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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpfinvalf | Structured version Visualization version GIF version |
Description: Same as esumpfinval 30739, 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 30692 | . . . 4 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | xrge0base 30251 | . . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
3 | xrge00 30252 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
4 | xrge0cmn 20188 | . . . . . . 7 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
6 | xrge0tps 30590 | . . . . . . 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 2934 | . . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞) | |
12 | icossicc 12577 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
13 | esumpfinvalf.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
14 | 12, 13 | sseldi 3819 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
15 | eqid 2778 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
16 | 9, 10, 11, 14, 15 | fmptdF 30025 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
17 | c0ex 10372 | . . . . . . . 8 ⊢ 0 ∈ V | |
18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
19 | 16, 8, 18 | fdmfifsupp 8575 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
20 | xrge0topn 30591 | . . . . . . 7 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
21 | 20 | eqcomi 2787 | . . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
22 | xrhaus 21601 | . . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus | |
23 | ovex 6956 | . . . . . . . 8 ⊢ (0[,]+∞) ∈ V | |
24 | resthaus 21584 | . . . . . . . 8 ⊢ (((ordTop‘ ≤ ) ∈ Haus ∧ (0[,]+∞) ∈ V) → ((ordTop‘ ≤ ) ↾t (0[,]+∞)) ∈ Haus) | |
25 | 22, 23, 24 | mp2an 682 | . . . . . . 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 22356 | . . . . 5 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
28 | 27 | unieqd 4683 | . . . 4 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
29 | 1, 28 | syl5eq 2826 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
30 | ovex 6956 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V | |
31 | 30 | unisn 4689 | . . 3 ⊢ ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) |
32 | 29, 31 | syl6eq 2830 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
33 | nfcv 2934 | . . . 4 ⊢ Ⅎ𝑘(0[,)+∞) | |
34 | 9, 10, 33, 13, 15 | fmptdF 30025 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) |
35 | esumpfinvallem 30738 | . . 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 579 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
37 | rge0ssre 12598 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ | |
38 | ax-resscn 10331 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
39 | 37, 38 | sstri 3830 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ |
40 | 39, 13 | sseldi 3819 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
41 | 40 | sbt 2496 | . . . . 5 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
42 | sbim 2471 | . . . . . 6 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ)) | |
43 | sban 2475 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴)) | |
44 | 9 | sbf 2456 | . . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑) |
45 | 10 | clelsb3f 2938 | . . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝑘 ∈ 𝐴 ↔ 𝑙 ∈ 𝐴) |
46 | 44, 45 | anbi12i 620 | . . . . . . . 8 ⊢ (([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)) |
47 | 43, 46 | bitri 267 | . . . . . . 7 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ (𝜑 ∧ 𝑙 ∈ 𝐴)) |
48 | sbsbc 3656 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ [𝑙 / 𝑘]𝐵 ∈ ℂ) | |
49 | vex 3401 | . . . . . . . . 9 ⊢ 𝑙 ∈ V | |
50 | sbcel1g 4212 | . . . . . . . . 9 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) | |
51 | 49, 50 | ax-mp 5 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ) |
52 | 48, 51 | bitri 267 | . . . . . . 7 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ) |
53 | 47, 52 | imbi12i 342 | . . . . . 6 ⊢ (([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) |
54 | 42, 53 | bitri 267 | . . . . 5 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) |
55 | 41, 54 | mpbi 222 | . . . 4 ⊢ ((𝜑 ∧ 𝑙 ∈ 𝐴) → ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ) |
56 | 8, 55 | gsumfsum 20213 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵) |
57 | nfcv 2934 | . . . . 5 ⊢ Ⅎ𝑙𝐴 | |
58 | nfcv 2934 | . . . . 5 ⊢ Ⅎ𝑙𝐵 | |
59 | nfcsb1v 3767 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵 | |
60 | csbeq1a 3760 | . . . . 5 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵) | |
61 | 10, 57, 58, 59, 60 | cbvmptf 4985 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) |
62 | 61 | oveq2i 6935 | . . 3 ⊢ (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = (ℂfld Σg (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) |
63 | 60, 57, 10, 58, 59 | cbvsum 14837 | . . 3 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵 |
64 | 56, 62, 63 | 3eqtr4g 2839 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) |
65 | 32, 36, 64 | 3eqtr2d 2820 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 Ⅎwnf 1827 [wsb 2011 ∈ wcel 2107 Ⅎwnfc 2919 Vcvv 3398 [wsbc 3652 ⦋csb 3751 {csn 4398 ∪ cuni 4673 ↦ cmpt 4967 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 Fincfn 8243 ℂcc 10272 ℝcr 10273 0cc0 10274 +∞cpnf 10410 ≤ cle 10414 [,)cico 12493 [,]cicc 12494 Σcsu 14828 ↾s cress 16260 ↾t crest 16471 TopOpenctopn 16472 Σg cgsu 16491 ordTopcordt 16549 ℝ*𝑠cxrs 16550 CMndccmn 18583 ℂfldccnfld 20146 TopSpctps 21148 Hauscha 21524 tsums ctsu 22341 Σ*cesum 30691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-addf 10353 ax-mulf 10354 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-iin 4758 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-rp 12142 df-xadd 12262 df-ico 12497 df-icc 12498 df-fz 12648 df-fzo 12789 df-seq 13124 df-exp 13183 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-clim 14631 df-sum 14829 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-starv 16357 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-rest 16473 df-topn 16474 df-0g 16492 df-gsum 16493 df-topgen 16494 df-ordt 16551 df-xrs 16552 df-ps 17590 df-tsr 17591 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-submnd 17726 df-grp 17816 df-minusg 17817 df-cntz 18137 df-cmn 18585 df-abl 18586 df-mgp 18881 df-ur 18893 df-ring 18940 df-cring 18941 df-fbas 20143 df-fg 20144 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-cn 21443 df-haus 21531 df-fil 22062 df-fm 22154 df-flim 22155 df-flf 22156 df-tsms 22342 df-esum 30692 |
This theorem is referenced by: volfiniune 30895 |
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