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Theorem psrplusg 19583
Description: The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
psrplusg.s 𝑆 = (𝐼 mPwSer 𝑅)
psrplusg.b 𝐵 = (Base‘𝑆)
psrplusg.a + = (+g𝑅)
psrplusg.p = (+g𝑆)
Assertion
Ref Expression
psrplusg = ( ∘𝑓 + ↾ (𝐵 × 𝐵))

Proof of Theorem psrplusg
Dummy variables 𝑓 𝑔 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psrplusg.s . . . . 5 𝑆 = (𝐼 mPwSer 𝑅)
2 eqid 2760 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
3 psrplusg.a . . . . 5 + = (+g𝑅)
4 eqid 2760 . . . . 5 (.r𝑅) = (.r𝑅)
5 eqid 2760 . . . . 5 (TopOpen‘𝑅) = (TopOpen‘𝑅)
6 eqid 2760 . . . . 5 { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}
7 psrplusg.b . . . . . 6 𝐵 = (Base‘𝑆)
8 simpl 474 . . . . . 6 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐼 ∈ V)
91, 2, 6, 7, 8psrbas 19580 . . . . 5 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ((Base‘𝑅) ↑𝑚 { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin}))
10 eqid 2760 . . . . 5 ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = ( ∘𝑓 + ↾ (𝐵 × 𝐵))
11 eqid 2760 . . . . 5 (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥))))))) = (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))
12 eqid 2760 . . . . 5 (𝑥 ∈ (Base‘𝑅), 𝑓𝐵 ↦ (({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r𝑅)𝑓)) = (𝑥 ∈ (Base‘𝑅), 𝑓𝐵 ↦ (({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r𝑅)𝑓))
13 eqidd 2761 . . . . 5 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (∏t‘({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})) = (∏t‘({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})))
14 simpr 479 . . . . 5 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
151, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 8, 14psrval 19564 . . . 4 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓𝐵 ↦ (({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}))
1615fveq2d 6356 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (+g𝑆) = (+g‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓𝐵 ↦ (({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
17 psrplusg.p . . 3 = (+g𝑆)
18 fvex 6362 . . . . . 6 (Base‘𝑆) ∈ V
197, 18eqeltri 2835 . . . . 5 𝐵 ∈ V
2019, 19xpex 7127 . . . 4 (𝐵 × 𝐵) ∈ V
21 ofexg 7066 . . . 4 ((𝐵 × 𝐵) ∈ V → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) ∈ V)
22 psrvalstr 19565 . . . . 5 ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓𝐵 ↦ (({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}) Struct ⟨1, 9⟩
23 plusgid 16179 . . . . 5 +g = Slot (+g‘ndx)
24 snsstp2 4493 . . . . . 6 {⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩} ⊆ {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))⟩}
25 ssun1 3919 . . . . . 6 {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓𝐵 ↦ (({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})
2624, 25sstri 3753 . . . . 5 {⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓𝐵 ↦ (({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})
2722, 23, 26strfv 16109 . . . 4 (( ∘𝑓 + ↾ (𝐵 × 𝐵)) ∈ V → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = (+g‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓𝐵 ↦ (({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩})))
2820, 21, 27mp2b 10 . . 3 ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = (+g‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), ( ∘𝑓 + ↾ (𝐵 × 𝐵))⟩, ⟨(.r‘ndx), (𝑓𝐵, 𝑔𝐵 ↦ (𝑘 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ { ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑦𝑟𝑘} ↦ ((𝑓𝑥)(.r𝑅)(𝑔‘(𝑘𝑓𝑥)))))))⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑓𝐵 ↦ (({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {𝑥}) ∘𝑓 (.r𝑅)𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘({ ∈ (ℕ0𝑚 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)}))⟩}))
2916, 17, 283eqtr4g 2819 . 2 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))
30 reldmpsr 19563 . . . . . . 7 Rel dom mPwSer
3130ovprc 6846 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
321, 31syl5eq 2806 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
3332fveq2d 6356 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (+g𝑆) = (+g‘∅))
3423str0 16113 . . . 4 ∅ = (+g‘∅)
3533, 17, 343eqtr4g 2819 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → = ∅)
3632fveq2d 6356 . . . . . . . 8 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑆) = (Base‘∅))
37 base0 16114 . . . . . . . 8 ∅ = (Base‘∅)
3836, 7, 373eqtr4g 2819 . . . . . . 7 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
3938xpeq2d 5296 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = (𝐵 × ∅))
40 xp0 5710 . . . . . 6 (𝐵 × ∅) = ∅
4139, 40syl6eq 2810 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = ∅)
4241reseq2d 5551 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = ( ∘𝑓 + ↾ ∅))
43 res0 5555 . . . 4 ( ∘𝑓 + ↾ ∅) = ∅
4442, 43syl6eq 2810 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘𝑓 + ↾ (𝐵 × 𝐵)) = ∅)
4535, 44eqtr4d 2797 . 2 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → = ( ∘𝑓 + ↾ (𝐵 × 𝐵)))
4629, 45pm2.61i 176 1 = ( ∘𝑓 + ↾ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1632  wcel 2139  {crab 3054  Vcvv 3340  cun 3713  c0 4058  {csn 4321  {ctp 4325  cop 4327   class class class wbr 4804  cmpt 4881   × cxp 5264  ccnv 5265  cres 5268  cima 5269  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑓 cof 7060  𝑟 cofr 7061  𝑚 cmap 8023  Fincfn 8121  1c1 10129  cle 10267  cmin 10458  cn 11212  9c9 11269  0cn0 11484  ndxcnx 16056  Basecbs 16059  +gcplusg 16143  .rcmulr 16144  Scalarcsca 16146   ·𝑠 cvsca 16147  TopSetcts 16149  TopOpenctopn 16284  tcpt 16301   Σg cgsu 16303   mPwSer cmps 19553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-uz 11880  df-fz 12520  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-tset 16162  df-psr 19558
This theorem is referenced by:  psradd  19584  psrmulr  19586  psrsca  19591  psrvscafval  19592  psrplusgpropd  19808  ply1plusgfvi  19814
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