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Theorem imasvsca 16793
Description: The scalar multiplication operation of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
imasbas.u (𝜑𝑈 = (𝐹s 𝑅))
imasbas.v (𝜑𝑉 = (Base‘𝑅))
imasbas.f (𝜑𝐹:𝑉onto𝐵)
imasbas.r (𝜑𝑅𝑍)
imassca.g 𝐺 = (Scalar‘𝑅)
imasvsca.k 𝐾 = (Base‘𝐺)
imasvsca.q · = ( ·𝑠𝑅)
imasvsca.s = ( ·𝑠𝑈)
Assertion
Ref Expression
imasvsca (𝜑 = 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
Distinct variable groups:   𝑞,𝑝,𝑥,𝐹   𝑅,𝑝,𝑞,𝑥   𝑥,𝑈   𝑥,𝐵   𝜑,𝑝,𝑞,𝑥   𝐾,𝑝,𝑥   𝑉,𝑝,𝑞
Allowed substitution hints:   𝐵(𝑞,𝑝)   (𝑥,𝑞,𝑝)   · (𝑥,𝑞,𝑝)   𝑈(𝑞,𝑝)   𝐺(𝑥,𝑞,𝑝)   𝐾(𝑞)   𝑉(𝑥)   𝑍(𝑥,𝑞,𝑝)

Proof of Theorem imasvsca
Dummy variables 𝑢 𝑣 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . 3 (𝜑𝑈 = (𝐹s 𝑅))
2 imasbas.v . . 3 (𝜑𝑉 = (Base‘𝑅))
3 eqid 2824 . . 3 (+g𝑅) = (+g𝑅)
4 eqid 2824 . . 3 (.r𝑅) = (.r𝑅)
5 eqid 2824 . . 3 (Scalar‘𝑅) = (Scalar‘𝑅)
6 imasvsca.k . . . 4 𝐾 = (Base‘𝐺)
7 imassca.g . . . . 5 𝐺 = (Scalar‘𝑅)
87fveq2i 6664 . . . 4 (Base‘𝐺) = (Base‘(Scalar‘𝑅))
96, 8eqtri 2847 . . 3 𝐾 = (Base‘(Scalar‘𝑅))
10 imasvsca.q . . 3 · = ( ·𝑠𝑅)
11 eqid 2824 . . 3 (·𝑖𝑅) = (·𝑖𝑅)
12 eqid 2824 . . 3 (TopOpen‘𝑅) = (TopOpen‘𝑅)
13 eqid 2824 . . 3 (dist‘𝑅) = (dist‘𝑅)
14 eqid 2824 . . 3 (le‘𝑅) = (le‘𝑅)
15 imasbas.f . . . 4 (𝜑𝐹:𝑉onto𝐵)
16 imasbas.r . . . 4 (𝜑𝑅𝑍)
17 eqid 2824 . . . 4 (+g𝑈) = (+g𝑈)
181, 2, 15, 16, 3, 17imasplusg 16790 . . 3 (𝜑 → (+g𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩})
19 eqid 2824 . . . 4 (.r𝑈) = (.r𝑈)
201, 2, 15, 16, 4, 19imasmulr 16791 . . 3 (𝜑 → (.r𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩})
21 eqidd 2825 . . 3 (𝜑 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) = 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
22 eqidd 2825 . . 3 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩} = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩})
23 eqidd 2825 . . 3 (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
24 eqid 2824 . . . 4 (dist‘𝑈) = (dist‘𝑈)
251, 2, 15, 16, 13, 24imasds 16786 . . 3 (𝜑 → (dist‘𝑈) = (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑢 ∈ ℕ ran (𝑧 ∈ {𝑤 ∈ ((𝑉 × 𝑉) ↑m (1...𝑢)) ∣ ((𝐹‘(1st ‘(𝑤‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑤𝑢))) = 𝑦 ∧ ∀𝑣 ∈ (1...(𝑢 − 1))(𝐹‘(2nd ‘(𝑤𝑣))) = (𝐹‘(1st ‘(𝑤‘(𝑣 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑧))), ℝ*, < )))
26 eqidd 2825 . . 3 (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹) = ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹))
271, 2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 18, 20, 21, 22, 23, 25, 26, 15, 16imasval 16784 . 2 (𝜑𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}))
28 eqid 2824 . . 3 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
2928imasvalstr 16725 . 2 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩}) Struct ⟨1, 12⟩
30 vscaid 16635 . 2 ·𝑠 = Slot ( ·𝑠 ‘ndx)
31 snsstp2 4734 . . 3 {⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩} ⊆ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}
32 ssun2 4135 . . . 4 {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩})
33 ssun1 4134 . . . 4 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
3432, 33sstri 3962 . . 3 {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
3531, 34sstri 3962 . 2 {⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), (Scalar‘𝑅)⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ∪ {⟨(TopSet‘ndx), ((TopOpen‘𝑅) qTop 𝐹)⟩, ⟨(le‘ndx), ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹)⟩, ⟨(dist‘ndx), (dist‘𝑈)⟩})
36 fvex 6674 . . . 4 (Base‘𝑅) ∈ V
372, 36eqeltrdi 2924 . . 3 (𝜑𝑉 ∈ V)
386fvexi 6675 . . . . 5 𝐾 ∈ V
39 snex 5319 . . . . 5 {(𝐹𝑞)} ∈ V
4038, 39mpoex 7773 . . . 4 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V
4140rgenw 3145 . . 3 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V
42 iunexg 7659 . . 3 ((𝑉 ∈ V ∧ ∀𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V) → 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V)
4337, 41, 42sylancl 589 . 2 (𝜑 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))) ∈ V)
44 imasvsca.s . 2 = ( ·𝑠𝑈)
4527, 29, 30, 35, 43, 44strfv3 16532 1 (𝜑 = 𝑞𝑉 (𝑝𝐾, 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝 · 𝑞))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  cun 3917  {csn 4550  {ctp 4554  cop 4556   ciun 4905  ccnv 5541  ccom 5546  ontowfo 6341  cfv 6343  (class class class)co 7149  cmpo 7151  1c1 10536  2c2 11689  cdc 12095  ndxcnx 16480  Basecbs 16483  +gcplusg 16565  .rcmulr 16566  Scalarcsca 16568   ·𝑠 cvsca 16569  ·𝑖cip 16570  TopSetcts 16571  lecple 16572  distcds 16574  TopOpenctopn 16695   qTop cqtop 16776  s cimas 16777
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-sup 8903  df-inf 8904  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-z 11979  df-dec 12096  df-uz 12241  df-fz 12895  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-ip 16583  df-tset 16584  df-ple 16585  df-ds 16587  df-imas 16781
This theorem is referenced by:  imasip  16794  imastset  16795  imasle  16796  imasvscafn  16810  imasvscaval  16811  imasvscaf  16812
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