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Theorem psrplusg 20089
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 2818 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
3 psrplusg.a . . . . 5 + = (+g𝑅)
4 eqid 2818 . . . . 5 (.r𝑅) = (.r𝑅)
5 eqid 2818 . . . . 5 (TopOpen‘𝑅) = (TopOpen‘𝑅)
6 eqid 2818 . . . . 5 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
7 psrplusg.b . . . . . 6 𝐵 = (Base‘𝑆)
8 simpl 483 . . . . . 6 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐼 ∈ V)
91, 2, 6, 7, 8psrbas 20086 . . . . 5 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ((Base‘𝑅) ↑m { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}))
10 eqid 2818 . . . . 5 ( ∘f + ↾ (𝐵 × 𝐵)) = ( ∘f + ↾ (𝐵 × 𝐵))
11 eqid 2818 . . . . 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 2818 . . . . 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 2819 . . . . 5 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})) = (∏t‘({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} × {(TopOpen‘𝑅)})))
14 simpr 485 . . . . 5 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
151, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 8, 14psrval 20070 . . . 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 6667 . . 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)
187fvexi 6677 . . . . 5 𝐵 ∈ V
1918, 18xpex 7465 . . . 4 (𝐵 × 𝐵) ∈ V
20 ofexg 7402 . . . 4 ((𝐵 × 𝐵) ∈ V → ( ∘f + ↾ (𝐵 × 𝐵)) ∈ V)
21 psrvalstr 20071 . . . . 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⟩
22 plusgid 16584 . . . . 5 +g = Slot (+g‘ndx)
23 snsstp2 4742 . . . . . 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)))))))⟩}
24 ssun1 4145 . . . . . 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)}))⟩})
2523, 24sstri 3973 . . . . 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)}))⟩})
2621, 22, 25strfv 16519 . . . 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)}))⟩})))
2719, 20, 26mp2b 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)}))⟩}))
2816, 17, 273eqtr4g 2878 . 2 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → = ( ∘f + ↾ (𝐵 × 𝐵)))
29 reldmpsr 20069 . . . . . . 7 Rel dom mPwSer
3029ovprc 7183 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
311, 30syl5eq 2865 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
3231fveq2d 6667 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (+g𝑆) = (+g‘∅))
3322str0 16523 . . . 4 ∅ = (+g‘∅)
3432, 17, 333eqtr4g 2878 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → = ∅)
3531fveq2d 6667 . . . . . . . 8 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑆) = (Base‘∅))
36 base0 16524 . . . . . . . 8 ∅ = (Base‘∅)
3735, 7, 363eqtr4g 2878 . . . . . . 7 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
3837xpeq2d 5578 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = (𝐵 × ∅))
39 xp0 6008 . . . . . 6 (𝐵 × ∅) = ∅
4038, 39syl6eq 2869 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = ∅)
4140reseq2d 5846 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘f + ↾ (𝐵 × 𝐵)) = ( ∘f + ↾ ∅))
42 res0 5850 . . . 4 ( ∘f + ↾ ∅) = ∅
4341, 42syl6eq 2869 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘f + ↾ (𝐵 × 𝐵)) = ∅)
4434, 43eqtr4d 2856 . 2 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → = ( ∘f + ↾ (𝐵 × 𝐵)))
4528, 44pm2.61i 183 1 = ( ∘f + ↾ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  cun 3931  c0 4288  {csn 4557  {ctp 4561  cop 4563   class class class wbr 5057  cmpt 5137   × cxp 5546  ccnv 5547  cres 5550  cima 5551  cfv 6348  (class class class)co 7145  cmpo 7147  f cof 7396  r cofr 7397  m cmap 8395  Fincfn 8497  1c1 10526  cle 10664  cmin 10858  cn 11626  9c9 11687  0cn0 11885  ndxcnx 16468  Basecbs 16471  +gcplusg 16553  .rcmulr 16554  Scalarcsca 16556   ·𝑠 cvsca 16557  TopSetcts 16559  TopOpenctopn 16683  tcpt 16700   Σg cgsu 16702   mPwSer cmps 20059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-om 7570  df-1st 7678  df-2nd 7679  df-supp 7820  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fsupp 8822  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-plusg 16566  df-mulr 16567  df-sca 16569  df-vsca 16570  df-tset 16572  df-psr 20064
This theorem is referenced by:  psradd  20090  psrmulr  20092  psrsca  20097  psrvscafval  20098  psrplusgpropd  20332  ply1plusgfvi  20338
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