Theorem cnmpt1vsca 24308

Description: Continuity of scalar multiplication; analogue of cnmpt12f 23780 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 24305	. . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
5	2, 4	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ TopSp)
6		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
7		cnmpt1vsca.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹)
8	6, 7	istps 23048	. . . . . . 7 ⊢ (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
9	5, 8	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(Base‘𝐹)))
10		cnmpt1vsca.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾))
11		cnf2 23363	. . . . . 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 1394	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹))
13	12	fvmptelcdm 7098	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐹))
14		tlmtps 24302	. . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
15	2, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ TopSp)
16		eqid 2765	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
17		cnmpt1vsca.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
18	16, 17	istps 23048	. . . . . . 7 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
19	15, 18	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊)))
20		cnmpt1vsca.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽))
21		cnf2 23363	. . . . . 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 1394	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
23	22	fvmptelcdm 7098	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊))
24		eqid 2765	. . . . 5 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
25		cnmpt1vsca.t	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
26	16, 3, 6, 24, 25	scafval 20968	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
27	13, 23, 26	syl2anc 595	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
28	27	mpteq2dva 5197	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))
29	24, 17, 3, 7	vscacn 24300	. . . 4 ⊢ (𝑊 ∈ TopMod → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
30	2, 29	syl 18	. . 3 ⊢ (𝜑 → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
31	1, 10, 20, 30	cnmpt12f 23780	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) ∈ (𝐿 Cn 𝐽))
32	28, 31	eqeltrrd 2866	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ↦ cmpt 5185  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  Scalarcsca 17301   ·_𝑠 cvsca 17302  TopOpenctopn 17462   ·_sf cscaf 20948  TopOnctopon 23024  TopSpctps 23046   Cn ccn 23338   ×_t ctx 23674  TopModctlm 24272

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-topgen 17484  df-scaf 20950  df-top 23008  df-topon 23025  df-topsp 23047  df-bases 23060  df-cn 23341  df-tx 23676  df-tmd 24186  df-tgp 24187  df-trg 24274  df-tlm 24276

This theorem is referenced by:  tlmtgp  24310