Theorem cnmpt2vsca 24116

Description: Continuity of scalar multiplication; analogue of cnmpt22f 23596 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))
cnmpt2vsca.m	⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑌))
cnmpt2vsca.a	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))
cnmpt2vsca.b	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))

Assertion

Ref	Expression
cnmpt2vsca	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))

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

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   · (𝑥,𝑦)   𝐿(𝑦)   𝑀(𝑥,𝑦)

Proof of Theorem cnmpt2vsca

Step	Hyp	Ref	Expression
1		cnmpt1vsca.l	. . . . . . . . . 10 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑋))
2		cnmpt2vsca.m	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (TopOn‘𝑌))
3		txtopon 23512	. . . . . . . . . 10 ⊢ ((𝐿 ∈ (TopOn‘𝑋) ∧ 𝑀 ∈ (TopOn‘𝑌)) → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)))
4	1, 2, 3	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)))
5		cnmpt1vsca.w	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ∈ TopMod)
6		tlmtrg.f	. . . . . . . . . . . 12 ⊢ 𝐹 = (Scalar‘𝑊)
7	6	tlmscatps 24112	. . . . . . . . . . 11 ⊢ (𝑊 ∈ TopMod → 𝐹 ∈ TopSp)
8	5, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ TopSp)
9		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝐹) = (Base‘𝐹)
10		cnmpt1vsca.k	. . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘𝐹)
11	9, 10	istps 22855	. . . . . . . . . 10 ⊢ (𝐹 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝐹)))
12	8, 11	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘(Base‘𝐹)))
13		cnmpt2vsca.a	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾))
14		cnf2 23170	. . . . . . . . 9 ⊢ (((𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐾 ∈ (TopOn‘(Base‘𝐹)) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐿 ×t 𝑀) Cn 𝐾)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
15	4, 12, 13, 14	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
16		eqid 2731	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴)
17	16	fmpo 8006	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ (Base‘𝐹) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴):(𝑋 × 𝑌)⟶(Base‘𝐹))
18	15, 17	sylibr 234	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐴 ∈ (Base‘𝐹))
19	18	r19.21bi 3224	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐴 ∈ (Base‘𝐹))
20	19	r19.21bi 3224	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐴 ∈ (Base‘𝐹))
21		tlmtps 24109	. . . . . . . . . . 11 ⊢ (𝑊 ∈ TopMod → 𝑊 ∈ TopSp)
22	5, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ TopSp)
23		eqid 2731	. . . . . . . . . . 11 ⊢ (Base‘𝑊) = (Base‘𝑊)
24		cnmpt1vsca.j	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘𝑊)
25	23, 24	istps 22855	. . . . . . . . . 10 ⊢ (𝑊 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑊)))
26	22, 25	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘(Base‘𝑊)))
27		cnmpt2vsca.b	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
28		cnf2 23170	. . . . . . . . 9 ⊢ (((𝐿 ×t 𝑀) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐽 ∈ (TopOn‘(Base‘𝑊)) ∧ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐿 ×t 𝑀) Cn 𝐽)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
29	4, 26, 27, 28	syl3anc 1373	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
30		eqid 2731	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵)
31	30	fmpo 8006	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ (Base‘𝑊) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵):(𝑋 × 𝑌)⟶(Base‘𝑊))
32	29, 31	sylibr 234	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 𝐵 ∈ (Base‘𝑊))
33	32	r19.21bi 3224	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝑌 𝐵 ∈ (Base‘𝑊))
34	33	r19.21bi 3224	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → 𝐵 ∈ (Base‘𝑊))
35		eqid 2731	. . . . . 6 ⊢ ( ·sf ‘𝑊) = ( ·sf ‘𝑊)
36		cnmpt1vsca.t	. . . . . 6 ⊢ · = ( ·𝑠 ‘𝑊)
37	23, 6, 9, 35, 36	scafval 20820	. . . . 5 ⊢ ((𝐴 ∈ (Base‘𝐹) ∧ 𝐵 ∈ (Base‘𝑊)) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
38	20, 34, 37	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝑌) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
39	38	3impa 1109	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → (𝐴( ·sf ‘𝑊)𝐵) = (𝐴 · 𝐵))
40	39	mpoeq3dva 7429	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴( ·sf ‘𝑊)𝐵)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)))
41	35, 24, 6, 10	vscacn 24107	. . . 4 ⊢ (𝑊 ∈ TopMod → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
42	5, 41	syl 17	. . 3 ⊢ (𝜑 → ( ·sf ‘𝑊) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
43	1, 2, 13, 27, 42	cnmpt22f 23596	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴( ·sf ‘𝑊)𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))
44	40, 43	eqeltrrd 2832	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 · 𝐵)) ∈ ((𝐿 ×t 𝑀) Cn 𝐽))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   × cxp 5617  ⟶wf 6483  ‘cfv 6487  (class class class)co 7352   ∈ cmpo 7354  Basecbs 17126  Scalarcsca 17170   ·_𝑠 cvsca 17171  TopOpenctopn 17331   ·_sf cscaf 20800  TopOnctopon 22831  TopSpctps 22853   Cn ccn 23145   ×_t ctx 23481  TopModctlm 24079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-fv 6495  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-map 8758  df-topgen 17353  df-scaf 20802  df-top 22815  df-topon 22832  df-topsp 22854  df-bases 22867  df-cn 23148  df-tx 23483  df-tmd 23993  df-tgp 23994  df-trg 24081  df-tlm 24083

This theorem is referenced by: (None)