Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 30676 | . . . . 5 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
2 | xrge0cmn 20590 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
3 | 2 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
4 | xrge0tps 31189 | . . . . . 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 6882 | . . . . 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 22756 | . . . 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 6618 | . . . . . . . . . . 11 ⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴) | |
18 | 9, 17 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝐶⟶𝐴) |
19 | 18 | ffvelrnda 6854 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴) |
20 | 16, 19 | eqeltrrd 2917 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴) |
21 | 20 | ex 415 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝐶 → 𝐺 ∈ 𝐴)) |
22 | 15, 21 | ralrimi 3219 | . . . . . 6 ⊢ (𝜑 → ∀𝑛 ∈ 𝐶 𝐺 ∈ 𝐴) |
23 | esumf1o.f | . . . . . . . 8 ⊢ Ⅎ𝑛𝐹 | |
24 | 13, 23, 18 | feqmptdf 6738 | . . . . . . 7 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛))) |
25 | 15, 16 | mpteq2da 5163 | . . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝐶 ↦ (𝐹‘𝑛)) = (𝑛 ∈ 𝐶 ↦ 𝐺)) |
26 | 24, 25 | eqtrd 2859 | . . . . . 6 ⊢ (𝜑 → 𝐹 = (𝑛 ∈ 𝐶 ↦ 𝐺)) |
27 | eqidd 2825 | . . . . . 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 30405 | . . . . 5 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝐹) = (𝑛 ∈ 𝐶 ↦ 𝐷)) |
30 | 29 | oveq2d 7175 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ 𝐹)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷))) |
31 | 10, 30 | eqtrd 2859 | . . 3 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷))) |
32 | 31 | unieqd 4855 | . 2 ⊢ (𝜑 → ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷))) |
33 | df-esum 31291 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
34 | df-esum 31291 | . 2 ⊢ Σ*𝑛 ∈ 𝐶𝐷 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑛 ∈ 𝐶 ↦ 𝐷)) | |
35 | 32, 33, 34 | 3eqtr4g 2884 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑛 ∈ 𝐶𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 Ⅎwnf 1783 ∈ wcel 2113 Ⅎwnfc 2964 ∪ cuni 4841 ↦ cmpt 5149 ∘ ccom 5562 ⟶wf 6354 –1-1-onto→wf1o 6357 ‘cfv 6358 (class class class)co 7159 0cc0 10540 +∞cpnf 10675 [,]cicc 12744 ↾s cress 16487 ℝ*𝑠cxrs 16776 CMndccmn 18909 TopSpctps 21543 tsums ctsu 22737 Σ*cesum 31290 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 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-xadd 12511 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-tset 16587 df-ple 16588 df-ds 16590 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-ordt 16777 df-xrs 16778 df-ps 17813 df-tsr 17814 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-cntz 18450 df-cmn 18911 df-fbas 20545 df-fg 20546 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-ntr 21631 df-nei 21709 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-tsms 22738 df-esum 31291 |
This theorem is referenced by: esumc 31314 esumiun 31357 volmeas 31494 |
Copyright terms: Public domain | W3C validator |