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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcocn | Structured version Visualization version GIF version |
Description: Lemma for esummulc2 29471 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 1830 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | nfcv 2751 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
3 | esumcocn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | esumcocn.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐽)) | |
5 | xrge0tps 29316 | . . . . . . . 8 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
6 | xrge0base 29016 | . . . . . . . . 9 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | esumcocn.j | . . . . . . . . . 10 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
8 | xrge0topn 29317 | . . . . . . . . . 10 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
9 | 7, 8 | eqtr4i 2635 | . . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
10 | 6, 9 | tpsuni 20553 | . . . . . . . 8 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp → (0[,]+∞) = ∪ 𝐽) |
11 | 5, 10 | ax-mp 5 | . . . . . . 7 ⊢ (0[,]+∞) = ∪ 𝐽 |
12 | 11, 11 | cnf 20860 | . . . . . 6 ⊢ (𝐶 ∈ (𝐽 Cn 𝐽) → 𝐶:(0[,]+∞)⟶(0[,]+∞)) |
13 | 4, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶:(0[,]+∞)⟶(0[,]+∞)) |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶:(0[,]+∞)⟶(0[,]+∞)) |
15 | esumcocn.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
16 | 14, 15 | ffvelrnd 6268 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶‘𝐵) ∈ (0[,]+∞)) |
17 | xrge0cmn 19607 | . . . . . 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 18031 | . . . . . . . 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 1260 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦)))) |
25 | 24 | ralrimivv 2953 | . . . . . 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 29018 | . . . . . . . 8 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
28 | xrge00 29017 | . . . . . . . 8 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
29 | 6, 6, 27, 27, 28, 28 | ismhm 17160 | . . . . . . 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 217 | . . . . . 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 1325 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) MndHom (ℝ*𝑠 ↾s (0[,]+∞)))) |
32 | eqidd 2611 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
33 | 32, 15 | fmpt3d 6293 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
34 | 1, 2, 3, 15 | esumel 29436 | . . . . 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 21759 | . . . 4 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
36 | 13, 15 | cofmpt 28846 | . . . . 5 ⊢ (𝜑 → (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵))) |
37 | 36 | oveq2d 6565 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵))) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
38 | 35, 37 | eleqtrd 2690 | . . 3 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
39 | 1, 2, 3, 16, 38 | esumid 29433 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶‘𝐵) = (𝐶‘Σ*𝑘 ∈ 𝐴𝐵)) |
40 | 39 | eqcomd 2616 | 1 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∪ cuni 4372 ↦ cmpt 4643 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 ≤ cle 9954 +𝑒 cxad 11820 [,]cicc 12049 ↾s cress 15696 ↾t crest 15904 TopOpenctopn 15905 ordTopcordt 15982 ℝ*𝑠cxrs 15983 Mndcmnd 17117 MndHom cmhm 17156 CMndccmn 18016 TopSpctps 20519 Cn ccn 20838 tsums ctsu 21739 Σ*cesum 29416 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-xadd 11823 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-tset 15787 df-ple 15788 df-ds 15791 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-ordt 15984 df-xrs 15985 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-cntz 17573 df-cmn 18018 df-fbas 19564 df-fg 19565 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-ntr 20634 df-nei 20712 df-cn 20841 df-cnp 20842 df-haus 20929 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tsms 21740 df-esum 29417 |
This theorem is referenced by: esummulc1 29470 |
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