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Theorem psrplusgpropd 19427
 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 1057 . . . . . . . 8 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → 𝜑)
2 eqid 2610 . . . . . . . . . . 11 (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
3 eqid 2610 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
4 eqid 2610 . . . . . . . . . . 11 {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} = {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}
5 eqid 2610 . . . . . . . . . . 11 (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
6 simp2 1055 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)))
72, 3, 4, 5, 6psrelbas 19200 . . . . . . . . . 10 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑎:{𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
87ffvelrnda 6267 . . . . . . . . 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 2691 . . . . . . . 8 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → (𝑎𝑑) ∈ 𝐵)
12 simp3 1056 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)))
132, 3, 4, 5, 12psrelbas 19200 . . . . . . . . . 10 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑏:{𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
1413ffvelrnda 6267 . . . . . . . . 9 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → (𝑏𝑑) ∈ (Base‘𝑅))
1514, 10eleqtrrd 2691 . . . . . . . 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 6572 . . . . . . . 8 ((𝜑 ∧ ((𝑎𝑑) ∈ 𝐵 ∧ (𝑏𝑑) ∈ 𝐵)) → ((𝑎𝑑)(+g𝑅)(𝑏𝑑)) = ((𝑎𝑑)(+g𝑆)(𝑏𝑑)))
181, 11, 15, 17syl12anc 1316 . . . . . . 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 4672 . . . . . 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 5958 . . . . . . . 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 5958 . . . . . . . 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 6577 . . . . . . . . 9 (ℕ0𝑚 𝐼) ∈ V
2524rabex 4740 . . . . . . . 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 3784 . . . . . . 7 ({𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin} ∩ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) = {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}
28 eqidd 2611 . . . . . . 7 (((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0𝑚 𝐼) ∣ (𝑐 “ ℕ) ∈ Fin}) → (𝑎𝑑) = (𝑎𝑑))
29 eqidd 2611 . . . . . . 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 6802 . . . . . 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 6802 . . . . . 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 2654 . . . . 5 ((𝜑𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑎𝑓 (+g𝑅)𝑏) = (𝑎𝑓 (+g𝑆)𝑏))
3332mpt2eq3dva 6617 . . . 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 2646 . . . . . 6 (𝜑 → (Base‘𝑅) = (Base‘𝑆))
3635psrbaspropd 19426 . . . . 5 (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
37 mpt2eq12 6613 . . . . 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 691 . . . 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 2644 . . 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 7055 . . 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 7055 . . 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 2669 . 2 (𝜑 → ( ∘𝑓 (+g𝑅) ↾ ((Base‘(𝐼 mPwSer 𝑅)) × (Base‘(𝐼 mPwSer 𝑅)))) = ( ∘𝑓 (+g𝑆) ↾ ((Base‘(𝐼 mPwSer 𝑆)) × (Base‘(𝐼 mPwSer 𝑆)))))
43 eqid 2610 . . 3 (+g𝑅) = (+g𝑅)
44 eqid 2610 . . 3 (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑅))
452, 5, 43, 44psrplusg 19202 . 2 (+g‘(𝐼 mPwSer 𝑅)) = ( ∘𝑓 (+g𝑅) ↾ ((Base‘(𝐼 mPwSer 𝑅)) × (Base‘(𝐼 mPwSer 𝑅))))
46 eqid 2610 . . 3 (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆)
47 eqid 2610 . . 3 (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆))
48 eqid 2610 . . 3 (+g𝑆) = (+g𝑆)
49 eqid 2610 . . 3 (+g‘(𝐼 mPwSer 𝑆)) = (+g‘(𝐼 mPwSer 𝑆))
5046, 47, 48, 49psrplusg 19202 . 2 (+g‘(𝐼 mPwSer 𝑆)) = ( ∘𝑓 (+g𝑆) ↾ ((Base‘(𝐼 mPwSer 𝑆)) × (Base‘(𝐼 mPwSer 𝑆))))
5142, 45, 503eqtr4g 2669 1 (𝜑 → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑆)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   ∘𝑓 cof 6793   ↑𝑚 cmap 7744  Fincfn 7841  ℕcn 10897  ℕ0cn0 11169  Basecbs 15695  +gcplusg 15768   mPwSer cmps 19172 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-tset 15787  df-psr 19177 This theorem is referenced by:  ply1plusgpropd  19435
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