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Theorem phlssphl 21086
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 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹))
2 eqidd 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (+gβ€˜π‘‹) = (+gβ€˜π‘‹))
3 eqidd 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘‹))
4 eqidd 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘‹))
5 phllmod 21057 . . . 4 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
6 phlssphl.x . . . . 5 𝑋 = (π‘Š β†Ύs π‘ˆ)
7 eqid 2733 . . . . 5 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
8 eqid 2733 . . . . 5 (0gβ€˜π‘‹) = (0gβ€˜π‘‹)
9 phlssphl.s . . . . 5 𝑆 = (LSubSpβ€˜π‘Š)
106, 7, 8, 9lss0v 20521 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜π‘‹) = (0gβ€˜π‘Š))
115, 10sylan 581 . . 3 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜π‘‹) = (0gβ€˜π‘Š))
1211eqcomd 2739 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜π‘Š) = (0gβ€˜π‘‹))
13 eqidd 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘‹))
14 eqidd 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘‹)))
15 eqidd 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (+gβ€˜(Scalarβ€˜π‘‹)) = (+gβ€˜(Scalarβ€˜π‘‹)))
16 eqidd 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (.rβ€˜(Scalarβ€˜π‘‹)) = (.rβ€˜(Scalarβ€˜π‘‹)))
17 eqidd 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (*π‘Ÿβ€˜(Scalarβ€˜π‘‹)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘‹)))
18 eqidd 2734 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘‹)))
19 phllvec 21056 . . 3 (π‘Š ∈ PreHil β†’ π‘Š ∈ LVec)
206, 9lsslvec 20613 . . 3 ((π‘Š ∈ LVec ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ LVec)
2119, 20sylan 581 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ LVec)
22 eqid 2733 . . . . . 6 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
236, 22resssca 17232 . . . . 5 (π‘ˆ ∈ 𝑆 β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘‹))
2423eqcomd 2739 . . . 4 (π‘ˆ ∈ 𝑆 β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘Š))
2524adantl 483 . . 3 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘Š))
2622phlsrng 21058 . . . 4 (π‘Š ∈ PreHil β†’ (Scalarβ€˜π‘Š) ∈ *-Ring)
2726adantr 482 . . 3 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘Š) ∈ *-Ring)
2825, 27eqeltrd 2834 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘‹) ∈ *-Ring)
29 simpl 484 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ π‘Š ∈ PreHil)
30 eqid 2733 . . . . . . 7 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
316, 30ressbasss 17129 . . . . . 6 (Baseβ€˜π‘‹) βŠ† (Baseβ€˜π‘Š)
3231sseli 3944 . . . . 5 (π‘₯ ∈ (Baseβ€˜π‘‹) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
3331sseli 3944 . . . . 5 (𝑦 ∈ (Baseβ€˜π‘‹) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
34 eqid 2733 . . . . . 6 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
35 eqid 2733 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
3622, 34, 30, 35ipcl 21060 . . . . 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 1161 . . . 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 6850 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
3938eleq2d 2820 . . . . . 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 483 . . . . 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 1134 . . . 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 257 . . 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 2733 . . . . . . . 8 (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘‹)
446, 34, 43ssipeq 21083 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘Š))
4544oveqd 7378 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ (π‘₯(Β·π‘–β€˜π‘‹)𝑦) = (π‘₯(Β·π‘–β€˜π‘Š)𝑦))
4645eleq1d 2819 . . . . 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 483 . . . 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 1134 . . 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 257 . 2 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹)) β†’ (π‘₯(Β·π‘–β€˜π‘‹)𝑦) ∈ (Baseβ€˜(Scalarβ€˜π‘‹)))
50293ad2ant1 1134 . . . . 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 482 . . . . . . 7 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ π‘Š ∈ LMod)
52513ad2ant1 1134 . . . . . 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 6850 . . . . . . . . 9 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
5453eleq2d 2820 . . . . . . . 8 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ↔ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘Š))))
5554biimpa 478 . . . . . . 7 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹))) β†’ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
56553adant3 1133 . . . . . 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 1134 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
58573ad2ant3 1136 . . . . . 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 2733 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
6030, 22, 59, 35lmodvscl 20383 . . . . . 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 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‘₯) ∈ (Baseβ€˜π‘Š))
62333ad2ant2 1135 . . . . . 6 ((π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹)) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
63623ad2ant3 1136 . . . . 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 3944 . . . . . . 7 (𝑧 ∈ (Baseβ€˜π‘‹) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
65643ad2ant3 1136 . . . . . 6 ((π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹)) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
66653ad2ant3 1136 . . . . 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 2733 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
68 eqid 2733 . . . . . 6 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
6922, 34, 30, 67, 68ipdir 21066 . . . . 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 1373 . . . 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 2733 . . . . . . 7 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
7222, 34, 30, 35, 59, 71ipass 21072 . . . . . 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 1373 . . . . 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 7376 . . . 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 2773 . . 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 17179 . . . . . . . . 9 (π‘ˆ ∈ 𝑆 β†’ (+gβ€˜π‘Š) = (+gβ€˜π‘‹))
7776eqcomd 2739 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (+gβ€˜π‘‹) = (+gβ€˜π‘Š))
786, 59ressvsca 17233 . . . . . . . . . 10 (π‘ˆ ∈ 𝑆 β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘‹))
7978eqcomd 2739 . . . . . . . . 9 (π‘ˆ ∈ 𝑆 β†’ ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘Š))
8079oveqd 7378 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (π‘ž( ·𝑠 β€˜π‘‹)π‘₯) = (π‘ž( ·𝑠 β€˜π‘Š)π‘₯))
81 eqidd 2734 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ 𝑦 = 𝑦)
8277, 80, 81oveq123d 7382 . . . . . . 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 2734 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ 𝑧 = 𝑧)
8444, 82, 83oveq123d 7382 . . . . . 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 6850 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (+gβ€˜(Scalarβ€˜π‘‹)) = (+gβ€˜(Scalarβ€˜π‘Š)))
8624fveq2d 6850 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (.rβ€˜(Scalarβ€˜π‘‹)) = (.rβ€˜(Scalarβ€˜π‘Š)))
87 eqidd 2734 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ π‘ž = π‘ž)
8844oveqd 7378 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (π‘₯(Β·π‘–β€˜π‘‹)𝑧) = (π‘₯(Β·π‘–β€˜π‘Š)𝑧))
8986, 87, 88oveq123d 7382 . . . . . . 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 7378 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (𝑦(Β·π‘–β€˜π‘‹)𝑧) = (𝑦(Β·π‘–β€˜π‘Š)𝑧))
9185, 89, 90oveq123d 7382 . . . . . 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 2749 . . . . 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 483 . . . 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 1134 . . 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 257 . 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 483 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘Š))
9796oveqdr 7389 . . . . 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 6850 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
9998adantl 483 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
10099adantr 482 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹)) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
10197, 100eqeq12d 2749 . . . 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 2733 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
10322, 34, 30, 102, 7ipeq0 21065 . . . . . 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 597 . . . . 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 1118 . 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 6850 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (*π‘Ÿβ€˜(Scalarβ€˜π‘‹)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘Š)))
109108fveq1d 6848 . . . . . 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 483 . . . . 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 1134 . . . 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 2733 . . . . . 6 (*π‘Ÿβ€˜(Scalarβ€˜π‘Š)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘Š))
11322, 34, 30, 112ipcj 21061 . . . . 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 1161 . . . 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 2773 . . 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 6850 . . . . . 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 7378 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ (𝑦(Β·π‘–β€˜π‘‹)π‘₯) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯))
118116, 117eqeq12d 2749 . . . . 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 483 . . . 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 1134 . . 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 257 . 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 21081 1 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  β€˜cfv 6500  (class class class)co 7361  Basecbs 17091   β†Ύs cress 17120  +gcplusg 17141  .rcmulr 17142  *π‘Ÿcstv 17143  Scalarcsca 17144   ·𝑠 cvsca 17145  Β·π‘–cip 17146  0gc0g 17329  *-Ringcsr 20346  LModclmod 20365  LSubSpclss 20436  LVecclvec 20607  PreHilcphl 21051
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 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-en 8890  df-dom 8891  df-sdom 8892  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-sca 17157  df-vsca 17158  df-ip 17159  df-0g 17331  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-minusg 18760  df-sbg 18761  df-subg 18933  df-ghm 19014  df-mgp 19905  df-ur 19922  df-ring 19974  df-subrg 20262  df-lmod 20367  df-lss 20437  df-lsp 20477  df-lmhm 20527  df-lvec 20608  df-sra 20678  df-rgmod 20679  df-phl 21053
This theorem is referenced by:  cphsscph  24638
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