π‘Š ∈ PreHil)
125sselda 3977 . . . . . 6 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑉)
13 eqid 2726 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
14 eqid 2726 . . . . . . 7 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
15 eqid 2726 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
1613, 14, 1, 15, 8ip0l 21525 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘₯ ∈ 𝑉) β†’ ((0gβ€˜π‘Š)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
1711, 12, 16syl2anc 583 . . . . 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 21557 . . . 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 1340 . . 3 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ (0gβ€˜π‘Š) ∈ ( βŠ₯ β€˜π‘†))
2120ne0d 4330 . 2 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘†) β‰  βˆ…)
225adantr 480 . . . 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 480 . . . . 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 1191 . . . . . 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 1192 . . . . . . 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 3975 . . . . . 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 2726 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
28 eqid 2726 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
291, 13, 27, 28lmodvscl 20722 . . . . . 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 1368 . . . . 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 1193 . . . . . 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 3975 . . . . 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 2726 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
341, 33lmodvacl 20719 . . . . 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 1368 . . . 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 712 . . . . . . 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 480 . . . . . . 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 480 . . . . . . 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 712 . . . . . . 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 2726 . . . . . . . 8 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
4113, 14, 1, 33, 40ipdir 21528 . . . . . . 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 1369 . . . . . 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 480 . . . . . . . . 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 480 . . . . . . . . 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 2726 . . . . . . . . . 10 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
4613, 14, 1, 28, 27, 45ipass 21534 . . . . . . . . 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 1369 . . . . . . . 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 21558 . . . . . . . . . 10 ((𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ π‘₯ ∈ 𝑆) β†’ (𝑦(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
4925, 48sylan 579 . . . . . . . . 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 7420 . . . . . . . 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 480 . . . . . . . . . 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 20712 . . . . . . . . . 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 20191 . . . . . . . . 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 583 . . . . . . . 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 2770 . . . . . . 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 21558 . . . . . . . 8 ((𝑧 ∈ ( βŠ₯ β€˜π‘†) ∧ π‘₯ ∈ 𝑆) β†’ (𝑧(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
5831, 57sylan 579 . . . . . . 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 7422 . . . . . 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 20713 . . . . . . 7 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Grp)
6128, 15grpidcl 18893 . . . . . . . 8 ((Scalarβ€˜π‘Š) ∈ Grp β†’ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
6228, 40, 15grplid 18895 . . . . . . . 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 684 . . . . . . 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 2770 . . . . 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 21557 . . . 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 1340 . . 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 20779 . 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 1340 1 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘†) ∈ 𝐿)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055   βŠ† wss 3943  βˆ…c0 4317  β€˜cfv 6536  (class class class)co 7404  Basecbs 17151  +gcplusg 17204  .rcmulr 17205  Scalarcsca 17207   ·𝑠 cvsca 17208  Β·π‘–cip 17209  0gc0g 17392  Grpcgrp 18861  Ringcrg 20136  LModclmod 20704  LSubSpclss 20776  PreHilcphl 21513  ocvcocv 21549
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-sets 17104  df-slot 17122  df-ndx 17134  df-base 17152  df-plusg 17217  df-sca 17220  df-vsca 17221  df-ip 17222  df-0g 17394  df-mgm 18571  df-sgrp 18650  df-mnd 18666  df-grp 18864  df-minusg 18865  df-ghm 19137  df-cmn 19700  df-abl 19701  df-mgp 20038  df-rng 20056  df-ur 20085  df-ring 20138  df-lmod 20706  df-lss 20777  df-lmhm 20868  df-lvec 20949  df-sra 21019  df-rgmod 21020  df-phl 21515  df-ocv 21552
This theorem is referenced by:  ocvin  21563  ocvlsp  21565  csslss  21580  pjdm2  21602  pjff  21603  pjf2  21605  pjfo  21606  ocvpj  21608  pjthlem2  25317  pjth  25318
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