MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psrplusgpropd Structured version   Visualization version   GIF version

Theorem psrplusgpropd 19373
Description: Property deduction for power series addition. (Contributed by Stefan O'Rear, 27-Mar-2015.) (Revised by Mario Carneiro, 3-Oct-2015.)
Hypotheses
Ref Expression
psrplusgpropd.b1 (𝜑𝐵 = (Base‘𝑅))
psrplusgpropd.b2 (𝜑𝐵 = (Base‘𝑆))
psrplusgpropd.p ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))
Assertion
Ref Expression
psrplusgpropd (𝜑 → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑆)))
Distinct variable groups:   𝜑,𝑦,𝑥   𝑥,𝐵,𝑦   𝑦,𝑅,𝑥   𝑦,𝑆,𝑥
Allowed substitution hints:   𝐼(𝑥,𝑦)

Proof of Theorem psrplusgpropd
Dummy variables 𝑎 𝑏 𝑑 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl1 1056 . . . . . . . 8 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → 𝜑)
2 eqid 2609 . . . . . . . . . . 11 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
3 eqid 2609 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
4 eqid 2609 . . . . . . . . . . 11 {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} = {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}
5 eqid 2609 . . . . . . . . . . 11 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
6 simp2 1054 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)))
72, 3, 4, 5, 6psrelbas 19146 . . . . . . . . . 10 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑎:{𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
87ffvelrnda 6252 . . . . . . . . 9 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → (𝑎𝑑) ∈ (Base‘𝑅))
9 psrplusgpropd.b1 . . . . . . . . . 10 (𝜑𝐵 = (Base‘𝑅))
101, 9syl 17 . . . . . . . . 9 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → 𝐵 = (Base‘𝑅))
118, 10eleqtrrd 2690 . . . . . . . 8 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → (𝑎𝑑) ∈ 𝐵)
12 simp3 1055 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)))
132, 3, 4, 5, 12psrelbas 19146 . . . . . . . . . 10 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑏:{𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
1413ffvelrnda 6252 . . . . . . . . 9 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → (𝑏𝑑) ∈ (Base‘𝑅))
1514, 10eleqtrrd 2690 . . . . . . . 8 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → (𝑏𝑑) ∈ 𝐵)
16 psrplusgpropd.p . . . . . . . . 9 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑆)𝑦))
1716oveqrspc2v 6550 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝑑) ∈ 𝐵 ∧ (𝑏𝑑) ∈ 𝐵)) → ((𝑎𝑑)(+g𝑅)(𝑏𝑑)) = ((𝑎𝑑)(+g𝑆)(𝑏𝑑)))
181, 11, 15, 17syl12anc 1315 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → ((𝑎𝑑)(+g𝑅)(𝑏𝑑)) = ((𝑎𝑑)(+g𝑆)(𝑏𝑑)))
1918mpteq2dva 4666 . . . . . 6 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} ↦ ((𝑎𝑑)(+g𝑅)(𝑏𝑑))) = (𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} ↦ ((𝑎𝑑)(+g𝑆)(𝑏𝑑))))
20 ffn 5944 . . . . . . . 8 (𝑎:{𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅) → 𝑎 Fn {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin})
217, 20syl 17 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑎 Fn {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin})
22 ffn 5944 . . . . . . . 8 (𝑏:{𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅) → 𝑏 Fn {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin})
2313, 22syl 17 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑏 Fn {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin})
24 ovex 6555 . . . . . . . . 9 (ℕ0𝑚 𝐼) ∈ V
2524rabex 4735 . . . . . . . 8 {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} ∈ V
2625a1i 11 . . . . . . 7 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} ∈ V)
27 inidm 3783 . . . . . . 7 ({𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} ∩ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) = {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}
28 eqidd 2610 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → (𝑎𝑑) = (𝑎𝑑))
29 eqidd 2610 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → (𝑏𝑑) = (𝑏𝑑))
3021, 23, 26, 26, 27, 28, 29offval 6779 . . . . . 6 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑎𝑓 (+g𝑅)𝑏) = (𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} ↦ ((𝑎𝑑)(+g𝑅)(𝑏𝑑))))
3121, 23, 26, 26, 27, 28, 29offval 6779 . . . . . 6 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑎𝑓 (+g𝑆)𝑏) = (𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} ↦ ((𝑎𝑑)(+g𝑆)(𝑏𝑑))))
3219, 30, 313eqtr4d 2653 . . . . 5 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑎𝑓 (+g𝑅)𝑏) = (𝑎𝑓 (+g𝑆)𝑏))
3332mpt2eq3dva 6595 . . . 4 (𝜑 → (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎𝑓 (+g𝑅)𝑏)) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎𝑓 (+g𝑆)𝑏)))
34 psrplusgpropd.b2 . . . . . . 7 (𝜑𝐵 = (Base‘𝑆))
359, 34eqtr3d 2645 . . . . . 6 (𝜑 → (Base‘𝑅) = (Base‘𝑆))
3635psrbaspropd 19372 . . . . 5 (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
37 mpt2eq12 6591 . . . . 5 (((Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)) ∧ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆))) → (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎𝑓 (+g𝑆)𝑏)) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑆)) ↦ (𝑎𝑓 (+g𝑆)𝑏)))
3836, 36, 37syl2anc 690 . . . 4 (𝜑 → (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎𝑓 (+g𝑆)𝑏)) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑆)) ↦ (𝑎𝑓 (+g𝑆)𝑏)))
3933, 38eqtrd 2643 . . 3 (𝜑 → (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎𝑓 (+g𝑅)𝑏)) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑆)) ↦ (𝑎𝑓 (+g𝑆)𝑏)))
40 ofmres 7032 . . 3 ( ∘𝑓 (+g𝑅) ↾ ((Base‘(𝐼 mPwSer 𝑅)) × (Base‘(𝐼 mPwSer 𝑅)))) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎𝑓 (+g𝑅)𝑏))
41 ofmres 7032 . . 3 ( ∘𝑓 (+g𝑆) ↾ ((Base‘(𝐼 mPwSer 𝑆)) × (Base‘(𝐼 mPwSer 𝑆)))) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑆)) ↦ (𝑎𝑓 (+g𝑆)𝑏))
4239, 40, 413eqtr4g 2668 . 2 (𝜑 → ( ∘𝑓 (+g𝑅) ↾ ((Base‘(𝐼 mPwSer 𝑅)) × (Base‘(𝐼 mPwSer 𝑅)))) = ( ∘𝑓 (+g𝑆) ↾ ((Base‘(𝐼 mPwSer 𝑆)) × (Base‘(𝐼 mPwSer 𝑆)))))
43 eqid 2609 . . 3 (+g𝑅) = (+g𝑅)
44 eqid 2609 . . 3 (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑅))
452, 5, 43, 44psrplusg 19148 . 2 (+g‘(𝐼 mPwSer 𝑅)) = ( ∘𝑓 (+g𝑅) ↾ ((Base‘(𝐼 mPwSer 𝑅)) × (Base‘(𝐼 mPwSer 𝑅))))
46 eqid 2609 . . 3 (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆)
47 eqid 2609 . . 3 (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆))
48 eqid 2609 . . 3 (+g𝑆) = (+g𝑆)
49 eqid 2609 . . 3 (+g‘(𝐼 mPwSer 𝑆)) = (+g‘(𝐼 mPwSer 𝑆))
5046, 47, 48, 49psrplusg 19148 . 2 (+g‘(𝐼 mPwSer 𝑆)) = ( ∘𝑓 (+g𝑆) ↾ ((Base‘(𝐼 mPwSer 𝑆)) × (Base‘(𝐼 mPwSer 𝑆))))
5142, 45, 503eqtr4g 2668 1 (𝜑 → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  {crab 2899  Vcvv 3172  cmpt 4637   × cxp 5026  ccnv 5027  cres 5030  cima 5031   Fn wfn 5785  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  𝑓 cof 6770  𝑚 cmap 7721  Fincfn 7818  cn 10867  0cn0 11139  Basecbs 15641  +gcplusg 15714   mPwSer cmps 19118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-of 6772  df-om 6935  df-1st 7036  df-2nd 7037  df-supp 7160  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-map 7723  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-fsupp 8136  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-2 10926  df-3 10927  df-4 10928  df-5 10929  df-6 10930  df-7 10931  df-8 10932  df-9 10933  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-struct 15643  df-ndx 15644  df-slot 15645  df-base 15646  df-plusg 15727  df-mulr 15728  df-sca 15730  df-vsca 15731  df-tset 15733  df-psr 19123
This theorem is referenced by:  ply1plusgpropd  19381
  Copyright terms: Public domain W3C validator