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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumcocn | Structured version Visualization version GIF version |
Description: Lemma for esummulc2 34063 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 1912 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | nfcv 2903 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
3 | esumcocn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
4 | esumcocn.1 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (𝐽 Cn 𝐽)) | |
5 | xrge0tps 33903 | . . . . . . . 8 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp | |
6 | xrge0base 32999 | . . . . . . . . 9 ⊢ (0[,]+∞) = (Base‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
7 | esumcocn.j | . . . . . . . . . 10 ⊢ 𝐽 = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
8 | xrge0topn 33904 | . . . . . . . . . 10 ⊢ (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) = ((ordTop‘ ≤ ) ↾t (0[,]+∞)) | |
9 | 7, 8 | eqtr4i 2766 | . . . . . . . . 9 ⊢ 𝐽 = (TopOpen‘(ℝ*𝑠 ↾s (0[,]+∞))) |
10 | 6, 9 | tpsuni 22958 | . . . . . . . 8 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ∈ TopSp → (0[,]+∞) = ∪ 𝐽) |
11 | 5, 10 | ax-mp 5 | . . . . . . 7 ⊢ (0[,]+∞) = ∪ 𝐽 |
12 | 11, 11 | cnf 23270 | . . . . . 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 | ffvelcdmd 7105 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶‘𝐵) ∈ (0[,]+∞)) |
17 | xrge0cmn 21444 | . . . . . 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 19830 | . . . . . . . 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 1121 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 ∈ (0[,]+∞) ∧ 𝑦 ∈ (0[,]+∞)) → (𝐶‘(𝑥 +𝑒 𝑦)) = ((𝐶‘𝑥) +𝑒 (𝐶‘𝑦)))) |
25 | 24 | ralrimivv 3198 | . . . . . 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 33001 | . . . . . . . 8 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
28 | xrge00 33000 | . . . . . . . 8 ⊢ 0 = (0g‘(ℝ*𝑠 ↾s (0[,]+∞))) | |
29 | 6, 6, 27, 27, 28, 28 | ismhm 18811 | . . . . . . 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 228 | . . . . . 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 1376 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) MndHom (ℝ*𝑠 ↾s (0[,]+∞)))) |
32 | eqidd 2736 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
33 | 32, 15 | fmpt3d 7136 | . . . . 5 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶(0[,]+∞)) |
34 | 1, 2, 3, 15 | esumel 34028 | . . . . 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 24170 | . . . 4 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)))) |
36 | 13, 15 | cofmpt 7152 | . . . . 5 ⊢ (𝜑 → (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵)) = (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵))) |
37 | 36 | oveq2d 7447 | . . . 4 ⊢ (𝜑 → ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝐶 ∘ (𝑘 ∈ 𝐴 ↦ 𝐵))) = ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
38 | 35, 37 | eleqtrd 2841 | . . 3 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) ∈ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ (𝐶‘𝐵)))) |
39 | 1, 2, 3, 16, 38 | esumid 34025 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴(𝐶‘𝐵) = (𝐶‘Σ*𝑘 ∈ 𝐴𝐵)) |
40 | 39 | eqcomd 2741 | 1 ⊢ (𝜑 → (𝐶‘Σ*𝑘 ∈ 𝐴𝐵) = Σ*𝑘 ∈ 𝐴(𝐶‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∪ cuni 4912 ↦ cmpt 5231 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 ≤ cle 11294 +𝑒 cxad 13150 [,]cicc 13387 ↾s cress 17274 ↾t crest 17467 TopOpenctopn 17468 ordTopcordt 17546 ℝ*𝑠cxrs 17547 Mndcmnd 18760 MndHom cmhm 18807 CMndccmn 19813 TopSpctps 22954 Cn ccn 23248 tsums ctsu 24150 Σ*cesum 34008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-xadd 13153 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-tset 17317 df-ple 17318 df-ds 17320 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-ordt 17548 df-xrs 17549 df-mre 17631 df-mrc 17632 df-acs 17634 df-ps 18624 df-tsr 18625 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-cntz 19348 df-cmn 19815 df-fbas 21379 df-fg 21380 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-ntr 23044 df-nei 23122 df-cn 23251 df-cnp 23252 df-haus 23339 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tsms 24151 df-esum 34009 |
This theorem is referenced by: esummulc1 34062 |
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