Theorem psrplusgpropd 19606

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	⊢ (𝜑 → 𝐵 = (Base‘𝑅))
psrplusgpropd.b2	⊢ (𝜑 → 𝐵 = (Base‘𝑆))
psrplusgpropd.p	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)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.

Step	Hyp	Ref	Expression
1		simpl1 1064	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) → 𝜑)
2		eqid 2622	. . . . . . . . . . 11 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅)
3		eqid 2622	. . . . . . . . . . 11 ⊢ (Base‘𝑅) = (Base‘𝑅)
4		eqid 2622	. . . . . . . . . . 11 ⊢ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin} = {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}
5		eqid 2622	. . . . . . . . . . 11 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅))
6		simp2 1062	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)))
7	2, 3, 4, 5, 6	psrelbas 19379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑎:{𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
8	7	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) → (𝑎‘𝑑) ∈ (Base‘𝑅))
9		psrplusgpropd.b1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = (Base‘𝑅))
10	1, 9	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) → 𝐵 = (Base‘𝑅))
11	8, 10	eleqtrrd 2704	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) → (𝑎‘𝑑) ∈ 𝐵)
12		simp3 1063	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)))
13	2, 3, 4, 5, 12	psrelbas 19379	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑏:{𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅))
14	13	ffvelrnda 6359	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) → (𝑏‘𝑑) ∈ (Base‘𝑅))
15	14, 10	eleqtrrd 2704	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) → (𝑏‘𝑑) ∈ 𝐵)
16		psrplusgpropd.p	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑆)𝑦))
17	16	oveqrspc2v 6673	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑎‘𝑑) ∈ 𝐵 ∧ (𝑏‘𝑑) ∈ 𝐵)) → ((𝑎‘𝑑)(+g‘𝑅)(𝑏‘𝑑)) = ((𝑎‘𝑑)(+g‘𝑆)(𝑏‘𝑑)))
18	1, 11, 15, 17	syl12anc 1324	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) → ((𝑎‘𝑑)(+g‘𝑅)(𝑏‘𝑑)) = ((𝑎‘𝑑)(+g‘𝑆)(𝑏‘𝑑)))
19	18	mpteq2dva 4744	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin} ↦ ((𝑎‘𝑑)(+g‘𝑅)(𝑏‘𝑑))) = (𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin} ↦ ((𝑎‘𝑑)(+g‘𝑆)(𝑏‘𝑑))))
20		ffn 6045	. . . . . . . 8 ⊢ (𝑎:{𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅) → 𝑎 Fn {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin})
21	7, 20	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑎 Fn {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin})
22		ffn 6045	. . . . . . . 8 ⊢ (𝑏:{𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}⟶(Base‘𝑅) → 𝑏 Fn {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin})
23	13, 22	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → 𝑏 Fn {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin})
24		ovex 6678	. . . . . . . . 9 ⊢ (ℕ0 ↑𝑚 𝐼) ∈ V
25	24	rabex 4813	. . . . . . . 8 ⊢ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin} ∈ V
26	25	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin} ∈ V)
27		inidm 3822	. . . . . . 7 ⊢ ({𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin} ∩ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) = {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}
28		eqidd 2623	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) → (𝑎‘𝑑) = (𝑎‘𝑑))
29		eqidd 2623	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) ∧ 𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin}) → (𝑏‘𝑑) = (𝑏‘𝑑))
30	21, 23, 26, 26, 27, 28, 29	offval 6904	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑎 ∘𝑓 (+g‘𝑅)𝑏) = (𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin} ↦ ((𝑎‘𝑑)(+g‘𝑅)(𝑏‘𝑑))))
31	21, 23, 26, 26, 27, 28, 29	offval 6904	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑎 ∘𝑓 (+g‘𝑆)𝑏) = (𝑑 ∈ {𝑐 ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡𝑐 “ ℕ) ∈ Fin} ↦ ((𝑎‘𝑑)(+g‘𝑆)(𝑏‘𝑑))))
32	19, 30, 31	3eqtr4d 2666	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅))) → (𝑎 ∘𝑓 (+g‘𝑅)𝑏) = (𝑎 ∘𝑓 (+g‘𝑆)𝑏))
33	32	mpt2eq3dva 6719	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎 ∘𝑓 (+g‘𝑅)𝑏)) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎 ∘𝑓 (+g‘𝑆)𝑏)))
34		psrplusgpropd.b2	. . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝑆))
35	9, 34	eqtr3d 2658	. . . . . 6 ⊢ (𝜑 → (Base‘𝑅) = (Base‘𝑆))
36	35	psrbaspropd 19605	. . . . 5 ⊢ (𝜑 → (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑆)))
37		mpt2eq12 6715	. . . . 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)) ↦ (𝑎 ∘𝑓 (+g‘𝑆)𝑏)) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑆)) ↦ (𝑎 ∘𝑓 (+g‘𝑆)𝑏)))
38	36, 36, 37	syl2anc 693	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎 ∘𝑓 (+g‘𝑆)𝑏)) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑆)) ↦ (𝑎 ∘𝑓 (+g‘𝑆)𝑏)))
39	33, 38	eqtrd 2656	. . 3 ⊢ (𝜑 → (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎 ∘𝑓 (+g‘𝑅)𝑏)) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑆)) ↦ (𝑎 ∘𝑓 (+g‘𝑆)𝑏)))
40		ofmres 7164	. . 3 ⊢ ( ∘𝑓 (+g‘𝑅) ↾ ((Base‘(𝐼 mPwSer 𝑅)) × (Base‘(𝐼 mPwSer 𝑅)))) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑅)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑅)) ↦ (𝑎 ∘𝑓 (+g‘𝑅)𝑏))
41		ofmres 7164	. . 3 ⊢ ( ∘𝑓 (+g‘𝑆) ↾ ((Base‘(𝐼 mPwSer 𝑆)) × (Base‘(𝐼 mPwSer 𝑆)))) = (𝑎 ∈ (Base‘(𝐼 mPwSer 𝑆)), 𝑏 ∈ (Base‘(𝐼 mPwSer 𝑆)) ↦ (𝑎 ∘𝑓 (+g‘𝑆)𝑏))
42	39, 40, 41	3eqtr4g 2681	. 2 ⊢ (𝜑 → ( ∘𝑓 (+g‘𝑅) ↾ ((Base‘(𝐼 mPwSer 𝑅)) × (Base‘(𝐼 mPwSer 𝑅)))) = ( ∘𝑓 (+g‘𝑆) ↾ ((Base‘(𝐼 mPwSer 𝑆)) × (Base‘(𝐼 mPwSer 𝑆)))))
43		eqid 2622	. . 3 ⊢ (+g‘𝑅) = (+g‘𝑅)
44		eqid 2622	. . 3 ⊢ (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑅))
45	2, 5, 43, 44	psrplusg 19381	. 2 ⊢ (+g‘(𝐼 mPwSer 𝑅)) = ( ∘𝑓 (+g‘𝑅) ↾ ((Base‘(𝐼 mPwSer 𝑅)) × (Base‘(𝐼 mPwSer 𝑅))))
46		eqid 2622	. . 3 ⊢ (𝐼 mPwSer 𝑆) = (𝐼 mPwSer 𝑆)
47		eqid 2622	. . 3 ⊢ (Base‘(𝐼 mPwSer 𝑆)) = (Base‘(𝐼 mPwSer 𝑆))
48		eqid 2622	. . 3 ⊢ (+g‘𝑆) = (+g‘𝑆)
49		eqid 2622	. . 3 ⊢ (+g‘(𝐼 mPwSer 𝑆)) = (+g‘(𝐼 mPwSer 𝑆))
50	46, 47, 48, 49	psrplusg 19381	. 2 ⊢ (+g‘(𝐼 mPwSer 𝑆)) = ( ∘𝑓 (+g‘𝑆) ↾ ((Base‘(𝐼 mPwSer 𝑆)) × (Base‘(𝐼 mPwSer 𝑆))))
51	42, 45, 50	3eqtr4g 2681	1 ⊢ (𝜑 → (+g‘(𝐼 mPwSer 𝑅)) = (+g‘(𝐼 mPwSer 𝑆)))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113   ↾ cres 5116   “ cima 5117   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ∘_𝑓 cof 6895   ↑_𝑚 cmap 7857  Fincfn 7955  ℕcn 11020  ℕ₀cn0 11292  Basecbs 15857  +_gcplusg 15941   mPwSer cmps 19351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-tset 15960  df-psr 19356

This theorem is referenced by:  ply1plusgpropd  19614