Theorem psrvscafval 19390

Description: The scalar multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
psrvsca.s	⊢ 𝑆 = (𝐼 mPwSer 𝑅)
psrvsca.n	⊢ ∙ = ( ·𝑠 ‘𝑆)
psrvsca.k	⊢ 𝐾 = (Base‘𝑅)
psrvsca.b	⊢ 𝐵 = (Base‘𝑆)
psrvsca.m	⊢ · = (.r‘𝑅)
psrvsca.d	⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}

Assertion

Ref	Expression
psrvscafval	⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))

Distinct variable groups:   𝑥,𝑓,𝐵   𝑓,ℎ,𝐼,𝑥   𝑓,𝐾,𝑥   𝐷,𝑓,𝑥   𝑅,𝑓,𝑥   · ,𝑓,𝑥   ∙ ,𝑓,𝑥

Allowed substitution hints:   𝐵(ℎ)   𝐷(ℎ)   𝑅(ℎ)   𝑆(𝑥,𝑓,ℎ)   ∙ (ℎ)   · (ℎ)   𝐾(ℎ)

Proof of Theorem psrvscafval

Dummy variables 𝑔 𝑘 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psrvsca.s	. . . . 5 ⊢ 𝑆 = (𝐼 mPwSer 𝑅)
2		psrvsca.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑅)
3		eqid 2622	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		psrvsca.m	. . . . 5 ⊢ · = (.r‘𝑅)
5		eqid 2622	. . . . 5 ⊢ (TopOpen‘𝑅) = (TopOpen‘𝑅)
6		psrvsca.d	. . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑𝑚 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin}
7		psrvsca.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
8		simpl 473	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐼 ∈ V)
9	1, 2, 6, 7, 8	psrbas 19378	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = (𝐾 ↑𝑚 𝐷))
10		eqid 2622	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
11	1, 7, 3, 10	psrplusg 19381	. . . . 5 ⊢ (+g‘𝑆) = ( ∘𝑓 (+g‘𝑅) ↾ (𝐵 × 𝐵))
12		eqid 2622	. . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆)
13	1, 7, 4, 12, 6	psrmulr 19384	. . . . 5 ⊢ (.r‘𝑆) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ (𝑘 ∈ 𝐷 ↦ (𝑅 Σg (𝑥 ∈ {𝑦 ∈ 𝐷 ∣ 𝑦 ∘𝑟 ≤ 𝑘} ↦ ((𝑓‘𝑥) · (𝑔‘(𝑘 ∘𝑓 − 𝑥)))))))
14		eqid 2622	. . . . 5 ⊢ (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))
15		eqidd 2623	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (∏t‘(𝐷 × {(TopOpen‘𝑅)})) = (∏t‘(𝐷 × {(TopOpen‘𝑅)})))
16		simpr 477	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
17	1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 8, 16	psrval 19362	. . . 4 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))⟩}))
18	17	fveq2d 6195	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))⟩})))
19		psrvsca.n	. . 3 ⊢ ∙ = ( ·𝑠 ‘𝑆)
20		fvex 6201	. . . . . 6 ⊢ (Base‘𝑅) ∈ V
21	2, 20	eqeltri 2697	. . . . 5 ⊢ 𝐾 ∈ V
22		fvex 6201	. . . . . 6 ⊢ (Base‘𝑆) ∈ V
23	7, 22	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
24	21, 23	mpt2ex 7247	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) ∈ V
25		psrvalstr 19363	. . . . 5 ⊢ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))⟩}) Struct ⟨1, 9⟩
26		vscaid 16016	. . . . 5 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
27		snsstp2 4348	. . . . . 6 ⊢ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩} ⊆ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))⟩}
28		ssun2 3777	. . . . . 6 ⊢ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))⟩})
29	27, 28	sstri 3612	. . . . 5 ⊢ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩} ⊆ ({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))⟩})
30	25, 26, 29	strfv 15907	. . . 4 ⊢ ((𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) ∈ V → (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) = ( ·𝑠 ‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))⟩})))
31	24, 30	ax-mp 5	. . 3 ⊢ (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) = ( ·𝑠 ‘({⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), (+g‘𝑆)⟩, ⟨(.r‘ndx), (.r‘𝑆)⟩} ∪ {⟨(Scalar‘ndx), 𝑅⟩, ⟨( ·𝑠 ‘ndx), (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))⟩, ⟨(TopSet‘ndx), (∏t‘(𝐷 × {(TopOpen‘𝑅)}))⟩}))
32	18, 19, 31	3eqtr4g 2681	. 2 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)))
33		eqid 2622	. . . . . 6 ⊢ ∅ = ∅
34		fn0 6011	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
35	33, 34	mpbir 221	. . . . 5 ⊢ ∅ Fn ∅
36		reldmpsr 19361	. . . . . . . . . 10 ⊢ Rel dom mPwSer
37	36	ovprc 6683	. . . . . . . . 9 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
38	1, 37	syl5eq 2668	. . . . . . . 8 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
39	38	fveq2d 6195	. . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘∅))
40		df-vsca 15958	. . . . . . . 8 ⊢ ·𝑠 = Slot 6
41	40	str0 15911	. . . . . . 7 ⊢ ∅ = ( ·𝑠 ‘∅)
42	39, 19, 41	3eqtr4g 2681	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ = ∅)
43	36, 1, 7	elbasov 15921	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
44	43	con3i 150	. . . . . . . . 9 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ¬ 𝑓 ∈ 𝐵)
45	44	eq0rdv 3979	. . . . . . . 8 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
46	45	xpeq2d 5139	. . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐾 × 𝐵) = (𝐾 × ∅))
47		xp0 5552	. . . . . . 7 ⊢ (𝐾 × ∅) = ∅
48	46, 47	syl6eq 2672	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐾 × 𝐵) = ∅)
49	42, 48	fneq12d 5983	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∙ Fn (𝐾 × 𝐵) ↔ ∅ Fn ∅))
50	35, 49	mpbiri 248	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ Fn (𝐾 × 𝐵))
51		fnov 6768	. . . 4 ⊢ ( ∙ Fn (𝐾 × 𝐵) ↔ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ (𝑥 ∙ 𝑓)))
52	50, 51	sylib 208	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ (𝑥 ∙ 𝑓)))
53	44	pm2.21d 118	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑓 ∈ 𝐵 → ((𝐷 × {𝑥}) ∘𝑓 · 𝑓) = (𝑥 ∙ 𝑓)))
54	53	a1d 25	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑥 ∈ 𝐾 → (𝑓 ∈ 𝐵 → ((𝐷 × {𝑥}) ∘𝑓 · 𝑓) = (𝑥 ∙ 𝑓))))
55	54	3imp 1256	. . . 4 ⊢ ((¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑥 ∈ 𝐾 ∧ 𝑓 ∈ 𝐵) → ((𝐷 × {𝑥}) ∘𝑓 · 𝑓) = (𝑥 ∙ 𝑓))
56	55	mpt2eq3dva 6719	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)) = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ (𝑥 ∙ 𝑓)))
57	52, 56	eqtr4d 2659	. 2 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓)))
58	32, 57	pm2.61i 176	1 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘𝑓 · 𝑓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200   ∪ cun 3572  ∅c0 3915  {csn 4177  {ctp 4181  ⟨cop 4183   × cxp 5112  ^◡ccnv 5113   “ cima 5117   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   ∘_𝑓 cof 6895   ↑_𝑚 cmap 7857  Fincfn 7955  1c1 9937  ℕcn 11020  6c6 11074  9c9 11077  ℕ₀cn0 11292  ndxcnx 15854  Basecbs 15857  +_gcplusg 15941  ._rcmulr 15942  Scalarcsca 15944   ·_𝑠 cvsca 15945  TopSetcts 15947  TopOpenctopn 16082  ∏_tcpt 16099   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:  psrvsca  19391