Theorem cnmpt1vsca 21997

Description: Continuity of scalar multiplication; analogue of cnmpt12f 21469 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	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋))
2		cnmpt1vsca.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ TopMod)
3		tlmtrg.f	. . . . . . . . . 10 ⊢ 𝐹 = (Scalar‘𝑊)
4	3	tlmscatps 21994	. . . . . . . . 9 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
5	2, 4	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ TopSp)
6		eqid 2622	. . . . . . . . 9 ⊢ (Base‘𝐹) = (Base‘𝐹)
7		cnmpt1vsca.k	. . . . . . . . 9 ⊢ 𝐾 = (TopOpen‘𝐹)
8	6, 7	istps 20738	. . . . . . . 8 ⊢ (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
9	5, 8	sylib 208	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(Base‘𝐹)))
10		cnmpt1vsca.a	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾))
11		cnf2 21053	. . . . . . 7 ⊢ ((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 1326	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹))
13		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴)
14	13	fmpt 6381	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐴 ∈ (Base‘𝐹) ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹))
15	12, 14	sylibr 224	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐴 ∈ (Base‘𝐹))
16	15	r19.21bi 2932	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐹))
17		tlmtps 21991	. . . . . . . . 9 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
18	2, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ TopSp)
19		eqid 2622	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊)
20		cnmpt1vsca.j	. . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘𝑊)
21	19, 20	istps 20738	. . . . . . . 8 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
22	18, 21	sylib 208	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊)))
23		cnmpt1vsca.b	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽))
24		cnf2 21053	. . . . . . 7 ⊢ ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
25	1, 22, 23, 24	syl3anc 1326	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
26		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
27	26	fmpt 6381	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ (Base‘𝑊) ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
28	25, 27	sylibr 224	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ (Base‘𝑊))
29	28	r19.21bi 2932	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊))
30		eqid 2622	. . . . 5 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
31		cnmpt1vsca.t	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
32	19, 3, 6, 30, 31	scafval 18882	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
33	16, 29, 32	syl2anc 693	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
34	33	mpteq2dva 4744	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))
35	30, 20, 3, 7	vscacn 21989	. . . 4 ⊢ (𝑊 ∈ TopMod → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
36	2, 35	syl 17	. . 3 ⊢ (𝜑 → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
37	1, 10, 23, 36	cnmpt12f 21469	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) ∈ (𝐿 Cn 𝐽))
38	34, 37	eqeltrrd 2702	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  Scalarcsca 15944   ·_𝑠 cvsca 15945  TopOpenctopn 16082   ·_sf cscaf 18864  TopOnctopon 20715  TopSpctps 20736   Cn ccn 21028   ×_t ctx 21363  TopModctlm 21961

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-slot 15861  df-base 15863  df-topgen 16104  df-scaf 18866  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-tx 21365  df-tmd 21876  df-tgp 21877  df-trg 21963  df-tlm 21965

This theorem is referenced by:  tlmtgp  21999