fprodcnlem - Mathbox for Glauco Siliprandi

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Theorem fprodcnlem 43030

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 1918	. . . . 5 ⊢ Ⅎ𝑘 𝑥 ∈ 𝑋
3	1, 2	nfan 1903	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋)
4		nfcsb1v 3853	. . . 4 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵
5		fprodcnlem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		fprodcnlem.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴)
7	5, 6	ssfid 8971	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ Fin)
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑍 ∈ Fin)
9		fprodcnlem.w	. . . . 5 ⊢ (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍))
10	9	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ (𝐴 ∖ 𝑍))
11	10	eldifbd 3896	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝑊 ∈ 𝑍)
12	6	sselda 3917	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴)
13	12	adantlr 711	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴)
14		fprodcnlem.j	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
15	14	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝑋))
16		fprodcnlem.k	. . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
17	16	cnfldtopon 23852	. . . . . . . . . 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 22308	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
21	15, 18, 19, 20	syl3anc 1369	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
22		eqid 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
23	22	fmpt 6966	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
24	21, 23	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
25	24	adantlr 711	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
26		simplr 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ 𝑋)
27		rspa 3130	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
28	25, 26, 27	syl2anc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
29	13, 28	syldan 590	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
30		csbeq1a 3842	. . . 4 ⊢ (𝑘 = 𝑊 → 𝐵 = ⦋𝑊 / 𝑘⦌𝐵)
31	10	eldifad 3895	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ 𝐴)
32		simpr 484	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → 𝑊 ∈ 𝐴)
33		nfcv 2906	. . . . . . 7 ⊢ Ⅎ𝑘𝑊
34		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑊 ∈ 𝐴
35	3, 34	nfan 1903	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴)
36	4	nfel1 2922	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ
37	35, 36	nfim 1900	. . . . . . 7 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
38		eleq1 2826	. . . . . . . . 9 ⊢ (𝑘 = 𝑊 → (𝑘 ∈ 𝐴 ↔ 𝑊 ∈ 𝐴))
39	38	anbi2d 628	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴)))
40	30	eleq1d 2823	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (𝐵 ∈ ℂ ↔ ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ))
41	39, 40	imbi12d 344	. . . . . . 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 3493	. . . . . 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 683	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
45	3, 4, 8, 10, 11, 29, 30, 44	fprodsplitsn 15627	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵 = (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵))
46	45	mpteq2dva 5170	. 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 3895	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐴)
49	1, 34	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑊 ∈ 𝐴)
50		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑋
51	50, 4	nfmpt 5177	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵)
52		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑘(𝐽 Cn 𝐾)
53	51, 52	nfel 2920	. . . . . . 7 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)
54	49, 53	nfim 1900	. . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
55	38	anbi2d 628	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑊 ∈ 𝐴)))
56	30	mpteq2dv 5172	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵))
57	56	eleq1d 2823	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
58	55, 57	imbi12d 344	. . . . . 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 3493	. . . . 5 ⊢ (𝑊 ∈ 𝐴 → ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
61	60	anabsi7 667	. . . 4 ⊢ ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
62	48, 61	mpdan 683	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
63	16	mulcn 23936	. . . 4 ⊢ · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾)
64	63	a1i 11	. . 3 ⊢ (𝜑 → · ∈ ((𝐾 ×_t 𝐾) Cn 𝐾))
65	14, 47, 62, 64	cnmpt12f 22725	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵)) ∈ (𝐽 Cn 𝐾))
66	46, 65	eqeltrd 2839	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 Ⅎwnf 1787 ∈ wcel 2108 ∀wral 3063 ⦋csb 3828 ∖ cdif 3880 ∪ cun 3881 ⊆ wss 3883 {csn 4558 ↦ cmpt 5153 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 Fincfn 8691 ℂcc 10800 · cmul 10807 ∏cprod 15543 TopOpenctopn 17049 ℂ_fldccnfld 20510 TopOnctopon 21967 Cn ccn 22283 ×_t ctx 22619

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-mulf 10882

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-fi 9100 df-sup 9131 df-inf 9132 df-oi 9199 df-card 9628 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-icc 13015 df-fz 13169 df-fzo 13312 df-seq 13650 df-exp 13711 df-hash 13973 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-clim 15125 df-prod 15544 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-starv 16903 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-hom 16912 df-cco 16913 df-rest 17050 df-topn 17051 df-0g 17069 df-gsum 17070 df-topgen 17071 df-pt 17072 df-prds 17075 df-xrs 17130 df-qtop 17135 df-imas 17136 df-xps 17138 df-mre 17212 df-mrc 17213 df-acs 17215 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-mulg 18616 df-cntz 18838 df-cmn 19303 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cn 22286 df-cnp 22287 df-tx 22621 df-hmeo 22814 df-xms 23381 df-ms 23382 df-tms 23383

This theorem is referenced by: fprodcn 43031

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