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Mirrors > Home > MPE Home > Th. List > iprodclim3 | Structured version Visualization version GIF version |
Description: The sequence of partial finite product of a converging infinite product converge to the infinite product of the series. Note that 𝑗 must not occur in 𝐴. (Contributed by Scott Fenton, 18-Dec-2017.) |
Ref | Expression |
---|---|
iprodclim3.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
iprodclim3.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
iprodclim3.3 | ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑦)) |
iprodclim3.4 | ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) |
iprodclim3.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
iprodclim3.6 | ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = ∏𝑘 ∈ (𝑀...𝑗)𝐴) |
Ref | Expression |
---|---|
iprodclim3 | ⊢ (𝜑 → 𝐹 ⇝ ∏𝑘 ∈ 𝑍 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iprodclim3.4 | . . 3 ⊢ (𝜑 → 𝐹 ∈ dom ⇝ ) | |
2 | climdm 14133 | . . 3 ⊢ (𝐹 ∈ dom ⇝ ↔ 𝐹 ⇝ ( ⇝ ‘𝐹)) | |
3 | 1, 2 | sylib 207 | . 2 ⊢ (𝜑 → 𝐹 ⇝ ( ⇝ ‘𝐹)) |
4 | prodfc 14514 | . . . 4 ⊢ ∏𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ 𝑍 𝐴 | |
5 | iprodclim3.1 | . . . . 5 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
6 | iprodclim3.2 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
7 | iprodclim3.3 | . . . . 5 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑦)) | |
8 | eqidd 2611 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) | |
9 | iprodclim3.5 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | |
10 | eqid 2610 | . . . . . . 7 ⊢ (𝑘 ∈ 𝑍 ↦ 𝐴) = (𝑘 ∈ 𝑍 ↦ 𝐴) | |
11 | 9, 10 | fmptd 6292 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝑍 ↦ 𝐴):𝑍⟶ℂ) |
12 | 11 | ffvelrnda 6267 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ) |
13 | 5, 6, 7, 8, 12 | iprod 14507 | . . . 4 ⊢ (𝜑 → ∏𝑚 ∈ 𝑍 ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)))) |
14 | 4, 13 | syl5eqr 2658 | . . 3 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)))) |
15 | seqex 12665 | . . . . . . 7 ⊢ seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ∈ V | |
16 | 15 | a1i 11 | . . . . . 6 ⊢ (𝜑 → seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ∈ V) |
17 | iprodclim3.6 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = ∏𝑘 ∈ (𝑀...𝑗)𝐴) | |
18 | fzssuz 12253 | . . . . . . . . . . . . . 14 ⊢ (𝑀...𝑗) ⊆ (ℤ≥‘𝑀) | |
19 | 18, 5 | sseqtr4i 3601 | . . . . . . . . . . . . 13 ⊢ (𝑀...𝑗) ⊆ 𝑍 |
20 | resmpt 5369 | . . . . . . . . . . . . 13 ⊢ ((𝑀...𝑗) ⊆ 𝑍 → ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)) | |
21 | 19, 20 | ax-mp 5 | . . . . . . . . . . . 12 ⊢ ((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗)) = (𝑘 ∈ (𝑀...𝑗) ↦ 𝐴) |
22 | 21 | fveq1i 6104 | . . . . . . . . . . 11 ⊢ (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) |
23 | fvres 6117 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → (((𝑘 ∈ 𝑍 ↦ 𝐴) ↾ (𝑀...𝑗))‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) | |
24 | 22, 23 | syl5reqr 2659 | . . . . . . . . . 10 ⊢ (𝑚 ∈ (𝑀...𝑗) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚)) |
25 | 24 | prodeq2i 14488 | . . . . . . . . 9 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) |
26 | prodfc 14514 | . . . . . . . . 9 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ (𝑀...𝑗) ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ (𝑀...𝑗)𝐴 | |
27 | 25, 26 | eqtri 2632 | . . . . . . . 8 ⊢ ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ∏𝑘 ∈ (𝑀...𝑗)𝐴 |
28 | eqidd 2611 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚)) | |
29 | simpr 476 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ 𝑍) | |
30 | 29, 5 | syl6eleq 2698 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → 𝑗 ∈ (ℤ≥‘𝑀)) |
31 | elfzuz 12209 | . . . . . . . . . . . 12 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ (ℤ≥‘𝑀)) | |
32 | 31, 5 | syl6eleqr 2699 | . . . . . . . . . . 11 ⊢ (𝑚 ∈ (𝑀...𝑗) → 𝑚 ∈ 𝑍) |
33 | 32, 12 | sylan2 490 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ) |
34 | 33 | adantlr 747 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ 𝑍) ∧ 𝑚 ∈ (𝑀...𝑗)) → ((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) ∈ ℂ) |
35 | 28, 30, 34 | fprodser 14518 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∏𝑚 ∈ (𝑀...𝑗)((𝑘 ∈ 𝑍 ↦ 𝐴)‘𝑚) = (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗)) |
36 | 27, 35 | syl5eqr 2658 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → ∏𝑘 ∈ (𝑀...𝑗)𝐴 = (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗)) |
37 | 17, 36 | eqtr2d 2645 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))‘𝑗) = (𝐹‘𝑗)) |
38 | 5, 16, 1, 6, 37 | climeq 14146 | . . . . 5 ⊢ (𝜑 → (seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥 ↔ 𝐹 ⇝ 𝑥)) |
39 | 38 | iotabidv 5789 | . . . 4 ⊢ (𝜑 → (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥) = (℩𝑥𝐹 ⇝ 𝑥)) |
40 | df-fv 5812 | . . . 4 ⊢ ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))) = (℩𝑥seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴)) ⇝ 𝑥) | |
41 | df-fv 5812 | . . . 4 ⊢ ( ⇝ ‘𝐹) = (℩𝑥𝐹 ⇝ 𝑥) | |
42 | 39, 40, 41 | 3eqtr4g 2669 | . . 3 ⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , (𝑘 ∈ 𝑍 ↦ 𝐴))) = ( ⇝ ‘𝐹)) |
43 | 14, 42 | eqtrd 2644 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 𝐴 = ( ⇝ ‘𝐹)) |
44 | 3, 43 | breqtrrd 4611 | 1 ⊢ (𝜑 → 𝐹 ⇝ ∏𝑘 ∈ 𝑍 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ↦ cmpt 4643 dom cdm 5038 ↾ cres 5040 ℩cio 5766 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 · cmul 9820 ℤcz 11254 ℤ≥cuz 11563 ...cfz 12197 seqcseq 12663 ⇝ cli 14063 ∏cprod 14474 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-prod 14475 |
This theorem is referenced by: (None) |
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