fprodcnlem - Mathbox for Glauco Siliprandi

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Theorem fprodcnlem 42241

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.)

**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**
Step	Hyp	Ref	Expression
1		fprodcnlem.1	. . . . 5 ⊢ Ⅎ𝑘𝜑
2		nfv 1915	. . . . 5 ⊢ Ⅎ𝑘 𝑥 ∈ 𝑋
3	1, 2	nfan 1900	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋)
4		nfcsb1v 3852	. . . 4 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵
5		fprodcnlem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		fprodcnlem.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴)
7	5, 6	ssfid 8725	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Fin)
8	7	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑍 ∈ Fin)
9		fprodcnlem.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍))
10	9	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ (𝐴 ∖ 𝑍))
11	10	eldifbd 3894	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝑊 ∈ 𝑍)
12	6	sselda 3915	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴)
13	12	adantlr 714	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴)
14		fprodcnlem.j	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
15	14	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝑋))
16		fprodcnlem.k	. . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
17	16	cnfldtopon 23388	. . . . . . . . . 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 21854	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
21	15, 18, 19, 20	syl3anc 1368	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
22		eqid 2798	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
23	22	fmpt 6851	. . . . . . . 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 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
26		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ 𝑋)
27		rspa 3171	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
28	25, 26, 27	syl2anc 587	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
29	13, 28	syldan 594	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
30		csbeq1a 3842	. . . 4 ⊢ (𝑘 = 𝑊 → 𝐵 = ⦋𝑊 / 𝑘⦌𝐵)
31	10	eldifad 3893	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ 𝐴)
32		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → 𝑊 ∈ 𝐴)
33		nfcv 2955	. . . . . . 7 ⊢ Ⅎ𝑘𝑊
34		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑊 ∈ 𝐴
35	3, 34	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴)
36	4	nfel1 2971	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ
37	35, 36	nfim 1897	. . . . . . 7 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
38		eleq1 2877	. . . . . . . . 9 ⊢ (𝑘 = 𝑊 → (𝑘 ∈ 𝐴 ↔ 𝑊 ∈ 𝐴))
39	38	anbi2d 631	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴)))
40	30	eleq1d 2874	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (𝐵 ∈ ℂ ↔ ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ))
41	39, 40	imbi12d 348	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)))
42	33, 37, 41, 28	vtoclgf 3513	. . . . . 6 ⊢ (𝑊 ∈ 𝐴 → (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ))
43	32, 42	mpcom 38	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
44	31, 43	mpdan 686	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
45	3, 4, 8, 10, 11, 29, 30, 44	fprodsplitsn 15335	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵 = (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵))
46	45	mpteq2dva 5125	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) = (𝑥 ∈ 𝑋 ↦ (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵)))
47		fprodcnlem.p	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ 𝑍 𝐵) ∈ (𝐽 Cn 𝐾))
48	9	eldifad 3893	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐴)
49	1, 34	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑊 ∈ 𝐴)
50		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑋
51	50, 4	nfmpt 5127	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵)
52		nfcv 2955	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐽 Cn 𝐾)
53	51, 52	nfel 2969	. . . . . . 7 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)
54	49, 53	nfim 1897	. . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
55	38	anbi2d 631	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑊 ∈ 𝐴)))
56	30	mpteq2dv 5126	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵))
57	56	eleq1d 2874	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
58	55, 57	imbi12d 348	. . . . . 6 ⊢ (𝑘 = 𝑊 → (((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) ↔ ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))))
59	19	idi 1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾))
60	33, 54, 58, 59	vtoclgf 3513	. . . . 5 ⊢ (𝑊 ∈ 𝐴 → ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
61	60	anabsi7 670	. . . 4 ⊢ ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
62	48, 61	mpdan 686	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
63	16	mulcn 23472	. . . 4 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
64	63	a1i 11	. . 3 ⊢ (𝜑 → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
65	14, 47, 62, 64	cnmpt12f 22271	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵)) ∈ (𝐽 Cn 𝐾))
66	46, 65	eqeltrd 2890	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 Ⅎwnf 1785 ∈ wcel 2111 ∀wral 3106 ⦋csb 3828 ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 {csn 4525 ↦ cmpt 5110 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 · cmul 10531 ∏cprod 15251 TopOpenctopn 16687 ℂ_fldccnfld 20091 TopOnctopon 21515 Cn ccn 21829 ×_t ctx 22165

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-prod 15252 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929

This theorem is referenced by: fprodcn 42242

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