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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumf1o | Structured version Visualization version GIF version |
Description: Re-index an extended sum using a bijection. (Contributed by Thierry Arnoux, 6-Apr-2017.) |
Ref | Expression |
---|---|
esumf1o.0 | ⊢ Ⅎ𝑛𝜑 |
esumf1o.b | ⊢ Ⅎ𝑛𝐵 |
esumf1o.d | ⊢ Ⅎ𝑘𝐷 |
esumf1o.a | ⊢ Ⅎ𝑛𝐴 |
esumf1o.c | ⊢ Ⅎ𝑛𝐶 |
esumf1o.f | ⊢ Ⅎ𝑛𝐹 |
esumf1o.1 | ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷) |
esumf1o.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumf1o.3 | ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) |
esumf1o.4 | ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺) |
esumf1o.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
esumf1o | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑛 ∈ 𝐶𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrge0base 32173 | . . . . 5 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
2 | xrge0cmn 20979 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
4 | xrge0tps 32910 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
5 | 4 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
6 | esumf1o.2 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
7 | esumf1o.5 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
8 | 7 | fmpttd 7111 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
9 | esumf1o.3 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) | |
10 | 1, 3, 5, 6, 8, 9 | tsmsf1o 23640 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝐹))) |
11 | esumf1o.b | . . . . . 6 ⊢ Ⅎ𝑛𝐵 | |
12 | esumf1o.d | . . . . . 6 ⊢ Ⅎ𝑘𝐷 | |
13 | esumf1o.c | . . . . . 6 ⊢ Ⅎ𝑛𝐶 | |
14 | esumf1o.a | . . . . . 6 ⊢ Ⅎ𝑛𝐴 | |
15 | esumf1o.0 | . . . . . 6 ⊢ Ⅎ𝑛𝜑 | |
16 | esumf1o.4 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺) | |
17 | f1of 6830 | . . . . . . . . . . 11 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴) | |
18 | 9, 17 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴) |
19 | 18 | ffvelcdmda 7083 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴) |
20 | 16, 19 | eqeltrrd 2834 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴) |
21 | 20 | ex 413 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝐶 → 𝐺 ∈ 𝐴)) |
22 | 15, 21 | ralrimi 3254 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 𝐺 ∈ 𝐴) |
23 | esumf1o.f | . . . . . . . 8 ⊢ Ⅎ𝑛𝐹 | |
24 | 13, 23, 18 | feqmptdf 6959 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))) |
25 | 15, 16 | mpteq2da 5245 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝐶 ↦ 𝐺)) |
26 | 24, 25 | eqtrd 2772 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐶 ↦ 𝐺)) |
27 | eqidd 2733 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
28 | esumf1o.1 | . . . . . 6 ⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷) | |
29 | 11, 12, 13, 14, 15, 22, 26, 27, 28 | fmptcof2 31869 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝐹) = (𝑛 ∈ 𝐶 ↦ 𝐷)) |
30 | 29 | oveq2d 7421 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝐹)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷))) |
31 | 10, 30 | eqtrd 2772 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷))) |
32 | 31 | unieqd 4921 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷))) |
33 | df-esum 33014 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
34 | df-esum 33014 | . 2 ⊢ Σ*𝑛 ∈ 𝐶𝐷 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷)) | |
35 | 32, 33, 34 | 3eqtr4g 2797 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑛 ∈ 𝐶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 ∪ cuni 4907 ↦ cmpt 5230 ∘ ccom 5679 ⟶wf 6536 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 0cc0 11106 +∞cpnf 11241 [,]cicc 13323 ↾s cress 17169 ℝ*𝑠cxrs 17442 CMndccmn 19642 TopSpctps 22425 tsums ctsu 23621 Σ*cesum 33013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-xadd 13089 df-icc 13327 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-tset 17212 df-ple 17213 df-ds 17215 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-ordt 17443 df-xrs 17444 df-ps 18515 df-tsr 18516 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-cntz 19175 df-cmn 19644 df-fbas 20933 df-fg 20934 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-ntr 22515 df-nei 22593 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-tsms 23622 df-esum 33014 |
This theorem is referenced by: esumc 33037 esumiun 33080 volmeas 33217 |
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