Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumc | Structured version Visualization version GIF version |
Description: Convert from the collection form to the class-variable form of a sum. (Contributed by Thierry Arnoux, 10-May-2017.) |
Ref | Expression |
---|---|
esumc.0 | ⊢ Ⅎ𝑘𝐷 |
esumc.1 | ⊢ Ⅎ𝑘𝜑 |
esumc.2 | ⊢ Ⅎ𝑘𝐴 |
esumc.3 | ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵) |
esumc.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumc.5 | ⊢ (𝜑 → Fun ◡(𝑘 ∈ 𝐴 ↦ 𝐶)) |
esumc.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) |
esumc.7 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ 𝑊) |
Ref | Expression |
---|---|
esumc | ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | esumc.1 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
2 | esumc.0 | . . 3 ⊢ Ⅎ𝑘𝐷 | |
3 | nfcv 2974 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
4 | nfre1 3303 | . . . 4 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝑧 = 𝐶 | |
5 | 4 | nfab 2981 | . . 3 ⊢ Ⅎ𝑘{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} |
6 | esumc.2 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
7 | nfmpt1 5155 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶) | |
8 | esumc.3 | . . 3 ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵) | |
9 | esumc.4 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
10 | elex 3510 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
12 | 6, 11 | abrexexd 30196 | . . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ∈ V) |
13 | esumc.7 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ 𝑊) | |
14 | 13 | ex 413 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ 𝑊)) |
15 | 1, 14 | ralrimi 3213 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊) |
16 | 6 | fnmptf 6477 | . . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊 → (𝑘 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) Fn 𝐴) |
18 | esumc.5 | . . . 4 ⊢ (𝜑 → Fun ◡(𝑘 ∈ 𝐴 ↦ 𝐶)) | |
19 | eqid 2818 | . . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶) | |
20 | 19 | rnmpt 5820 | . . . . 5 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐶) = {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝐴 ↦ 𝐶) = {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}) |
22 | dff1o2 6613 | . . . 4 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ↔ ((𝑘 ∈ 𝐴 ↦ 𝐶) Fn 𝐴 ∧ Fun ◡(𝑘 ∈ 𝐴 ↦ 𝐶) ∧ ran (𝑘 ∈ 𝐴 ↦ 𝐶) = {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶})) | |
23 | 17, 18, 21, 22 | syl3anbrc 1335 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}) |
24 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴) | |
25 | 6 | fvmpt2f 6762 | . . . 4 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐶 ∈ 𝑊) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶) |
26 | 24, 13, 25 | syl2anc 584 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶) |
27 | vex 3495 | . . . . . 6 ⊢ 𝑦 ∈ V | |
28 | eqeq1 2822 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑦 = 𝐶)) | |
29 | 28 | rexbidv 3294 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐶 ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶)) |
30 | 27, 29 | elab 3664 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶) |
31 | 8 | reximi 3240 | . . . . 5 ⊢ (∃𝑘 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) |
32 | 30, 31 | sylbi 218 | . . . 4 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) |
33 | nfcv 2974 | . . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞) | |
34 | 2, 33 | nfel 2989 | . . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞) |
35 | esumc.6 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
36 | eleq1 2897 | . . . . . . . 8 ⊢ (𝐷 = 𝐵 → (𝐷 ∈ (0[,]+∞) ↔ 𝐵 ∈ (0[,]+∞))) | |
37 | 35, 36 | syl5ibrcom 248 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞))) |
38 | 37 | ex 413 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞)))) |
39 | 1, 34, 38 | rexlimd 3314 | . . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝐴 𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞))) |
40 | 39 | imp 407 | . . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
41 | 32, 40 | sylan2 592 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}) → 𝐷 ∈ (0[,]+∞)) |
42 | 1, 2, 3, 5, 6, 7, 8, 12, 23, 26, 41 | esumf1o 31208 | . 2 ⊢ (𝜑 → Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷 = Σ*𝑘 ∈ 𝐴𝐵) |
43 | 42 | eqcomd 2824 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 Ⅎwnf 1775 ∈ wcel 2105 {cab 2796 Ⅎwnfc 2958 ∀wral 3135 ∃wrex 3136 Vcvv 3492 ↦ cmpt 5137 ◡ccnv 5547 ran crn 5549 Fun wfun 6342 Fn wfn 6343 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 0cc0 10525 +∞cpnf 10660 [,]cicc 12729 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 |
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-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-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-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-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-xadd 12496 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-ps 17798 df-tsr 17799 df-mgm 17840 df-sgrp 17889 df-mnd 17900 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-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-tsms 22662 df-esum 31186 |
This theorem is referenced by: esumrnmpt 31210 esum2dlem 31250 measvunilem 31370 omssubadd 31457 |
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