Theorem psrvscafval 20170

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 2821	. . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅)
4		psrvsca.m	. . . . 5 ⊢ · = (.r‘𝑅)
5		eqid 2821	. . . . 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 485	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐼 ∈ V)
9	1, 2, 6, 7, 8	psrbas 20158	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = (𝐾 ↑m 𝐷))
10		eqid 2821	. . . . . 6 ⊢ (+g‘𝑆) = (+g‘𝑆)
11	1, 7, 3, 10	psrplusg 20161	. . . . 5 ⊢ (+g‘𝑆) = ( ∘f (+g‘𝑅) ↾ (𝐵 × 𝐵))
12		eqid 2821	. . . . . 6 ⊢ (.r‘𝑆) = (.r‘𝑆)
13	1, 7, 4, 12, 6	psrmulr 20164	. . . . 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 2821	. . . . 5 ⊢ (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓)) = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓))
15		eqidd 2822	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (∏t‘(𝐷 × {(TopOpen‘𝑅)})) = (∏t‘(𝐷 × {(TopOpen‘𝑅)})))
16		simpr 487	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑅 ∈ V)
17	1, 2, 3, 4, 5, 6, 9, 11, 13, 14, 15, 8, 16	psrval 20142	. . . 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 6674	. . 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	2	fvexi 6684	. . . . 5 ⊢ 𝐾 ∈ V
21	7	fvexi 6684	. . . . 5 ⊢ 𝐵 ∈ V
22	20, 21	mpoex 7777	. . . 4 ⊢ (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓)) ∈ V
23		psrvalstr 20143	. . . . 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⟩
24		vscaid 16635	. . . . 5 ⊢ ·𝑠 = Slot ( ·𝑠 ‘ndx)
25		snsstp2 4750	. . . . . 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)}))⟩}
26		ssun2 4149	. . . . . 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)}))⟩})
27	25, 26	sstri 3976	. . . . 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)}))⟩})
28	23, 24, 27	strfv 16531	. . . 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)}))⟩})))
29	22, 28	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)}))⟩}))
30	18, 19, 29	3eqtr4g 2881	. 2 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓)))
31		eqid 2821	. . . . . 6 ⊢ ∅ = ∅
32		fn0 6479	. . . . . 6 ⊢ (∅ Fn ∅ ↔ ∅ = ∅)
33	31, 32	mpbir 233	. . . . 5 ⊢ ∅ Fn ∅
34		reldmpsr 20141	. . . . . . . . . 10 ⊢ Rel dom mPwSer
35	34	ovprc 7194	. . . . . . . . 9 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 mPwSer 𝑅) = ∅)
36	1, 35	syl5eq 2868	. . . . . . . 8 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝑆 = ∅)
37	36	fveq2d 6674	. . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ·𝑠 ‘𝑆) = ( ·𝑠 ‘∅))
38		df-vsca 16582	. . . . . . . 8 ⊢ ·𝑠 = Slot 6
39	38	str0 16535	. . . . . . 7 ⊢ ∅ = ( ·𝑠 ‘∅)
40	37, 19, 39	3eqtr4g 2881	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ = ∅)
41	34, 1, 7	elbasov 16545	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐵 → (𝐼 ∈ V ∧ 𝑅 ∈ V))
42	41	con3i 157	. . . . . . . . 9 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ¬ 𝑓 ∈ 𝐵)
43	42	eq0rdv 4357	. . . . . . . 8 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → 𝐵 = ∅)
44	43	xpeq2d 5585	. . . . . . 7 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐾 × 𝐵) = (𝐾 × ∅))
45		xp0 6015	. . . . . . 7 ⊢ (𝐾 × ∅) = ∅
46	44, 45	syl6eq 2872	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐾 × 𝐵) = ∅)
47	40, 46	fneq12d 6448	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ( ∙ Fn (𝐾 × 𝐵) ↔ ∅ Fn ∅))
48	33, 47	mpbiri 260	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ Fn (𝐾 × 𝐵))
49		fnov 7282	. . . 4 ⊢ ( ∙ Fn (𝐾 × 𝐵) ↔ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ (𝑥 ∙ 𝑓)))
50	48, 49	sylib 220	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ (𝑥 ∙ 𝑓)))
51	42	pm2.21d 121	. . . . . 6 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑓 ∈ 𝐵 → ((𝐷 × {𝑥}) ∘f · 𝑓) = (𝑥 ∙ 𝑓)))
52	51	a1d 25	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑥 ∈ 𝐾 → (𝑓 ∈ 𝐵 → ((𝐷 × {𝑥}) ∘f · 𝑓) = (𝑥 ∙ 𝑓))))
53	52	3imp 1107	. . . 4 ⊢ ((¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) ∧ 𝑥 ∈ 𝐾 ∧ 𝑓 ∈ 𝐵) → ((𝐷 × {𝑥}) ∘f · 𝑓) = (𝑥 ∙ 𝑓))
54	53	mpoeq3dva 7231	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓)) = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ (𝑥 ∙ 𝑓)))
55	50, 54	eqtr4d 2859	. 2 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓)))
56	30, 55	pm2.61i 184	1 ⊢ ∙ = (𝑥 ∈ 𝐾, 𝑓 ∈ 𝐵 ↦ ((𝐷 × {𝑥}) ∘f · 𝑓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3142  Vcvv 3494   ∪ cun 3934  ∅c0 4291  {csn 4567  {ctp 4571  ⟨cop 4573   × cxp 5553  ^◡ccnv 5554   “ cima 5558   Fn wfn 6350  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158   ∘_f cof 7407   ↑_m cmap 8406  Fincfn 8509  1c1 10538  ℕcn 11638  6c6 11697  9c9 11700  ℕ₀cn0 11898  ndxcnx 16480  Basecbs 16483  +_gcplusg 16565  ._rcmulr 16566  Scalarcsca 16568   ·_𝑠 cvsca 16569  TopSetcts 16571  TopOpenctopn 16695  ∏_tcpt 16712   mPwSer cmps 20131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-plusg 16578  df-mulr 16579  df-sca 16581  df-vsca 16582  df-tset 16584  df-psr 20136

This theorem is referenced by:  psrvsca  20171