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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 30430 | . . . . 5 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
2 | xrge0cmn 20304 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
4 | xrge0tps 30861 | . . . . . 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 6700 | . . . . 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 22471 | . . . 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 6441 | . . . . . . . . . . 11 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴) | |
18 | 9, 17 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴) |
19 | 18 | ffvelrnda 6674 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴) |
20 | 16, 19 | eqeltrrd 2860 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴) |
21 | 20 | ex 405 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝐶 → 𝐺 ∈ 𝐴)) |
22 | 15, 21 | ralrimi 3159 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 𝐺 ∈ 𝐴) |
23 | esumf1o.f | . . . . . . . 8 ⊢ Ⅎ𝑛𝐹 | |
24 | 13, 23, 18 | feqmptdf 6562 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))) |
25 | 15, 16 | mpteq2da 5017 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝐶 ↦ 𝐺)) |
26 | 24, 25 | eqtrd 2807 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐶 ↦ 𝐺)) |
27 | eqidd 2772 | . . . . . 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 30181 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝐹) = (𝑛 ∈ 𝐶 ↦ 𝐷)) |
30 | 29 | oveq2d 6990 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝐹)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷))) |
31 | 10, 30 | eqtrd 2807 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷))) |
32 | 31 | unieqd 4718 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷))) |
33 | df-esum 30963 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
34 | df-esum 30963 | . 2 ⊢ Σ*𝑛 ∈ 𝐶𝐷 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷)) | |
35 | 32, 33, 34 | 3eqtr4g 2832 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑛 ∈ 𝐶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 Ⅎwnf 1747 ∈ wcel 2051 Ⅎwnfc 2909 ∪ cuni 4708 ↦ cmpt 5004 ∘ ccom 5407 ⟶wf 6181 –1-1-onto→wf1o 6184 ‘cfv 6185 (class class class)co 6974 0cc0 10333 +∞cpnf 10469 [,]cicc 12555 ↾s cress 16338 ℝ*𝑠cxrs 16627 CMndccmn 18678 TopSpctps 21259 tsums ctsu 22452 Σ*cesum 30962 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-fal 1521 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-oi 8767 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-xadd 12323 df-icc 12559 df-fz 12707 df-fzo 12848 df-seq 13183 df-hash 13504 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-tset 16438 df-ple 16439 df-ds 16441 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-ordt 16628 df-xrs 16629 df-ps 17680 df-tsr 17681 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-cntz 18230 df-cmn 18680 df-fbas 20259 df-fg 20260 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-ntr 21347 df-nei 21425 df-fil 22173 df-fm 22265 df-flim 22266 df-flf 22267 df-tsms 22453 df-esum 30963 |
This theorem is referenced by: esumc 30986 esumiun 31029 volmeas 31167 |
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