MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ocvlss Structured version   Visualization version   GIF version

Theorem ocvlss 21609
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 21607 . . 3 ( βŠ₯ β€˜π‘†) βŠ† 𝑉
43a1i 11 . 2 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘†) βŠ† 𝑉)
5 simpr 483 . . . 4 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ 𝑆 βŠ† 𝑉)
6 phllmod 21567 . . . . . 6 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
76adantr 479 . . . . 5 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ π‘Š ∈ LMod)
8 eqid 2727 . . . . . 6 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
91, 8lmod0vcl 20779 . . . . 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 765 . . . . . 6 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ 𝑆) β†’ π‘Š ∈ PreHil)
125sselda 3980 . . . . . 6 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑉)
13 eqid 2727 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
14 eqid 2727 . . . . . . 7 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
15 eqid 2727 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
1613, 14, 1, 15, 8ip0l 21573 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘₯ ∈ 𝑉) β†’ ((0gβ€˜π‘Š)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
1711, 12, 16syl2anc 582 . . . . 5 (((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) ∧ π‘₯ ∈ 𝑆) β†’ ((0gβ€˜π‘Š)(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
1817ralrimiva 3142 . . . 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 21605 . . . 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 4337 . 2 ((π‘Š ∈ PreHil ∧ 𝑆 βŠ† 𝑉) β†’ ( βŠ₯ β€˜π‘†) β‰  βˆ…)
225adantr 479 . . . 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 479 . . . . 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 3978 . . . . . 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 2727 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
28 eqid 2727 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
291, 13, 27, 28lmodvscl 20766 . . . . . 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 3978 . . . . 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 2727 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
341, 33lmodvacl 20763 . . . . 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 713 . . . . . . 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 479 . . . . . . 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 479 . . . . . . 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 713 . . . . . . 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 2727 . . . . . . . 8 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
4113, 14, 1, 33, 40ipdir 21576 . . . . . . 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 479 . . . . . . . . 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 479 . . . . . . . . 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 2727 . . . . . . . . . 10 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
4613, 14, 1, 28, 27, 45ipass 21582 . . . . . . . . 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 21606 . . . . . . . . . 10 ((𝑦 ∈ ( βŠ₯ β€˜π‘†) ∧ π‘₯ ∈ 𝑆) β†’ (𝑦(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
4925, 48sylan 578 . . . . . . . . 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 7440 . . . . . . . 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 479 . . . . . . . . . 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 20756 . . . . . . . . . 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 20235 . . . . . . . . 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 582 . . . . . . . 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 2771 . . . . . . 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 21606 . . . . . . . 8 ((𝑧 ∈ ( βŠ₯ β€˜π‘†) ∧ π‘₯ ∈ 𝑆) β†’ (𝑧(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)))
5831, 57sylan 578 . . . . . . 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 7442 . . . . . 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 20757 . . . . . . 7 (π‘Š ∈ LMod β†’ (Scalarβ€˜π‘Š) ∈ Grp)
6128, 15grpidcl 18927 . . . . . . . 8 ((Scalarβ€˜π‘Š) ∈ Grp β†’ (0gβ€˜(Scalarβ€˜π‘Š)) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
6228, 40, 15grplid 18929 . . . . . . . 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 685 . . . . . . 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 2771 . . . . 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 3142 . . . 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 21605 . . . 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 3199 . 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 20823 . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2936  βˆ€wral 3057   βŠ† wss 3947  βˆ…c0 4324  β€˜cfv 6551  (class class class)co 7424  Basecbs 17185  +gcplusg 17238  .rcmulr 17239  Scalarcsca 17241   ·𝑠 cvsca 17242  Β·π‘–cip 17243  0gc0g 17426  Grpcgrp 18895  Ringcrg 20178  LModclmod 20748  LSubSpclss 20820  PreHilcphl 21561  ocvcocv 21597
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-cnex 11200  ax-resscn 11201  ax-1cn 11202  ax-icn 11203  ax-addcl 11204  ax-addrcl 11205  ax-mulcl 11206  ax-mulrcl 11207  ax-mulcom 11208  ax-addass 11209  ax-mulass 11210  ax-distr 11211  ax-i2m1 11212  ax-1ne0 11213  ax-1rid 11214  ax-rnegex 11215  ax-rrecex 11216  ax-cnre 11217  ax-pre-lttri 11218  ax-pre-lttrn 11219  ax-pre-ltadd 11220  ax-pre-mulgt0 11221
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-nel 3043  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-ord 6375  df-on 6376  df-lim 6377  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7875  df-2nd 7998  df-frecs 8291  df-wrecs 8322  df-recs 8396  df-rdg 8435  df-er 8729  df-en 8969  df-dom 8970  df-sdom 8971  df-pnf 11286  df-mnf 11287  df-xr 11288  df-ltxr 11289  df-le 11290  df-sub 11482  df-neg 11483  df-nn 12249  df-2 12311  df-3 12312  df-4 12313  df-5 12314  df-6 12315  df-7 12316  df-8 12317  df-sets 17138  df-slot 17156  df-ndx 17168  df-base 17186  df-plusg 17251  df-sca 17254  df-vsca 17255  df-ip 17256  df-0g 17428  df-mgm 18605  df-sgrp 18684  df-mnd 18700  df-grp 18898  df-minusg 18899  df-ghm 19173  df-cmn 19742  df-abl 19743  df-mgp 20080  df-rng 20098  df-ur 20127  df-ring 20180  df-lmod 20750  df-lss 20821  df-lmhm 20912  df-lvec 20993  df-sra 21063  df-rgmod 21064  df-phl 21563  df-ocv 21600
This theorem is referenced by:  ocvin  21611  ocvlsp  21613  csslss  21628  pjdm2  21650  pjff  21651  pjf2  21653  pjfo  21654  ocvpj  21656  pjthlem2  25384  pjth  25385
  Copyright terms: Public domain W3C validator