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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpfinvalf | Structured version Visualization version GIF version |
Description: Same as esumpfinval 34055, minus distinct variable restrictions. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Proof shortened by AV, 25-Jul-2019.) |
Ref | Expression |
---|---|
esumpfinvalf.1 | ⊢ Ⅎ𝑘𝐴 |
esumpfinvalf.2 | ⊢ Ⅎ𝑘𝜑 |
esumpfinvalf.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
esumpfinvalf.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
Ref | Expression |
---|---|
esumpfinvalf | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-esum 34008 | . . . 4 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | xrge0base 32998 | . . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
3 | xrge00 32999 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
4 | xrge0cmn 21443 | . . . . . . 7 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
6 | xrge0tps 33902 | . . . . . . 7 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
8 | esumpfinvalf.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
9 | esumpfinvalf.2 | . . . . . . 7 ⊢ Ⅎ𝑘𝜑 | |
10 | esumpfinvalf.1 | . . . . . . 7 ⊢ Ⅎ𝑘𝐴 | |
11 | nfcv 2902 | . . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞) | |
12 | icossicc 13472 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
13 | esumpfinvalf.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
14 | 12, 13 | sselid 3992 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
15 | eqid 2734 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
16 | 9, 10, 11, 14, 15 | fmptdF 32672 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
17 | c0ex 11252 | . . . . . . . 8 ⊢ 0 ∈ V | |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
19 | 16, 8, 18 | fdmfifsupp 9412 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
20 | xrge0topn 33903 | . . . . . . 7 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
21 | 20 | eqcomi 2743 | . . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
22 | xrhaus 23408 | . . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus | |
23 | ovex 7463 | . . . . . . . 8 ⊢ (0[,]+∞) ∈ V | |
24 | resthaus 23391 | . . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Haus) | |
25 | 22, 23, 24 | mp2an 692 | . . . . . . 7 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Haus |
26 | 25 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Haus) |
27 | 2, 3, 5, 7, 8, 16, 19, 21, 26 | haustsmsid 24164 | . . . . 5 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
28 | 27 | unieqd 4924 | . . . 4 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
29 | 1, 28 | eqtrid 2786 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
30 | ovex 7463 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V | |
31 | 30 | unisn 4930 | . . 3 ⊢ ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) |
32 | 29, 31 | eqtrdi 2790 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
33 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑘(0[,)+∞) | |
34 | 9, 10, 33, 13, 15 | fmptdF 32672 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) |
35 | esumpfinvallem 34054 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) | |
36 | 8, 34, 35 | syl2anc 584 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
37 | rge0ssre 13492 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ | |
38 | ax-resscn 11209 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
39 | 37, 38 | sstri 4004 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ |
40 | 39, 13 | sselid 3992 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
41 | 40 | sbt 2063 | . . . . 5 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
42 | sbim 2301 | . . . . . 6 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ)) | |
43 | sban 2077 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴)) | |
44 | 9 | sbf 2268 | . . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑) |
45 | 10 | clelsb1fw 2906 | . . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴) |
46 | 44, 45 | anbi12i 628 | . . . . . . . 8 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)) |
47 | 43, 46 | bitri 275 | . . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)) |
48 | sbsbc 3794 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ [𝑙 / 𝑘]𝐵 ∈ ℂ) | |
49 | sbcel1g 4421 | . . . . . . . . 9 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) | |
50 | 49 | elv 3482 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ) |
51 | 48, 50 | bitri 275 | . . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ) |
52 | 47, 51 | imbi12i 350 | . . . . . 6 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) |
53 | 42, 52 | bitri 275 | . . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) |
54 | 41, 53 | mpbi 230 | . . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ) |
55 | 8, 54 | gsumfsum 21469 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵) |
56 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑙𝐴 | |
57 | nfcv 2902 | . . . . 5 ⊢ Ⅎ𝑙𝐵 | |
58 | nfcsb1v 3932 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵 | |
59 | csbeq1a 3921 | . . . . 5 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵) | |
60 | 10, 56, 57, 58, 59 | cbvmptf 5256 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) |
61 | 60 | oveq2i 7441 | . . 3 ⊢ (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = (ℂfld Σg (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) |
62 | 59, 57, 58 | cbvsum 15727 | . . 3 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵 |
63 | 55, 61, 62 | 3eqtr4g 2799 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) |
64 | 32, 36, 63 | 3eqtr2d 2780 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 Ⅎwnf 1779 [wsb 2061 ∈ wcel 2105 Ⅎwnfc 2887 Vcvv 3477 [wsbc 3790 ⦋csb 3907 {csn 4630 ∪ cuni 4911 ↦ cmpt 5230 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 Fincfn 8983 ℂcc 11150 ℝcr 11151 0cc0 11152 +∞cpnf 11289 ≤ cle 11293 [,)cico 13385 [,]cicc 13386 Σcsu 15718 ↾s cress 17273 ↾t crest 17466 TopOpenctopn 17467 Σg cgsu 17486 ordTopcordt 17545 ℝ*𝑠cxrs 17546 CMndccmn 19812 ℂfldccnfld 21381 TopSpctps 22953 Hauscha 23331 tsums ctsu 24149 Σ*cesum 34007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-xadd 13152 df-ico 13389 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-ordt 17547 df-xrs 17548 df-ps 18623 df-tsr 18624 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18966 df-minusg 18967 df-cntz 19347 df-cmn 19814 df-abl 19815 df-mgp 20152 df-ur 20199 df-ring 20252 df-cring 20253 df-fbas 21378 df-fg 21379 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-cn 23250 df-haus 23338 df-fil 23869 df-fm 23961 df-flim 23962 df-flf 23963 df-tsms 24150 df-esum 34008 |
This theorem is referenced by: volfiniune 34210 |
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