Theorem cnmpt1vsca 24546

Description: Continuity of scalar multiplication; analogue of cnmpt12f 24018 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 24543	. . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
5	2, 4	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ TopSp)
6		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
7		cnmpt1vsca.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹)
8	6, 7	istps 23285	. . . . . . 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 23600	. . . . . 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 1399	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹))
13	12	fvmptelcdm 7140	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐹))
14		tlmtps 24540	. . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
15	2, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ TopSp)
16		eqid 2763	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
17		cnmpt1vsca.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
18	16, 17	istps 23285	. . . . . . 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 23600	. . . . . 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 1399	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
23	22	fvmptelcdm 7140	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊))
24		eqid 2763	. . . . 5 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
25		cnmpt1vsca.t	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
26	16, 3, 6, 24, 25	scafval 21189	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
27	13, 23, 26	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
28	27	mpteq2dva 5202	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))
29	24, 17, 3, 7	vscacn 24538	. . . 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 24018	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) ∈ (𝐿 Cn 𝐽))
32	28, 31	eqeltrrd 2865	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ↦ cmpt 5190  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  Scalarcsca 17463   ·_𝑠 cvsca 17464  TopOpenctopn 17624   ·_sf cscaf 21169  TopOnctopon 23261  TopSpctps 23283   Cn ccn 23575   ×_t ctx 23912  TopModctlm 24510

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-map 8880  df-topgen 17646  df-scaf 21171  df-top 23245  df-topon 23262  df-topsp 23284  df-bases 23297  df-cn 23578  df-tx 23914  df-tmd 24424  df-tgp 24425  df-trg 24512  df-tlm 24514

This theorem is used by:  tlmtgp  24548