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Theorem phlssphl 21211
Description: A subspace of an inner product space (pre-Hilbert space) is an inner product space. (Contributed by AV, 25-Sep-2022.)
Hypotheses
Ref Expression
phlssphl.x 𝑋 = (π‘Š β†Ύs π‘ˆ)
phlssphl.s 𝑆 = (LSubSpβ€˜π‘Š)
Assertion
Ref Expression
phlssphl ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ PreHil)
Proof of Theorem phlssphl
Dummy variables π‘ž π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹))
2 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (+gβ€˜π‘‹) = (+gβ€˜π‘‹))
3 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘‹))
4 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘‹))
5 phllmod 21182 . . . 4 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
6 phlssphl.x . . . . 5 𝑋 = (π‘Š β†Ύs π‘ˆ)
7 eqid 2732 . . . . 5 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
8 eqid 2732 . . . . 5 (0gβ€˜π‘‹) = (0gβ€˜π‘‹)
9 phlssphl.s . . . . 5 𝑆 = (LSubSpβ€˜π‘Š)
106, 7, 8, 9lss0v 20626 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜π‘‹) = (0gβ€˜π‘Š))
115, 10sylan 580 . . 3 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜π‘‹) = (0gβ€˜π‘Š))
1211eqcomd 2738 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜π‘Š) = (0gβ€˜π‘‹))
13 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘‹))
14 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘‹)))
15 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (+gβ€˜(Scalarβ€˜π‘‹)) = (+gβ€˜(Scalarβ€˜π‘‹)))
16 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (.rβ€˜(Scalarβ€˜π‘‹)) = (.rβ€˜(Scalarβ€˜π‘‹)))
17 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (*π‘Ÿβ€˜(Scalarβ€˜π‘‹)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘‹)))
18 eqidd 2733 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘‹)))
19 phllvec 21181 . . 3 (π‘Š ∈ PreHil β†’ π‘Š ∈ LVec)
206, 9lsslvec 20718 . . 3 ((π‘Š ∈ LVec ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ LVec)
2119, 20sylan 580 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ LVec)
22 eqid 2732 . . . . . 6 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
236, 22resssca 17287 . . . . 5 (π‘ˆ ∈ 𝑆 β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘‹))
2423eqcomd 2738 . . . 4 (π‘ˆ ∈ 𝑆 β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘Š))
2524adantl 482 . . 3 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘Š))
2622phlsrng 21183 . . . 4 (π‘Š ∈ PreHil β†’ (Scalarβ€˜π‘Š) ∈ *-Ring)
2726adantr 481 . . 3 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘Š) ∈ *-Ring)
2825, 27eqeltrd 2833 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘‹) ∈ *-Ring)
29 simpl 483 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ π‘Š ∈ PreHil)
30 eqid 2732 . . . . . . 7 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
316, 30ressbasss 17182 . . . . . 6 (Baseβ€˜π‘‹) βŠ† (Baseβ€˜π‘Š)
3231sseli 3978 . . . . 5 (π‘₯ ∈ (Baseβ€˜π‘‹) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
3331sseli 3978 . . . . 5 (𝑦 ∈ (Baseβ€˜π‘‹) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
34 eqid 2732 . . . . . 6 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
35 eqid 2732 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
3622, 34, 30, 35ipcl 21185 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
3729, 32, 33, 36syl3an 1160 . . . 4 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
3824fveq2d 6895 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
3938eleq2d 2819 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ ((π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ↔ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘Š))))
4039adantl 482 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ ((π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ↔ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘Š))))
41403ad2ant1 1133 . . . 4 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ ((π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ↔ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘Š))))
4237, 41mpbird 256 . . 3 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹)))
43 eqid 2732 . . . . . . . 8 (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘‹)
446, 34, 43ssipeq 21208 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘Š))
4544oveqd 7425 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ (π‘₯(Β·π‘–β€˜π‘‹)𝑦) = (π‘₯(Β·π‘–β€˜π‘Š)𝑦))
4645eleq1d 2818 . . . . 5 (π‘ˆ ∈ 𝑆 β†’ ((π‘₯(Β·π‘–β€˜π‘‹)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ↔ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹))))
4746adantl 482 . . . 4 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ ((π‘₯(Β·π‘–β€˜π‘‹)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ↔ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹))))
48473ad2ant1 1133 . . 3 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ ((π‘₯(Β·π‘–β€˜π‘‹)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ↔ (π‘₯(Β·π‘–β€˜π‘Š)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹))))
4942, 48mpbird 256 . 2 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ (π‘₯(Β·π‘–β€˜π‘‹)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹)))
50293ad2ant1 1133 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ π‘Š ∈ PreHil)
515adantr 481 . . . . . . 7 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ π‘Š ∈ LMod)
52513ad2ant1 1133 . . . . . 6 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ π‘Š ∈ LMod)
5325fveq2d 6895 . . . . . . . . 9 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
5453eleq2d 2819 . . . . . . . 8 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ↔ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘Š))))
5554biimpa 477 . . . . . . 7 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹))) β†’ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
56553adant3 1132 . . . . . 6 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
57323ad2ant1 1133 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
58573ad2ant3 1135 . . . . . 6 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
59 eqid 2732 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
6030, 22, 59, 35lmodvscl 20488 . . . . . 6 ((π‘Š ∈ LMod ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (π‘ž( ·𝑠 β€˜π‘Š)π‘₯) ∈ (Baseβ€˜π‘Š))
6152, 56, 58, 60syl3anc 1371 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ (π‘ž( ·𝑠 β€˜π‘Š)π‘₯) ∈ (Baseβ€˜π‘Š))
62333ad2ant2 1134 . . . . . 6 ((π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹)) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
63623ad2ant3 1135 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
6431sseli 3978 . . . . . . 7 (𝑧 ∈ (Baseβ€˜π‘‹) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
65643ad2ant3 1135 . . . . . 6 ((π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹)) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
66653ad2ant3 1135 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
67 eqid 2732 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
68 eqid 2732 . . . . . 6 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
6922, 34, 30, 67, 68ipdir 21191 . . . . 5 ((π‘Š ∈ PreHil ∧ ((π‘ž( ·𝑠 β€˜π‘Š)π‘₯) ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)𝑧) = (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(Β·π‘–β€˜π‘Š)𝑧)(+gβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)𝑧)))
7050, 61, 63, 66, 69syl13anc 1372 . . . 4 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)𝑧) = (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(Β·π‘–β€˜π‘Š)𝑧)(+gβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)𝑧)))
71 eqid 2732 . . . . . . 7 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
7222, 34, 30, 35, 59, 71ipass 21197 . . . . . 6 ((π‘Š ∈ PreHil ∧ (π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ ((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(Β·π‘–β€˜π‘Š)𝑧) = (π‘ž(.rβ€˜(Scalarβ€˜π‘Š))(π‘₯(Β·π‘–β€˜π‘Š)𝑧)))
7350, 56, 58, 66, 72syl13anc 1372 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ ((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(Β·π‘–β€˜π‘Š)𝑧) = (π‘ž(.rβ€˜(Scalarβ€˜π‘Š))(π‘₯(Β·π‘–β€˜π‘Š)𝑧)))
7473oveq1d 7423 . . . 4 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(Β·π‘–β€˜π‘Š)𝑧)(+gβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)𝑧)) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘Š))(π‘₯(Β·π‘–β€˜π‘Š)𝑧))(+gβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)𝑧)))
7570, 74eqtrd 2772 . . 3 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)𝑧) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘Š))(π‘₯(Β·π‘–β€˜π‘Š)𝑧))(+gβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)𝑧)))
766, 67ressplusg 17234 . . . . . . . . 9 (π‘ˆ ∈ 𝑆 β†’ (+gβ€˜π‘Š) = (+gβ€˜π‘‹))
7776eqcomd 2738 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (+gβ€˜π‘‹) = (+gβ€˜π‘Š))
786, 59ressvsca 17288 . . . . . . . . . 10 (π‘ˆ ∈ 𝑆 β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘‹))
7978eqcomd 2738 . . . . . . . . 9 (π‘ˆ ∈ 𝑆 β†’ ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘Š))
8079oveqd 7425 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (π‘ž( ·𝑠 β€˜π‘‹)π‘₯) = (π‘ž( ·𝑠 β€˜π‘Š)π‘₯))
81 eqidd 2733 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ 𝑦 = 𝑦)
8277, 80, 81oveq123d 7429 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ ((π‘ž( ·𝑠 β€˜π‘‹)π‘₯)(+gβ€˜π‘‹)𝑦) = ((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦))
83 eqidd 2733 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ 𝑧 = 𝑧)
8444, 82, 83oveq123d 7429 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ (((π‘ž( ·𝑠 β€˜π‘‹)π‘₯)(+gβ€˜π‘‹)𝑦)(Β·π‘–β€˜π‘‹)𝑧) = (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)𝑧))
8524fveq2d 6895 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (+gβ€˜(Scalarβ€˜π‘‹)) = (+gβ€˜(Scalarβ€˜π‘Š)))
8624fveq2d 6895 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (.rβ€˜(Scalarβ€˜π‘‹)) = (.rβ€˜(Scalarβ€˜π‘Š)))
87 eqidd 2733 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ π‘ž = π‘ž)
8844oveqd 7425 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (π‘₯(Β·π‘–β€˜π‘‹)𝑧) = (π‘₯(Β·π‘–β€˜π‘Š)𝑧))
8986, 87, 88oveq123d 7429 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (π‘ž(.rβ€˜(Scalarβ€˜π‘‹))(π‘₯(Β·π‘–β€˜π‘‹)𝑧)) = (π‘ž(.rβ€˜(Scalarβ€˜π‘Š))(π‘₯(Β·π‘–β€˜π‘Š)𝑧)))
9044oveqd 7425 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (𝑦(Β·π‘–β€˜π‘‹)𝑧) = (𝑦(Β·π‘–β€˜π‘Š)𝑧))
9185, 89, 90oveq123d 7429 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ ((π‘ž(.rβ€˜(Scalarβ€˜π‘‹))(π‘₯(Β·π‘–β€˜π‘‹)𝑧))(+gβ€˜(Scalarβ€˜π‘‹))(𝑦(Β·π‘–β€˜π‘‹)𝑧)) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘Š))(π‘₯(Β·π‘–β€˜π‘Š)𝑧))(+gβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)𝑧)))
9284, 91eqeq12d 2748 . . . . 5 (π‘ˆ ∈ 𝑆 β†’ ((((π‘ž( ·𝑠 β€˜π‘‹)π‘₯)(+gβ€˜π‘‹)𝑦)(Β·π‘–β€˜π‘‹)𝑧) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘‹))(π‘₯(Β·π‘–β€˜π‘‹)𝑧))(+gβ€˜(Scalarβ€˜π‘‹))(𝑦(Β·π‘–β€˜π‘‹)𝑧)) ↔ (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)𝑧) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘Š))(π‘₯(Β·π‘–β€˜π‘Š)𝑧))(+gβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)𝑧))))
9392adantl 482 . . . 4 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ ((((π‘ž( ·𝑠 β€˜π‘‹)π‘₯)(+gβ€˜π‘‹)𝑦)(Β·π‘–β€˜π‘‹)𝑧) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘‹))(π‘₯(Β·π‘–β€˜π‘‹)𝑧))(+gβ€˜(Scalarβ€˜π‘‹))(𝑦(Β·π‘–β€˜π‘‹)𝑧)) ↔ (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)𝑧) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘Š))(π‘₯(Β·π‘–β€˜π‘Š)𝑧))(+gβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)𝑧))))
94933ad2ant1 1133 . . 3 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ ((((π‘ž( ·𝑠 β€˜π‘‹)π‘₯)(+gβ€˜π‘‹)𝑦)(Β·π‘–β€˜π‘‹)𝑧) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘‹))(π‘₯(Β·π‘–β€˜π‘‹)𝑧))(+gβ€˜(Scalarβ€˜π‘‹))(𝑦(Β·π‘–β€˜π‘‹)𝑧)) ↔ (((π‘ž( ·𝑠 β€˜π‘Š)π‘₯)(+gβ€˜π‘Š)𝑦)(Β·π‘–β€˜π‘Š)𝑧) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘Š))(π‘₯(Β·π‘–β€˜π‘Š)𝑧))(+gβ€˜(Scalarβ€˜π‘Š))(𝑦(Β·π‘–β€˜π‘Š)𝑧))))
9575, 94mpbird 256 . 2 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ∧ (π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹))) β†’ (((π‘ž( ·𝑠 β€˜π‘‹)π‘₯)(+gβ€˜π‘‹)𝑦)(Β·π‘–β€˜π‘‹)𝑧) = ((π‘ž(.rβ€˜(Scalarβ€˜π‘‹))(π‘₯(Β·π‘–β€˜π‘‹)𝑧))(+gβ€˜(Scalarβ€˜π‘‹))(𝑦(Β·π‘–β€˜π‘‹)𝑧)))
9644adantl 482 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘Š))
9796oveqdr 7436 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹)) β†’ (π‘₯(Β·π‘–β€˜π‘‹)π‘₯) = (π‘₯(Β·π‘–β€˜π‘Š)π‘₯))
9824fveq2d 6895 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
9998adantl 482 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
10099adantr 481 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹)) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
10197, 100eqeq12d 2748 . . . 4 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹)) β†’ ((π‘₯(Β·π‘–β€˜π‘‹)π‘₯) = (0gβ€˜(Scalarβ€˜π‘‹)) ↔ (π‘₯(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
102 eqid 2732 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
10322, 34, 30, 102, 7ipeq0 21190 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ ((π‘₯(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ π‘₯ = (0gβ€˜π‘Š)))
10429, 32, 103syl2an 596 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹)) β†’ ((π‘₯(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)) ↔ π‘₯ = (0gβ€˜π‘Š)))
105104biimpd 228 . . . 4 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹)) β†’ ((π‘₯(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š)) β†’ π‘₯ = (0gβ€˜π‘Š)))
106101, 105sylbid 239 . . 3 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹)) β†’ ((π‘₯(Β·π‘–β€˜π‘‹)π‘₯) = (0gβ€˜(Scalarβ€˜π‘‹)) β†’ π‘₯ = (0gβ€˜π‘Š)))
1071063impia 1117 . 2 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ (π‘₯(Β·π‘–β€˜π‘‹)π‘₯) = (0gβ€˜(Scalarβ€˜π‘‹))) β†’ π‘₯ = (0gβ€˜π‘Š))
10824fveq2d 6895 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (*π‘Ÿβ€˜(Scalarβ€˜π‘‹)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘Š)))
109108fveq1d 6893 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = ((*π‘Ÿβ€˜(Scalarβ€˜π‘Š))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)))
110109adantl 482 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = ((*π‘Ÿβ€˜(Scalarβ€˜π‘Š))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)))
1111103ad2ant1 1133 . . . 4 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = ((*π‘Ÿβ€˜(Scalarβ€˜π‘Š))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)))
112 eqid 2732 . . . . . 6 (*π‘Ÿβ€˜(Scalarβ€˜π‘Š)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘Š))
11322, 34, 30, 112ipcj 21186 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ ((*π‘Ÿβ€˜(Scalarβ€˜π‘Š))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯))
11429, 32, 33, 113syl3an 1160 . . . 4 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ ((*π‘Ÿβ€˜(Scalarβ€˜π‘Š))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯))
115111, 114eqtrd 2772 . . 3 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯))
11645fveq2d 6895 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘‹)𝑦)) = ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)))
11744oveqd 7425 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ (𝑦(Β·π‘–β€˜π‘‹)π‘₯) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯))
118116, 117eqeq12d 2748 . . . . 5 (π‘ˆ ∈ 𝑆 β†’ (((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘‹)𝑦)) = (𝑦(Β·π‘–β€˜π‘‹)π‘₯) ↔ ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯)))
119118adantl 482 . . . 4 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘‹)𝑦)) = (𝑦(Β·π‘–β€˜π‘‹)π‘₯) ↔ ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯)))
1201193ad2ant1 1133 . . 3 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ (((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘‹)𝑦)) = (𝑦(Β·π‘–β€˜π‘‹)π‘₯) ↔ ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘Š)𝑦)) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯)))
121115, 120mpbird 256 . 2 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ ((*π‘Ÿβ€˜(Scalarβ€˜π‘‹))β€˜(π‘₯(Β·π‘–β€˜π‘‹)𝑦)) = (𝑦(Β·π‘–β€˜π‘‹)π‘₯))
1221, 2, 3, 4, 12, 13, 14, 15, 16, 17, 18, 21, 28, 49, 95, 107, 121isphld 21206 1 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143   β†Ύs cress 17172  +gcplusg 17196  .rcmulr 17197  *π‘Ÿcstv 17198  Scalarcsca 17199   ·𝑠 cvsca 17200  Β·π‘–cip 17201  0gc0g 17384  *-Ringcsr 20451  LModclmod 20470  LSubSpclss 20541  LVecclvec 20712  PreHilcphl 21176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  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 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-ip 17214  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-ghm 19089  df-mgp 19987  df-ur 20004  df-ring 20057  df-subrg 20316  df-lmod 20472  df-lss 20542  df-lsp 20582  df-lmhm 20632  df-lvec 20713  df-sra 20784  df-rgmod 20785  df-phl 21178
This theorem is referenced by:  cphsscph  24767
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