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Theorem imassca 17565
Description: The scalar field 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)
Assertion
Ref Expression
imassca (𝜑𝐺 = (Scalar‘𝑈))

Proof of Theorem imassca
Dummy variables 𝑔 𝑖 𝑛 𝑝 𝑞 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassca.g . . . 4 𝐺 = (Scalar‘𝑅)
21fvexi 6920 . . 3 𝐺 ∈ V
3 eqid 2734 . . . . 5 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)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), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)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)⟩})
43imasvalstr 17497 . . . 4 (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)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⟩
5 scaid 17360 . . . 4 Scalar = Slot (Scalar‘ndx)
6 snsstp1 4820 . . . . . 6 {⟨(Scalar‘ndx), 𝐺⟩} ⊆ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}
7 ssun2 4188 . . . . . 6 {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩})
86, 7sstri 4004 . . . . 5 {⟨(Scalar‘ndx), 𝐺⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩})
9 ssun1 4187 . . . . 5 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞)))⟩, ⟨(·𝑖‘ndx), 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩}⟩}) ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)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)⟩})
108, 9sstri 4004 . . . 4 {⟨(Scalar‘ndx), 𝐺⟩} ⊆ (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)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)⟩})
114, 5, 10strfv 17237 . . 3 (𝐺 ∈ V → 𝐺 = (Scalar‘(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)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)⟩})))
122, 11ax-mp 5 . 2 𝐺 = (Scalar‘(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)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)⟩}))
13 imasbas.u . . . 4 (𝜑𝑈 = (𝐹s 𝑅))
14 imasbas.v . . . 4 (𝜑𝑉 = (Base‘𝑅))
15 eqid 2734 . . . 4 (+g𝑅) = (+g𝑅)
16 eqid 2734 . . . 4 (.r𝑅) = (.r𝑅)
17 eqid 2734 . . . 4 (Base‘𝐺) = (Base‘𝐺)
18 eqid 2734 . . . 4 ( ·𝑠𝑅) = ( ·𝑠𝑅)
19 eqid 2734 . . . 4 (·𝑖𝑅) = (·𝑖𝑅)
20 eqid 2734 . . . 4 (TopOpen‘𝑅) = (TopOpen‘𝑅)
21 eqid 2734 . . . 4 (dist‘𝑅) = (dist‘𝑅)
22 eqid 2734 . . . 4 (le‘𝑅) = (le‘𝑅)
23 imasbas.f . . . . 5 (𝜑𝐹:𝑉onto𝐵)
24 imasbas.r . . . . 5 (𝜑𝑅𝑍)
25 eqid 2734 . . . . 5 (+g𝑈) = (+g𝑈)
2613, 14, 23, 24, 15, 25imasplusg 17563 . . . 4 (𝜑 → (+g𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(+g𝑅)𝑞))⟩})
27 eqid 2734 . . . . 5 (.r𝑈) = (.r𝑈)
2813, 14, 23, 24, 16, 27imasmulr 17564 . . . 4 (𝜑 → (.r𝑈) = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝐹‘(𝑝(.r𝑅)𝑞))⟩})
29 eqidd 2735 . . . 4 (𝜑 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞))) = 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)𝑞))))
30 eqidd 2735 . . . 4 (𝜑 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩} = 𝑝𝑉 𝑞𝑉 {⟨⟨(𝐹𝑝), (𝐹𝑞)⟩, (𝑝(·𝑖𝑅)𝑞)⟩})
31 eqidd 2735 . . . 4 (𝜑 → ((TopOpen‘𝑅) qTop 𝐹) = ((TopOpen‘𝑅) qTop 𝐹))
32 eqid 2734 . . . . 5 (dist‘𝑈) = (dist‘𝑈)
3313, 14, 23, 24, 21, 32imasds 17559 . . . 4 (𝜑 → (dist‘𝑈) = (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg ((dist‘𝑅) ∘ 𝑔))), ℝ*, < )))
34 eqidd 2735 . . . 4 (𝜑 → ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹) = ((𝐹 ∘ (le‘𝑅)) ∘ 𝐹))
3513, 14, 15, 16, 1, 17, 18, 19, 20, 21, 22, 26, 28, 29, 30, 31, 33, 34, 23, 24imasval 17557 . . 3 (𝜑𝑈 = (({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)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)⟩}))
3635fveq2d 6910 . 2 (𝜑 → (Scalar‘𝑈) = (Scalar‘(({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g𝑈)⟩, ⟨(.r‘ndx), (.r𝑈)⟩} ∪ {⟨(Scalar‘ndx), 𝐺⟩, ⟨( ·𝑠 ‘ndx), 𝑞𝑉 (𝑝 ∈ (Base‘𝐺), 𝑥 ∈ {(𝐹𝑞)} ↦ (𝐹‘(𝑝( ·𝑠𝑅)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)⟩})))
3712, 36eqtr4id 2793 1 (𝜑𝐺 = (Scalar‘𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  Vcvv 3477  cun 3960  {csn 4630  {ctp 4634  cop 4636   ciun 4995  ccnv 5687  ccom 5692  ontowfo 6560  cfv 6562  (class class class)co 7430  cmpo 7432  1c1 11153  2c2 12318  cdc 12730  ndxcnx 17226  Basecbs 17244  +gcplusg 17297  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  ·𝑖cip 17302  TopSetcts 17303  lecple 17304  distcds 17306  TopOpenctopn 17467   qTop cqtop 17549  s cimas 17550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-struct 17180  df-slot 17215  df-ndx 17227  df-base 17245  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-imas 17554
This theorem is referenced by:  quss  17592  xpssca  17622  imaslmod  33360  imaslmhm  33364  algextdeglem8  33729
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