fprodcnlem - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fprodcnlem		Structured version Visualization version GIF version

Theorem fprodcnlem 46200

Description: A finite product of functions to complex numbers from a common topological space is continuous. Induction step. (Contributed by Glauco Siliprandi, 8-Apr-2021.) Avoid ax-mulf 11176. (Revised by GG, 19-Apr-2025.)

**Hypotheses**
Ref	Expression
fprodcnlem.1	⊢ Ⅎ𝑘𝜑
fprodcnlem.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
fprodcnlem.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
fprodcnlem.a	⊢ (𝜑 → 𝐴 ∈ Fin)
fprodcnlem.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
fprodcnlem.z	⊢ (𝜑 → 𝑍 ⊆ 𝐴)
fprodcnlem.w	⊢ (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍))
fprodcnlem.p	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑍 𝐵) ∈ (𝐽 Cn 𝐾))

**Assertion**
Ref	Expression
fprodcnlem	⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾))

Distinct variable groups: 𝐴,𝑘 𝑘,𝐽,𝑥 𝑘,𝐾,𝑥 𝑘,𝑊,𝑥 𝑘,𝑋,𝑥 𝑘,𝑍 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑥) 𝐵(𝑥,𝑘) 𝑍(𝑥)

Proof of Theorem fprodcnlem Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fprodcnlem.1	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfv 1941	. . . . 5 ⊢ Ⅎ𝑘 𝑥 ∈ 𝑋
3	1, 2	nfan 1926	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋)
4		nfcsb1v 3885	. . . 4 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵
5		fprodcnlem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		fprodcnlem.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴)
7	5, 6	ssfid 9225	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Fin)
8	7	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑍 ∈ Fin)
9		fprodcnlem.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍))
10	9	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ (𝐴 ∖ 𝑍))
11	10	eldifbd 3926	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝑊 ∈ 𝑍)
12	6	sselda 3945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴)
13	12	adantlr 727	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴)
14		fprodcnlem.j	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
15	14	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝑋))
16		fprodcnlem.k	. . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
17	16	cnfldtopon 24904	. . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ)
18	17	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐾 ∈ (TopOn‘ℂ))
19		fprodcnlem.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
20		cnf2 23371	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
21	15, 18, 19, 20	syl3anc 1396	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
22		eqid 2769	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
23	22	fmpt 7103	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
24	21, 23	sylibr 237	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
25	24	adantlr 727	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
26		simplr 780	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ 𝑋)
27		rspa 3260	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
28	25, 26, 27	syl2anc 595	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
29	13, 28	syldan 602	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
30		csbeq1a 3875	. . . 4 ⊢ (𝑘 = 𝑊 → 𝐵 = ⦋𝑊 / 𝑘⦌𝐵)
31	10	eldifad 3925	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ 𝐴)
32		nfv 1941	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑊 ∈ 𝐴
33	3, 32	nfan 1926	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴)
34	4	nfel1 2947	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ
35	33, 34	nfim 1923	. . . . . . 7 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
36		eleq1 2857	. . . . . . . . 9 ⊢ (𝑘 = 𝑊 → (𝑘 ∈ 𝐴 ↔ 𝑊 ∈ 𝐴))
37	36	anbi2d 641	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴)))
38	30	eleq1d 2854	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (𝐵 ∈ ℂ ↔ ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ))
39	37, 38	imbi12d 347	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)))
40	35, 39, 28	vtoclg1f 3544	. . . . . 6 ⊢ (𝑊 ∈ 𝐴 → (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ))
41	40	anabsi7 683	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
42	31, 41	mpdan 699	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
43	3, 4, 8, 10, 11, 29, 30, 42	fprodsplitsn 16039	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵 = (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵))
44	43	mpteq2dva 5205	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) = (𝑥 ∈ 𝑋 ↦ (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵)))
45		fprodcnlem.p	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑍 𝐵) ∈ (𝐽 Cn 𝐾))
46	9	eldifad 3925	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐴)
47	1, 32	nfan 1926	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑊 ∈ 𝐴)
48		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑋
49	48, 4	nfmpt 5210	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵)
50	49	nfel1 2947	. . . . . . 7 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)
51	47, 50	nfim 1923	. . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
52	36	anbi2d 641	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑊 ∈ 𝐴)))
53	30	mpteq2dv 5206	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵))
54	53	eleq1d 2854	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
55	52, 54	imbi12d 347	. . . . . 6 ⊢ (𝑘 = 𝑊 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))))
56	51, 55, 19	vtoclg1f 3544	. . . . 5 ⊢ (𝑊 ∈ 𝐴 → ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
57	56	anabsi7 683	. . . 4 ⊢ ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
58	46, 57	mpdan 699	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
59	17	a1i 11	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
60	16	mpomulcn 24991	. . . 4 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
61	60	a1i 11	. . 3 ⊢ (𝜑 → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
62		oveq12 7417	. . 3 ⊢ ((𝑢 = ∏𝑘 ∈ 𝑍 𝐵 ∧ 𝑣 = ⦋𝑊 / 𝑘⦌𝐵) → (𝑢 · 𝑣) = (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵))
63	14, 45, 58, 59, 59, 61, 62	cnmpt12 23789	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵)) ∈ (𝐽 Cn 𝐾))
64	44, 63	eqeltrd 2869	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 Ⅎwnf 1810 ∈ wcel 2149 ∀wral 3085 ⦋csb 3861 ∖ cdif 3910 ∪ cun 3911 ⊆ wss 3913 {csn 4591 ↦ cmpt 5193 ⟶wf 6529 ‘cfv 6533 (class class class)co 7408 ∈ cmpo 7410 Fincfn 8939 ℂcc 11094 · cmul 11101 ∏cprod 15953 TopOpenctopn 17470 ℂ_fldccnfld 21487 TopOnctopon 23032 Cn ccn 23346 ×_t ctx 23682

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-inf2 9606 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-icc 13375 df-fz 13532 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-clim 15535 df-prod 15954 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17471 df-topn 17472 df-0g 17490 df-gsum 17491 df-topgen 17492 df-pt 17493 df-prds 17496 df-xrs 17552 df-qtop 17557 df-imas 17558 df-xps 17560 df-mre 17634 df-mrc 17635 df-acs 17637 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-mulg 19130 df-cntz 19383 df-cmn 19848 df-psmet 21479 df-xmet 21480 df-met 21481 df-bl 21482 df-mopn 21483 df-cnfld 21488 df-top 23016 df-topon 23033 df-topsp 23055 df-bases 23068 df-cn 23349 df-cnp 23350 df-tx 23684 df-hmeo 23877 df-xms 24442 df-ms 24443 df-tms 24444

This theorem is referenced by: fprodcn 46201

W3C validator