fprodcnlem - Mathbox for Glauco Siliprandi

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Theorem fprodcnlem 45570

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 11124. (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 1914	. . . . 5 ⊢ Ⅎ𝑘 𝑥 ∈ 𝑋
3	1, 2	nfan 1899	. . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋)
4		nfcsb1v 3883	. . . 4 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵
5		fprodcnlem.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
6		fprodcnlem.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴)
7	5, 6	ssfid 9188	. . . . 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 3924	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝑊 ∈ 𝑍)
12	6	sselda 3943	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴)
13	12	adantlr 715	. . . . 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 24646	. . . . . . . . . 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 23112	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
21	15, 18, 19, 20	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
22		eqid 2729	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
23	22	fmpt 7064	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ)
24	21, 23	sylibr 234	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
25	24	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ)
26		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ 𝑋)
27		rspa 3224	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ)
28	25, 26, 27	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
29	13, 28	syldan 591	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)
30		csbeq1a 3873	. . . 4 ⊢ (𝑘 = 𝑊 → 𝐵 = ⦋𝑊 / 𝑘⦌𝐵)
31	10	eldifad 3923	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ 𝐴)
32		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑊 ∈ 𝐴
33	3, 32	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴)
34	4	nfel1 2908	. . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ
35	33, 34	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
36		eleq1 2816	. . . . . . . . 9 ⊢ (𝑘 = 𝑊 → (𝑘 ∈ 𝐴 ↔ 𝑊 ∈ 𝐴))
37	36	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴)))
38	30	eleq1d 2813	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (𝐵 ∈ ℂ ↔ ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ))
39	37, 38	imbi12d 344	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)))
40	35, 39, 28	vtoclg1f 3533	. . . . . 6 ⊢ (𝑊 ∈ 𝐴 → (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ))
41	40	anabsi7 671	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
42	31, 41	mpdan 687	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ)
43	3, 4, 8, 10, 11, 29, 30, 42	fprodsplitsn 15931	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵 = (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵))
44	43	mpteq2dva 5195	. 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 3923	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐴)
47	1, 32	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑊 ∈ 𝐴)
48		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑋
49	48, 4	nfmpt 5200	. . . . . . . 8 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵)
50	49	nfel1 2908	. . . . . . 7 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)
51	47, 50	nfim 1896	. . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
52	36	anbi2d 630	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑊 ∈ 𝐴)))
53	30	mpteq2dv 5196	. . . . . . . 8 ⊢ (𝑘 = 𝑊 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵))
54	53	eleq1d 2813	. . . . . . 7 ⊢ (𝑘 = 𝑊 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾) ↔ (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
55	52, 54	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))))
56	51, 55, 19	vtoclg1f 3533	. . . . 5 ⊢ (𝑊 ∈ 𝐴 → ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)))
57	56	anabsi7 671	. . . 4 ⊢ ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
58	46, 57	mpdan 687	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))
59	17	a1i 11	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘ℂ))
60	16	mpomulcn 24734	. . . 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 7378	. . 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 23530	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵)) ∈ (𝐽 Cn 𝐾))
64	44, 63	eqeltrd 2828	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 ∀wral 3044 ⦋csb 3859 ∖ cdif 3908 ∪ cun 3909 ⊆ wss 3911 {csn 4585 ↦ cmpt 5183 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ∈ cmpo 7371 Fincfn 8895 ℂcc 11042 · cmul 11049 ∏cprod 15845 TopOpenctopn 17360 ℂ_fldccnfld 21240 TopOnctopon 22773 Cn ccn 23087 ×_t ctx 23423

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-prod 15846 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cn 23090 df-cnp 23091 df-tx 23425 df-hmeo 23618 df-xms 24184 df-ms 24185 df-tms 24186

This theorem is referenced by: fprodcn 45571

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