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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpfinvalf | Structured version Visualization version GIF version |
Description: Same as esumpfinval 31336, 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 31289 | . . . 4 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | xrge0base 30674 | . . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
3 | xrge00 30675 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
4 | xrge0cmn 20589 | . . . . . . 7 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
6 | xrge0tps 31187 | . . . . . . 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 2979 | . . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞) | |
12 | icossicc 12827 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
13 | esumpfinvalf.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
14 | 12, 13 | sseldi 3967 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
15 | eqid 2823 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
16 | 9, 10, 11, 14, 15 | fmptdF 30403 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
17 | c0ex 10637 | . . . . . . . 8 ⊢ 0 ∈ V | |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
19 | 16, 8, 18 | fdmfifsupp 8845 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
20 | xrge0topn 31188 | . . . . . . 7 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
21 | 20 | eqcomi 2832 | . . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
22 | xrhaus 21995 | . . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus | |
23 | ovex 7191 | . . . . . . . 8 ⊢ (0[,]+∞) ∈ V | |
24 | resthaus 21978 | . . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Haus) | |
25 | 22, 23, 24 | mp2an 690 | . . . . . . 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 22751 | . . . . 5 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
28 | 27 | unieqd 4854 | . . . 4 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
29 | 1, 28 | syl5eq 2870 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
30 | ovex 7191 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V | |
31 | 30 | unisn 4860 | . . 3 ⊢ ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) |
32 | 29, 31 | syl6eq 2874 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
33 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑘(0[,)+∞) | |
34 | 9, 10, 33, 13, 15 | fmptdF 30403 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) |
35 | esumpfinvallem 31335 | . . 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 586 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
37 | rge0ssre 12847 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ | |
38 | ax-resscn 10596 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
39 | 37, 38 | sstri 3978 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ |
40 | 39, 13 | sseldi 3967 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
41 | 40 | sbt 2071 | . . . . 5 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
42 | sbim 2311 | . . . . . 6 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ)) | |
43 | sban 2086 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴)) | |
44 | 9 | sbf 2271 | . . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑) |
45 | 10 | clelsb3fw 2983 | . . . . . . . . 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 277 | . . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)) |
48 | sbsbc 3778 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ [𝑙 / 𝑘]𝐵 ∈ ℂ) | |
49 | sbcel1g 4367 | . . . . . . . . 9 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) | |
50 | 49 | elv 3501 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ) |
51 | 48, 50 | bitri 277 | . . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ) |
52 | 47, 51 | imbi12i 353 | . . . . . 6 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) |
53 | 42, 52 | bitri 277 | . . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) |
54 | 41, 53 | mpbi 232 | . . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ) |
55 | 8, 54 | gsumfsum 20614 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵) |
56 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑙𝐴 | |
57 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑙𝐵 | |
58 | nfcsb1v 3909 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵 | |
59 | csbeq1a 3899 | . . . . 5 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵) | |
60 | 10, 56, 57, 58, 59 | cbvmptf 5167 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) |
61 | 60 | oveq2i 7169 | . . 3 ⊢ (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = (ℂfld Σg (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) |
62 | 59, 56, 10, 57, 58 | cbvsum 15054 | . . 3 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵 |
63 | 55, 61, 62 | 3eqtr4g 2883 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) |
64 | 32, 36, 63 | 3eqtr2d 2864 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 Ⅎwnf 1784 [wsb 2069 ∈ wcel 2114 Ⅎwnfc 2963 Vcvv 3496 [wsbc 3774 ⦋csb 3885 {csn 4569 ∪ cuni 4840 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 Fincfn 8511 ℂcc 10537 ℝcr 10538 0cc0 10539 +∞cpnf 10674 ≤ cle 10678 [,)cico 12743 [,]cicc 12744 Σcsu 15044 ↾s cress 16486 ↾t crest 16696 TopOpenctopn 16697 Σg cgsu 16716 ordTopcordt 16774 ℝ*𝑠cxrs 16775 CMndccmn 18908 ℂfldccnfld 20547 TopSpctps 21542 Hauscha 21918 tsums ctsu 22736 Σ*cesum 31288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-xadd 12511 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-ordt 16776 df-xrs 16777 df-ps 17812 df-tsr 17813 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-cn 21837 df-haus 21925 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-tsms 22737 df-esum 31289 |
This theorem is referenced by: volfiniune 31491 |
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