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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcocn | Structured version Visualization version GIF version |
Description: Lemma for esummulc2 31240 and co. Composing with a continuous function preserves extended sums. (Contributed by Thierry Arnoux, 29-Jun-2017.) |
Ref | Expression |
---|---|
esumcocn.j | ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) |
esumcocn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumcocn.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumcocn.1 | ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐽)) |
esumcocn.0 | ⊢ (𝜑 → (𝐶‘0) = 0) |
esumcocn.f | ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦))) |
Ref | Expression |
---|---|
esumcocn | ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1906 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | nfcv 2974 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
3 | esumcocn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | esumcocn.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐽)) | |
5 | xrge0tps 31084 | . . . . . . . 8 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
6 | xrge0base 30599 | . . . . . . . . 9 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | esumcocn.j | . . . . . . . . . 10 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
8 | xrge0topn 31085 | . . . . . . . . . 10 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
9 | 7, 8 | eqtr4i 2844 | . . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
10 | 6, 9 | tpsuni 21472 | . . . . . . . 8 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp → (0[,]+∞) = ∪ 𝐽) |
11 | 5, 10 | ax-mp 5 | . . . . . . 7 ⊢ (0[,]+∞) = ∪ 𝐽 |
12 | 11, 11 | cnf 21782 | . . . . . 6 ⊢ (𝐶 ∈ (𝐽 Cn 𝐽) → 𝐶:(0[,]+∞)⟶(0[,]+∞)) |
13 | 4, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶:(0[,]+∞)⟶(0[,]+∞)) |
14 | 13 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶:(0[,]+∞)⟶(0[,]+∞)) |
15 | esumcocn.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
16 | 14, 15 | ffvelrnd 6844 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶‘𝐵) ∈ (0[,]+∞)) |
17 | xrge0cmn 20515 | . . . . . 6 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd | |
18 | 17 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd) |
19 | 5 | a1i 11 | . . . . 5 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp) |
20 | cmnmnd 18851 | . . . . . . . 8 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) | |
21 | 17, 20 | ax-mp 5 | . . . . . . 7 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd |
22 | 21 | a1i 11 | . . . . . 6 ⊢ (𝜑 → (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) |
23 | esumcocn.f | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦))) | |
24 | 23 | 3expib 1114 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦)))) |
25 | 24 | ralrimivv 3187 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦))) |
26 | esumcocn.0 | . . . . . 6 ⊢ (𝜑 → (𝐶‘0) = 0) | |
27 | xrge0plusg 30601 | . . . . . . . 8 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
28 | xrge00 30600 | . . . . . . . 8 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
29 | 6, 6, 27, 27, 28, 28 | ismhm 17946 | . . . . . . 7 ⊢ (𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) MndHom (ℝ*𝑠 ↾s (0[,]+∞))) ↔ (((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) ∧ (𝐶:(0[,]+∞)⟶(0[,]+∞) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦)) ∧ (𝐶‘0) = 0))) |
30 | 29 | biimpri 229 | . . . . . 6 ⊢ ((((ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd ∧ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ Mnd) ∧ (𝐶:(0[,]+∞)⟶(0[,]+∞) ∧ ∀𝑥 ∈ (0[,]+∞)∀𝑦 ∈ (0[,]+∞)(𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦)) ∧ (𝐶‘0) = 0)) → 𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) MndHom (ℝ*𝑠 ↾s (0[,]+∞)))) |
31 | 22, 22, 13, 25, 26, 30 | syl23anc 1369 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) MndHom (ℝ*𝑠 ↾s (0[,]+∞)))) |
32 | eqidd 2819 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
33 | 32, 15 | fmpt3d 6872 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
34 | 1, 2, 3, 15 | esumel 31205 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵))) |
35 | 6, 9, 9, 18, 19, 18, 19, 31, 4, 3, 33, 34 | tsmsmhm 22681 | . . . 4 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
36 | 13, 15 | cofmpt 6886 | . . . . 5 ⊢ (𝜑 → (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵))) |
37 | 36 | oveq2d 7161 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵))) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
38 | 35, 37 | eleqtrd 2912 | . . 3 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
39 | 1, 2, 3, 16, 38 | esumid 31202 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶‘𝐵) = (𝐶‘Σ*𝑘 ∈ 𝐴𝐵)) |
40 | 39 | eqcomd 2824 | 1 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∪ cuni 4830 ↦ cmpt 5137 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 0cc0 10525 +∞cpnf 10660 ≤ cle 10664 +𝑒 cxad 12493 [,]cicc 12729 ↾s cress 16472 ↾t crest 16682 TopOpenctopn 16683 ordTopcordt 16760 ℝ*𝑠cxrs 16761 Mndcmnd 17899 MndHom cmhm 17942 CMndccmn 18835 TopSpctps 21468 Cn ccn 21760 tsums ctsu 22661 Σ*cesum 31185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-xadd 12496 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-tset 16572 df-ple 16573 df-ds 16575 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-ordt 16762 df-xrs 16763 df-mre 16845 df-mrc 16846 df-acs 16848 df-ps 17798 df-tsr 17799 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-cntz 18385 df-cmn 18837 df-fbas 20470 df-fg 20471 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-ntr 21556 df-nei 21634 df-cn 21763 df-cnp 21764 df-haus 21851 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-tsms 22662 df-esum 31186 |
This theorem is referenced by: esummulc1 31239 |
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