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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcocn | Structured version Visualization version GIF version |
Description: Lemma for esummulc2 30144 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 1843 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | nfcv 2764 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
3 | esumcocn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | esumcocn.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐽)) | |
5 | xrge0tps 29988 | . . . . . . . 8 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
6 | xrge0base 29685 | . . . . . . . . 9 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | esumcocn.j | . . . . . . . . . 10 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
8 | xrge0topn 29989 | . . . . . . . . . 10 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
9 | 7, 8 | eqtr4i 2647 | . . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
10 | 6, 9 | tpsuni 20740 | . . . . . . . 8 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp → (0[,]+∞) = ∪ 𝐽) |
11 | 5, 10 | ax-mp 5 | . . . . . . 7 ⊢ (0[,]+∞) = ∪ 𝐽 |
12 | 11, 11 | cnf 21050 | . . . . . 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 6360 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶‘𝐵) ∈ (0[,]+∞)) |
17 | xrge0cmn 19788 | . . . . . 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 18208 | . . . . . . . 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 1268 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦)))) |
25 | 24 | ralrimivv 2970 | . . . . . 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 29687 | . . . . . . . 8 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
28 | xrge00 29686 | . . . . . . . 8 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
29 | 6, 6, 27, 27, 28, 28 | ismhm 17337 | . . . . . . 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 218 | . . . . . 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 1333 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) MndHom (ℝ*𝑠 ↾s (0[,]+∞)))) |
32 | eqidd 2623 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
33 | 32, 15 | fmpt3d 6386 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
34 | 1, 2, 3, 15 | esumel 30109 | . . . . 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 21949 | . . . 4 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
36 | 13, 15 | cofmpt 6399 | . . . . 5 ⊢ (𝜑 → (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵))) |
37 | 36 | oveq2d 6666 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵))) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
38 | 35, 37 | eleqtrd 2703 | . . 3 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
39 | 1, 2, 3, 16, 38 | esumid 30106 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶‘𝐵) = (𝐶‘Σ*𝑘 ∈ 𝐴𝐵)) |
40 | 39 | eqcomd 2628 | 1 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∪ cuni 4436 ↦ cmpt 4729 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 ≤ cle 10075 +𝑒 cxad 11944 [,]cicc 12178 ↾s cress 15858 ↾t crest 16081 TopOpenctopn 16082 ordTopcordt 16159 ℝ*𝑠cxrs 16160 Mndcmnd 17294 MndHom cmhm 17333 CMndccmn 18193 TopSpctps 20736 Cn ccn 21028 tsums ctsu 21929 Σ*cesum 30089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-xadd 11947 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-tset 15960 df-ple 15961 df-ds 15964 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-ordt 16161 df-xrs 16162 df-mre 16246 df-mrc 16247 df-acs 16249 df-ps 17200 df-tsr 17201 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-cntz 17750 df-cmn 18195 df-fbas 19743 df-fg 19744 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-ntr 20824 df-nei 20902 df-cn 21031 df-cnp 21032 df-haus 21119 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 df-esum 30090 |
This theorem is referenced by: esummulc1 30143 |
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