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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcocn | Structured version Visualization version GIF version |
Description: Lemma for esummulc2 31343 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 1915 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | nfcv 2979 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
3 | esumcocn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | esumcocn.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐽)) | |
5 | xrge0tps 31187 | . . . . . . . 8 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
6 | xrge0base 30674 | . . . . . . . . 9 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | esumcocn.j | . . . . . . . . . 10 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
8 | xrge0topn 31188 | . . . . . . . . . 10 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
9 | 7, 8 | eqtr4i 2849 | . . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
10 | 6, 9 | tpsuni 21546 | . . . . . . . 8 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp → (0[,]+∞) = ∪ 𝐽) |
11 | 5, 10 | ax-mp 5 | . . . . . . 7 ⊢ (0[,]+∞) = ∪ 𝐽 |
12 | 11, 11 | cnf 21856 | . . . . . 6 ⊢ (𝐶 ∈ (𝐽 Cn 𝐽) → 𝐶:(0[,]+∞)⟶(0[,]+∞)) |
13 | 4, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶:(0[,]+∞)⟶(0[,]+∞)) |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶:(0[,]+∞)⟶(0[,]+∞)) |
15 | esumcocn.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
16 | 14, 15 | ffvelrnd 6854 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶‘𝐵) ∈ (0[,]+∞)) |
17 | xrge0cmn 20589 | . . . . . 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 18924 | . . . . . . . 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 1118 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦)))) |
25 | 24 | ralrimivv 3192 | . . . . . 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 30676 | . . . . . . . 8 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
28 | xrge00 30675 | . . . . . . . 8 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
29 | 6, 6, 27, 27, 28, 28 | ismhm 17960 | . . . . . . 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 230 | . . . . . 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 1373 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) MndHom (ℝ*𝑠 ↾s (0[,]+∞)))) |
32 | eqidd 2824 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
33 | 32, 15 | fmpt3d 6882 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
34 | 1, 2, 3, 15 | esumel 31308 | . . . . 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 22756 | . . . 4 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
36 | 13, 15 | cofmpt 6896 | . . . . 5 ⊢ (𝜑 → (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵))) |
37 | 36 | oveq2d 7174 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵))) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
38 | 35, 37 | eleqtrd 2917 | . . 3 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
39 | 1, 2, 3, 16, 38 | esumid 31305 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶‘𝐵) = (𝐶‘Σ*𝑘 ∈ 𝐴𝐵)) |
40 | 39 | eqcomd 2829 | 1 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∪ cuni 4840 ↦ cmpt 5148 ∘ ccom 5561 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 ≤ cle 10678 +𝑒 cxad 12508 [,]cicc 12744 ↾s cress 16486 ↾t crest 16696 TopOpenctopn 16697 ordTopcordt 16774 ℝ*𝑠cxrs 16775 Mndcmnd 17913 MndHom cmhm 17956 CMndccmn 18908 TopSpctps 21542 Cn ccn 21834 tsums ctsu 22736 Σ*cesum 31288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 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-q 12352 df-xadd 12511 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-tset 16586 df-ple 16587 df-ds 16589 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-ordt 16776 df-xrs 16777 df-mre 16859 df-mrc 16860 df-acs 16862 df-ps 17812 df-tsr 17813 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-submnd 17959 df-cntz 18449 df-cmn 18910 df-fbas 20544 df-fg 20545 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-ntr 21630 df-nei 21708 df-cn 21837 df-cnp 21838 df-haus 21925 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-tsms 22737 df-esum 31289 |
This theorem is referenced by: esummulc1 31342 |
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