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Theorem psrplusg 20096
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 2826 . . . . 5 (Base‘𝑅) = (Base‘𝑅)
3 psrplusg.a . . . . 5 + = (+g𝑅)
4 eqid 2826 . . . . 5 (.r𝑅) = (.r𝑅)
5 eqid 2826 . . . . 5 (TopOpen‘𝑅) = (TopOpen‘𝑅)
6 eqid 2826 . . . . 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 20093 . . . . 5 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ((Base‘𝑅) ↑m { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}))
10 eqid 2826 . . . . 5 ( ∘f + ↾ (𝐵 × 𝐵)) = ( ∘f + ↾ (𝐵 × 𝐵))
11 eqid 2826 . . . . 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 2826 . . . . 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 2827 . . . . 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 20077 . . . 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 6673 . . 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 6683 . . . . 5 𝐵 ∈ V
1918, 18xpex 7469 . . . 4 (𝐵 × 𝐵) ∈ V
20 ofexg 7407 . . . 4 ((𝐵 × 𝐵) ∈ V → ( ∘f + ↾ (𝐵 × 𝐵)) ∈ V)
21 psrvalstr 20078 . . . . 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 16591 . . . . 5 +g = Slot (+g‘ndx)
23 snsstp2 4749 . . . . . 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 4152 . . . . . 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 3980 . . . . 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 16526 . . . 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 2886 . 2 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → = ( ∘f + ↾ (𝐵 × 𝐵)))
29 reldmpsr 20076 . . . . . . 7 Rel dom mPwSer
3029ovprc 7188 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
311, 30syl5eq 2873 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
3231fveq2d 6673 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (+g𝑆) = (+g‘∅))
3322str0 16530 . . . 4 ∅ = (+g‘∅)
3432, 17, 333eqtr4g 2886 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → = ∅)
3531fveq2d 6673 . . . . . . . 8 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (Base‘𝑆) = (Base‘∅))
36 base0 16531 . . . . . . . 8 ∅ = (Base‘∅)
3735, 7, 363eqtr4g 2886 . . . . . . 7 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
3837xpeq2d 5584 . . . . . 6 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = (𝐵 × ∅))
39 xp0 6014 . . . . . 6 (𝐵 × ∅) = ∅
4038, 39syl6eq 2877 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐵 × 𝐵) = ∅)
4140reseq2d 5852 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘f + ↾ (𝐵 × 𝐵)) = ( ∘f + ↾ ∅))
42 res0 5856 . . . 4 ( ∘f + ↾ ∅) = ∅
4341, 42syl6eq 2877 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∘f + ↾ (𝐵 × 𝐵)) = ∅)
4434, 43eqtr4d 2864 . 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 1530  wcel 2107  {crab 3147  Vcvv 3500  cun 3938  c0 4295  {csn 4564  {ctp 4568  cop 4570   class class class wbr 5063  cmpt 5143   × cxp 5552  ccnv 5553  cres 5556  cima 5557  cfv 6354  (class class class)co 7150  cmpo 7152  f cof 7401  r cofr 7402  m cmap 8401  Fincfn 8503  1c1 10532  cle 10670  cmin 10864  cn 11632  9c9 11693  0cn0 11891  ndxcnx 16475  Basecbs 16478  +gcplusg 16560  .rcmulr 16561  Scalarcsca 16563   ·𝑠 cvsca 16564  TopSetcts 16566  TopOpenctopn 16690  tcpt 16707   Σg cgsu 16709   mPwSer cmps 20066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7403  df-om 7574  df-1st 7685  df-2nd 7686  df-supp 7827  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fsupp 8828  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-7 11699  df-8 11700  df-9 11701  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12888  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-tset 16579  df-psr 20071
This theorem is referenced by:  psradd  20097  psrmulr  20099  psrsca  20104  psrvscafval  20105  psrplusgpropd  20339  ply1plusgfvi  20345
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