Theorem cnmpt1vsca 24169

Description: Continuity of scalar multiplication; analogue of cnmpt12f 23641 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 24166	. . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
5	2, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ TopSp)
6		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝐹) = (Base‘𝐹)
7		cnmpt1vsca.k	. . . . . . . 8 ⊢ 𝐾 = (TopOpen‘𝐹)
8	6, 7	istps 22909	. . . . . . 7 ⊢ (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
9	5, 8	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(Base‘𝐹)))
10		cnmpt1vsca.a	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐿 Cn 𝐾))
11		cnf2 23224	. . . . . 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 1374	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶(Base‘𝐹))
13	12	fvmptelcdm 7059	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ (Base‘𝐹))
14		tlmtps 24163	. . . . . . . 8 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
15	2, 14	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ TopSp)
16		eqid 2737	. . . . . . . 8 ⊢ (Base‘𝑊) = (Base‘𝑊)
17		cnmpt1vsca.j	. . . . . . . 8 ⊢ 𝐽 = (TopOpen‘𝑊)
18	16, 17	istps 22909	. . . . . . 7 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
19	15, 18	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊)))
20		cnmpt1vsca.b	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐿 Cn 𝐽))
21		cnf2 23224	. . . . . 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 1374	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶(Base‘𝑊))
23	22	fvmptelcdm 7059	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ (Base‘𝑊))
24		eqid 2737	. . . . 5 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
25		cnmpt1vsca.t	. . . . 5 ⊢ · = ( ·𝑠 ‘𝑊)
26	16, 3, 6, 24, 25	scafval 20867	. . . 4 ⊢ ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
27	13, 23, 26	syl2anc 585	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
28	27	mpteq2dva 5179	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) = (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)))
29	24, 17, 3, 7	vscacn 24161	. . . 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 23641	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴( ·sf ‘𝑊)𝐵)) ∈ (𝐿 Cn 𝐽))
32	28, 31	eqeltrrd 2838	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 · 𝐵)) ∈ (𝐿 Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5167  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  Scalarcsca 17214   ·_𝑠 cvsca 17215  TopOpenctopn 17375   ·_sf cscaf 20847  TopOnctopon 22885  TopSpctps 22907   Cn ccn 23199   ×_t ctx 23535  TopModctlm 24133

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8768  df-topgen 17397  df-scaf 20849  df-top 22869  df-topon 22886  df-topsp 22908  df-bases 22921  df-cn 23202  df-tx 23537  df-tmd 24047  df-tgp 24048  df-trg 24135  df-tlm 24137

This theorem is referenced by:  tlmtgp  24171