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Theorem phlssphl 21431
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 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Baseβ€˜π‘‹) = (Baseβ€˜π‘‹))
2 eqidd 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (+gβ€˜π‘‹) = (+gβ€˜π‘‹))
3 eqidd 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘‹))
4 eqidd 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘‹))
5 phllmod 21402 . . . 4 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
6 phlssphl.x . . . . 5 𝑋 = (π‘Š β†Ύs π‘ˆ)
7 eqid 2730 . . . . 5 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
8 eqid 2730 . . . . 5 (0gβ€˜π‘‹) = (0gβ€˜π‘‹)
9 phlssphl.s . . . . 5 𝑆 = (LSubSpβ€˜π‘Š)
106, 7, 8, 9lss0v 20771 . . . 4 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜π‘‹) = (0gβ€˜π‘Š))
115, 10sylan 578 . . 3 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜π‘‹) = (0gβ€˜π‘Š))
1211eqcomd 2736 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜π‘Š) = (0gβ€˜π‘‹))
13 eqidd 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘‹))
14 eqidd 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘‹)))
15 eqidd 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (+gβ€˜(Scalarβ€˜π‘‹)) = (+gβ€˜(Scalarβ€˜π‘‹)))
16 eqidd 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (.rβ€˜(Scalarβ€˜π‘‹)) = (.rβ€˜(Scalarβ€˜π‘‹)))
17 eqidd 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (*π‘Ÿβ€˜(Scalarβ€˜π‘‹)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘‹)))
18 eqidd 2731 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘‹)))
19 phllvec 21401 . . 3 (π‘Š ∈ PreHil β†’ π‘Š ∈ LVec)
206, 9lsslvec 20864 . . 3 ((π‘Š ∈ LVec ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ LVec)
2119, 20sylan 578 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ LVec)
22 eqid 2730 . . . . . 6 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
236, 22resssca 17292 . . . . 5 (π‘ˆ ∈ 𝑆 β†’ (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘‹))
2423eqcomd 2736 . . . 4 (π‘ˆ ∈ 𝑆 β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘Š))
2524adantl 480 . . 3 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘‹) = (Scalarβ€˜π‘Š))
2622phlsrng 21403 . . . 4 (π‘Š ∈ PreHil β†’ (Scalarβ€˜π‘Š) ∈ *-Ring)
2726adantr 479 . . 3 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘Š) ∈ *-Ring)
2825, 27eqeltrd 2831 . 2 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Scalarβ€˜π‘‹) ∈ *-Ring)
29 simpl 481 . . . . 5 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ π‘Š ∈ PreHil)
30 eqid 2730 . . . . . . 7 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
316, 30ressbasss 17187 . . . . . 6 (Baseβ€˜π‘‹) βŠ† (Baseβ€˜π‘Š)
3231sseli 3977 . . . . 5 (π‘₯ ∈ (Baseβ€˜π‘‹) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
3331sseli 3977 . . . . 5 (𝑦 ∈ (Baseβ€˜π‘‹) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
34 eqid 2730 . . . . . 6 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
35 eqid 2730 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
3622, 34, 30, 35ipcl 21405 . . . . 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 1158 . . . 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 6894 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
3938eleq2d 2817 . . . . . 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 480 . . . . 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 1131 . . . 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 2730 . . . . . . . 8 (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘‹)
446, 34, 43ssipeq 21428 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘Š))
4544oveqd 7428 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ (π‘₯(Β·π‘–β€˜π‘‹)𝑦) = (π‘₯(Β·π‘–β€˜π‘Š)𝑦))
4645eleq1d 2816 . . . . 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 480 . . . 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 1131 . . 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 1131 . . . . 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 479 . . . . . . 7 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ π‘Š ∈ LMod)
52513ad2ant1 1131 . . . . . 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 6894 . . . . . . . . 9 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Baseβ€˜(Scalarβ€˜π‘‹)) = (Baseβ€˜(Scalarβ€˜π‘Š)))
5453eleq2d 2817 . . . . . . . 8 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹)) ↔ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘Š))))
5554biimpa 475 . . . . . . 7 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘‹))) β†’ π‘ž ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
56553adant3 1130 . . . . . 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 1131 . . . . . . 7 ((π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
58573ad2ant3 1133 . . . . . 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 2730 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
6030, 22, 59, 35lmodvscl 20632 . . . . . 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 1369 . . . . 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 1132 . . . . . 6 ((π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹)) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
63623ad2ant3 1133 . . . . 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 3977 . . . . . . 7 (𝑧 ∈ (Baseβ€˜π‘‹) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
65643ad2ant3 1133 . . . . . 6 ((π‘₯ ∈ (Baseβ€˜π‘‹) ∧ 𝑦 ∈ (Baseβ€˜π‘‹) ∧ 𝑧 ∈ (Baseβ€˜π‘‹)) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
66653ad2ant3 1133 . . . . 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 2730 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
68 eqid 2730 . . . . . 6 (+gβ€˜(Scalarβ€˜π‘Š)) = (+gβ€˜(Scalarβ€˜π‘Š))
6922, 34, 30, 67, 68ipdir 21411 . . . . 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 1370 . . . 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 2730 . . . . . . 7 (.rβ€˜(Scalarβ€˜π‘Š)) = (.rβ€˜(Scalarβ€˜π‘Š))
7222, 34, 30, 35, 59, 71ipass 21417 . . . . . 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 1370 . . . . 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 7426 . . . 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 2770 . . 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 17239 . . . . . . . . 9 (π‘ˆ ∈ 𝑆 β†’ (+gβ€˜π‘Š) = (+gβ€˜π‘‹))
7776eqcomd 2736 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (+gβ€˜π‘‹) = (+gβ€˜π‘Š))
786, 59ressvsca 17293 . . . . . . . . . 10 (π‘ˆ ∈ 𝑆 β†’ ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘‹))
7978eqcomd 2736 . . . . . . . . 9 (π‘ˆ ∈ 𝑆 β†’ ( ·𝑠 β€˜π‘‹) = ( ·𝑠 β€˜π‘Š))
8079oveqd 7428 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (π‘ž( ·𝑠 β€˜π‘‹)π‘₯) = (π‘ž( ·𝑠 β€˜π‘Š)π‘₯))
81 eqidd 2731 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ 𝑦 = 𝑦)
8277, 80, 81oveq123d 7432 . . . . . . 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 2731 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ 𝑧 = 𝑧)
8444, 82, 83oveq123d 7432 . . . . . 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 6894 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (+gβ€˜(Scalarβ€˜π‘‹)) = (+gβ€˜(Scalarβ€˜π‘Š)))
8624fveq2d 6894 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (.rβ€˜(Scalarβ€˜π‘‹)) = (.rβ€˜(Scalarβ€˜π‘Š)))
87 eqidd 2731 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ π‘ž = π‘ž)
8844oveqd 7428 . . . . . . . 8 (π‘ˆ ∈ 𝑆 β†’ (π‘₯(Β·π‘–β€˜π‘‹)𝑧) = (π‘₯(Β·π‘–β€˜π‘Š)𝑧))
8986, 87, 88oveq123d 7432 . . . . . . 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 7428 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (𝑦(Β·π‘–β€˜π‘‹)𝑧) = (𝑦(Β·π‘–β€˜π‘Š)𝑧))
9185, 89, 90oveq123d 7432 . . . . . 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 2746 . . . . 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 480 . . . 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 1131 . . 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 480 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (Β·π‘–β€˜π‘‹) = (Β·π‘–β€˜π‘Š))
9796oveqdr 7439 . . . . 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 6894 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
9998adantl 480 . . . . . 6 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
10099adantr 479 . . . . 5 (((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘‹)) β†’ (0gβ€˜(Scalarβ€˜π‘‹)) = (0gβ€˜(Scalarβ€˜π‘Š)))
10197, 100eqeq12d 2746 . . . 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 2730 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
10322, 34, 30, 102, 7ipeq0 21410 . . . . . 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 594 . . . . 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 1115 . 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 6894 . . . . . . 7 (π‘ˆ ∈ 𝑆 β†’ (*π‘Ÿβ€˜(Scalarβ€˜π‘‹)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘Š)))
109108fveq1d 6892 . . . . . 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 480 . . . . 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 1131 . . . 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 2730 . . . . . 6 (*π‘Ÿβ€˜(Scalarβ€˜π‘Š)) = (*π‘Ÿβ€˜(Scalarβ€˜π‘Š))
11322, 34, 30, 112ipcj 21406 . . . . 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 1158 . . . 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 2770 . . 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 6894 . . . . . 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 7428 . . . . . 6 (π‘ˆ ∈ 𝑆 β†’ (𝑦(Β·π‘–β€˜π‘‹)π‘₯) = (𝑦(Β·π‘–β€˜π‘Š)π‘₯))
118116, 117eqeq12d 2746 . . . . 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 480 . . . 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 1131 . . 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 21426 1 ((π‘Š ∈ PreHil ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148   β†Ύs cress 17177  +gcplusg 17201  .rcmulr 17202  *π‘Ÿcstv 17203  Scalarcsca 17204   ·𝑠 cvsca 17205  Β·π‘–cip 17206  0gc0g 17389  *-Ringcsr 20595  LModclmod 20614  LSubSpclss 20686  LVecclvec 20857  PreHilcphl 21396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-sca 17217  df-vsca 17218  df-ip 17219  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-minusg 18859  df-sbg 18860  df-subg 19039  df-ghm 19128  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-subrg 20459  df-lmod 20616  df-lss 20687  df-lsp 20727  df-lmhm 20777  df-lvec 20858  df-sra 20930  df-rgmod 20931  df-phl 21398
This theorem is referenced by:  cphsscph  24999
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