Theorem cnmpt1vsca 23345

Description: Continuity of scalar multiplication; analogue of cnmpt12f 22817 which cannot be used directly because ·_𝑠 is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
tlmtrg.f	⊢ 𝐹 = (Scalar‘𝑊)
cnmpt1vsca.t	⊢ · = ( ·𝑠 ‘𝑊)
cnmpt1vsca.j	⊢ 𝐽 = (TopOpen‘𝑊)
cnmpt1vsca.k	⊢ 𝐾 = (TopOpen‘𝐹)
cnmpt1vsca.w	⊢ (𝜑 → 𝑊 ∈ TopMod)
cnmpt1vsca.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋))
cnmpt1vsca.a	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾))
cnmpt1vsca.b	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽))

Assertion

Ref	Expression
cnmpt1vsca	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾   𝑥,𝐿   𝜑,𝑥   𝑥,𝑊   𝑥,𝑋

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   · (𝑥)

Proof of Theorem cnmpt1vsca

Step	Hyp	Ref	Expression
1		cnmpt1vsca.l	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋))
2		cnmpt1vsca.w	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ TopMod)
3		tlmtrg.f	. . . . . . . . 9 ⊢ 𝐹 = (Scalar‘𝑊)
4	3	tlmscatps 23342	. . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ TopSp)
6		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
7		cnmpt1vsca.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹)
8	6, 7	istps 22083	. . . . . . 7 ⊢ (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
9	5, 8	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(Base‘𝐹)))
10		cnmpt1vsca.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾))
11		cnf2 22400	. . . . . 6 ⊢ ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐹)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹))
12	1, 9, 10, 11	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹))
13	12	fvmptelrn 6987	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐹))
14		tlmtps 23339	. . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
15	2, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ TopSp)
16		eqid 2738	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
17		cnmpt1vsca.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
18	16, 17	istps 22083	. . . . . . 7 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
19	15, 18	sylib 217	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊)))
20		cnmpt1vsca.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽))
21		cnf2 22400	. . . . . 6 ⊢ ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
22	1, 19, 20, 21	syl3anc 1370	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
23	22	fvmptelrn 6987	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊))
24		eqid 2738	. . . . 5 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
25		cnmpt1vsca.t	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
26	16, 3, 6, 24, 25	scafval 20142	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
27	13, 23, 26	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
28	27	mpteq2dva 5174	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))
29	24, 17, 3, 7	vscacn 23337	. . . 4 ⊢ (𝑊 ∈ TopMod → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
30	2, 29	syl 17	. . 3 ⊢ (𝜑 → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
31	1, 10, 20, 30	cnmpt12f 22817	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) ∈ (𝐿 Cn 𝐽))
32	28, 31	eqeltrrd 2840	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  Scalarcsca 16965   ·_𝑠 cvsca 16966  TopOpenctopn 17132   ·_sf cscaf 20124  TopOnctopon 22059  TopSpctps 22081   Cn ccn 22375   ×_t ctx 22711  TopModctlm 23309

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-topgen 17154  df-scaf 20126  df-top 22043  df-topon 22060  df-topsp 22082  df-bases 22096  df-cn 22378  df-tx 22713  df-tmd 23223  df-tgp 23224  df-trg 23311  df-tlm 23313

This theorem is referenced by:  tlmtgp  23347