| Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpfinvalf | Structured version Visualization version GIF version | ||
| Description: Same as esumpfinval 34076, 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 34029 | . . . 4 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
| 2 | xrge0base 33016 | . . . . . 6 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
| 3 | xrge00 33017 | . . . . . 6 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
| 4 | xrge0cmn 21426 | . . . . . . 7 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
| 5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
| 6 | xrge0tps 33941 | . . . . . . 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 2905 | . . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞) | |
| 12 | icossicc 13476 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
| 13 | esumpfinvalf.b | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) | |
| 14 | 12, 13 | sselid 3981 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
| 15 | eqid 2737 | . . . . . . 7 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
| 16 | 9, 10, 11, 14, 15 | fmptdF 32666 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
| 17 | c0ex 11255 | . . . . . . . 8 ⊢ 0 ∈ V | |
| 18 | 17 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ V) |
| 19 | 16, 8, 18 | fdmfifsupp 9415 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) finSupp 0) |
| 20 | xrge0topn 33942 | . . . . . . 7 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
| 21 | 20 | eqcomi 2746 | . . . . . 6 ⊢ ((ordTop‘ ≤ ) ↾t (0[,]+∞)) = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
| 22 | xrhaus 23393 | . . . . . . . 8 ⊢ (ordTop‘ ≤ ) ∈ Haus | |
| 23 | ovex 7464 | . . . . . . . 8 ⊢ (0[,]+∞) ∈ V | |
| 24 | resthaus 23376 | . . . . . . . 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 24149 | . . . . 5 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
| 28 | 27 | unieqd 4920 | . . . 4 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
| 29 | 1, 28 | eqtrid 2789 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))}) |
| 30 | ovex 7464 | . . . 4 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) ∈ V | |
| 31 | 30 | unisn 4926 | . . 3 ⊢ ∪ {((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))} = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) |
| 32 | 29, 31 | eqtrdi 2793 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = ((ℝ*𝑠 ↾s (0[,]+∞)) Σg (𝑘 ∈ 𝐴 ↦ 𝐵))) |
| 33 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑘(0[,)+∞) | |
| 34 | 9, 10, 33, 13, 15 | fmptdF 32666 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,)+∞)) |
| 35 | esumpfinvallem 34075 | . . 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 13496 | . . . . . . . 8 ⊢ (0[,)+∞) ⊆ ℝ | |
| 38 | ax-resscn 11212 | . . . . . . . 8 ⊢ ℝ ⊆ ℂ | |
| 39 | 37, 38 | sstri 3993 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℂ |
| 40 | 39, 13 | sselid 3981 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
| 41 | 40 | sbt 2066 | . . . . 5 ⊢ [𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
| 42 | sbim 2303 | . . . . . 6 ⊢ ([𝑙 / 𝑘]((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) ↔ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) → [𝑙 / 𝑘]𝐵 ∈ ℂ)) | |
| 43 | sban 2080 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘](𝜑 ∧ 𝑘 ∈ 𝐴) ↔ ([𝑙 / 𝑘]𝜑 ∧ [𝑙 / 𝑘]𝑘 ∈ 𝐴)) | |
| 44 | 9 | sbf 2271 | . . . . . . . . 9 ⊢ ([𝑙 / 𝑘]𝜑 ↔ 𝜑) |
| 45 | 10 | clelsb1fw 2909 | . . . . . . . . 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 3792 | . . . . . . . 8 ⊢ ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ [𝑙 / 𝑘]𝐵 ∈ ℂ) | |
| 49 | sbcel1g 4416 | . . . . . . . . 9 ⊢ (𝑙 ∈ V → ([𝑙 / 𝑘]𝐵 ∈ ℂ ↔ ⦋𝑙 / 𝑘⦌𝐵 ∈ ℂ)) | |
| 50 | 49 | elv 3485 | . . . . . . . 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 21452 | . . 3 ⊢ (𝜑 → (ℂfld Σg (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵) |
| 56 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑙𝐴 | |
| 57 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑙𝐵 | |
| 58 | nfcsb1v 3923 | . . . . 5 ⊢ Ⅎ𝑘⦋𝑙 / 𝑘⦌𝐵 | |
| 59 | csbeq1a 3913 | . . . . 5 ⊢ (𝑘 = 𝑙 → 𝐵 = ⦋𝑙 / 𝑘⦌𝐵) | |
| 60 | 10, 56, 57, 58, 59 | cbvmptf 5251 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵) |
| 61 | 60 | oveq2i 7442 | . . 3 ⊢ (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = (ℂfld Σg (𝑙 ∈ 𝐴 ↦ ⦋𝑙 / 𝑘⦌𝐵)) |
| 62 | 59, 57, 58 | cbvsum 15731 | . . 3 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑙 ∈ 𝐴 ⦋𝑙 / 𝑘⦌𝐵 |
| 63 | 55, 61, 62 | 3eqtr4g 2802 | . 2 ⊢ (𝜑 → (ℂfld Σg (𝑘 ∈ 𝐴 ↦ 𝐵)) = Σ𝑘 ∈ 𝐴 𝐵) |
| 64 | 32, 36, 63 | 3eqtr2d 2783 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ𝑘 ∈ 𝐴 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 Ⅎwnfc 2890 Vcvv 3480 [wsbc 3788 ⦋csb 3899 {csn 4626 ∪ cuni 4907 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℂcc 11153 ℝcr 11154 0cc0 11155 +∞cpnf 11292 ≤ cle 11296 [,)cico 13389 [,]cicc 13390 Σcsu 15722 ↾s cress 17274 ↾t crest 17465 TopOpenctopn 17466 Σg cgsu 17485 ordTopcordt 17544 ℝ*𝑠cxrs 17545 CMndccmn 19798 ℂfldccnfld 21364 TopSpctps 22938 Hauscha 23316 tsums ctsu 24134 Σ*cesum 34028 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-xadd 13155 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-ordt 17546 df-xrs 17547 df-ps 18611 df-tsr 18612 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-grp 18954 df-minusg 18955 df-cntz 19335 df-cmn 19800 df-abl 19801 df-mgp 20138 df-ur 20179 df-ring 20232 df-cring 20233 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-cn 23235 df-haus 23323 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tsms 24135 df-esum 34029 |
| This theorem is referenced by: volfiniune 34231 |
| Copyright terms: Public domain | W3C validator |