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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fprodcnlem | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fprodcnlem.1 | ⊢ Ⅎ𝑘𝜑 |
fprodcnlem.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
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)) |
Ref | Expression |
---|---|
fprodcnlem | ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fprodcnlem.1 | . . . . 5 ⊢ Ⅎ𝑘𝜑 | |
2 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑘 𝑥 ∈ 𝑋 | |
3 | 1, 2 | nfan 1902 | . . . 4 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ 𝑋) |
4 | nfcsb1v 3914 | . . . 4 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵 | |
5 | fprodcnlem.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
6 | fprodcnlem.z | . . . . . 6 ⊢ (𝜑 → 𝑍 ⊆ 𝐴) | |
7 | 5, 6 | ssfid 9250 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Fin) |
8 | 7 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑍 ∈ Fin) |
9 | fprodcnlem.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ (𝐴 ∖ 𝑍)) | |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ (𝐴 ∖ 𝑍)) |
11 | 10 | eldifbd 3957 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ¬ 𝑊 ∈ 𝑍) |
12 | 6 | sselda 3978 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴) |
13 | 12 | adantlr 713 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝐴) |
14 | fprodcnlem.j | . . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
15 | 14 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐽 ∈ (TopOn‘𝑋)) |
16 | fprodcnlem.k | . . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
17 | 16 | cnfldtopon 24228 | . . . . . . . . . 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 22682 | . . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘ℂ) ∧ (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐽 Cn 𝐾)) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ) | |
21 | 15, 18, 19, 20 | syl3anc 1371 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ 𝐵):𝑋⟶ℂ) |
22 | eqid 2731 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵) | |
23 | 22 | fmpt 7094 | . . . . . . . 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 713 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → ∀𝑥 ∈ 𝑋 𝐵 ∈ ℂ) |
26 | simplr 767 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) → 𝑥 ∈ 𝑋) | |
27 | rspa 3244 | . . . . . 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 3903 | . . . 4 ⊢ (𝑘 = 𝑊 → 𝐵 = ⦋𝑊 / 𝑘⦌𝐵) | |
31 | 10 | eldifad 3956 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑊 ∈ 𝐴) |
32 | simpr 485 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → 𝑊 ∈ 𝐴) | |
33 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑘𝑊 | |
34 | nfv 1917 | . . . . . . . . 9 ⊢ Ⅎ𝑘 𝑊 ∈ 𝐴 | |
35 | 3, 34 | nfan 1902 | . . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) |
36 | 4 | nfel1 2918 | . . . . . . . 8 ⊢ Ⅎ𝑘⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ |
37 | 35, 36 | nfim 1899 | . . . . . . 7 ⊢ Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ) |
38 | eleq1 2820 | . . . . . . . . 9 ⊢ (𝑘 = 𝑊 → (𝑘 ∈ 𝐴 ↔ 𝑊 ∈ 𝐴)) | |
39 | 38 | anbi2d 629 | . . . . . . . 8 ⊢ (𝑘 = 𝑊 → (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐴) ↔ ((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑊 ∈ 𝐴))) |
40 | 30 | eleq1d 2817 | . . . . . . . 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 3551 | . . . . . 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 685 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ⦋𝑊 / 𝑘⦌𝐵 ∈ ℂ) |
45 | 3, 4, 8, 10, 11, 29, 30, 44 | fprodsplitsn 15915 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵 = (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵)) |
46 | 45 | mpteq2dva 5241 | . 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 3956 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐴) |
49 | 1, 34 | nfan 1902 | . . . . . . 7 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑊 ∈ 𝐴) |
50 | nfcv 2902 | . . . . . . . . 9 ⊢ Ⅎ𝑘𝑋 | |
51 | 50, 4 | nfmpt 5248 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) |
52 | nfcv 2902 | . . . . . . . 8 ⊢ Ⅎ𝑘(𝐽 Cn 𝐾) | |
53 | 51, 52 | nfel 2916 | . . . . . . 7 ⊢ Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾) |
54 | 49, 53 | nfim 1899 | . . . . . 6 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
55 | 38 | anbi2d 629 | . . . . . . 7 ⊢ (𝑘 = 𝑊 → ((𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑊 ∈ 𝐴))) |
56 | 30 | mpteq2dv 5243 | . . . . . . . 8 ⊢ (𝑘 = 𝑊 → (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵)) |
57 | 56 | eleq1d 2817 | . . . . . . 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 3551 | . . . . 5 ⊢ (𝑊 ∈ 𝐴 → ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾))) |
61 | 60 | anabsi7 669 | . . . 4 ⊢ ((𝜑 ∧ 𝑊 ∈ 𝐴) → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
62 | 48, 61 | mpdan 685 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ⦋𝑊 / 𝑘⦌𝐵) ∈ (𝐽 Cn 𝐾)) |
63 | 16 | mulcn 24312 | . . . 4 ⊢ · ∈ ((𝐾 ×t 𝐾) Cn 𝐾) |
64 | 63 | a1i 11 | . . 3 ⊢ (𝜑 → · ∈ ((𝐾 ×t 𝐾) Cn 𝐾)) |
65 | 14, 47, 62, 64 | cnmpt12f 23099 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (∏𝑘 ∈ 𝑍 𝐵 · ⦋𝑊 / 𝑘⦌𝐵)) ∈ (𝐽 Cn 𝐾)) |
66 | 46, 65 | eqeltrd 2832 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ ∏𝑘 ∈ (𝑍 ∪ {𝑊})𝐵) ∈ (𝐽 Cn 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ∀wral 3060 ⦋csb 3889 ∖ cdif 3941 ∪ cun 3942 ⊆ wss 3944 {csn 4622 ↦ cmpt 5224 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 Fincfn 8922 ℂcc 11090 · cmul 11097 ∏cprod 15831 TopOpenctopn 17349 ℂfldccnfld 20878 TopOnctopon 22341 Cn ccn 22657 ×t ctx 22993 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-er 8686 df-map 8805 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9487 df-card 9916 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-icc 13313 df-fz 13467 df-fzo 13610 df-seq 13949 df-exp 14010 df-hash 14273 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-clim 15414 df-prod 15832 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17350 df-topn 17351 df-0g 17369 df-gsum 17370 df-topgen 17371 df-pt 17372 df-prds 17375 df-xrs 17430 df-qtop 17435 df-imas 17436 df-xps 17438 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-mulg 18923 df-cntz 19147 df-cmn 19614 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-cnfld 20879 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-cn 22660 df-cnp 22661 df-tx 22995 df-hmeo 23188 df-xms 23755 df-ms 23756 df-tms 23757 |
This theorem is referenced by: fprodcn 44089 |
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