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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 2968 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
4 | nfre1 3212 | . . . 4 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝑧 = 𝐶 | |
5 | 4 | nfab 2973 | . . 3 ⊢ Ⅎ𝑘{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} |
6 | esumc.2 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
7 | nfmpt1 4969 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶) | |
8 | esumc.3 | . . 3 ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵) | |
9 | esumc.4 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
10 | elex 3428 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
12 | 6, 11 | abrexexd 29894 | . . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ∈ V) |
13 | esumc.7 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ 𝑊) | |
14 | 13 | ex 403 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ 𝑊)) |
15 | 1, 14 | ralrimi 3165 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊) |
16 | 6 | fnmptf 6248 | . . . . 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 2824 | . . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶) | |
20 | 19 | rnmpt 5603 | . . . . 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 6382 | . . . 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 1449 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}) |
24 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴) | |
25 | 6 | fvmpt2f 6529 | . . . 4 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐶 ∈ 𝑊) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶) |
26 | 24, 13, 25 | syl2anc 581 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶) |
27 | vex 3416 | . . . . . 6 ⊢ 𝑦 ∈ V | |
28 | eqeq1 2828 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑦 = 𝐶)) | |
29 | 28 | rexbidv 3261 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐶 ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶)) |
30 | 27, 29 | elab 3570 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶) |
31 | 8 | reximi 3218 | . . . . 5 ⊢ (∃𝑘 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) |
32 | 30, 31 | sylbi 209 | . . . 4 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) |
33 | nfcv 2968 | . . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞) | |
34 | 2, 33 | nfel 2981 | . . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞) |
35 | esumc.6 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
36 | eleq1 2893 | . . . . . . . 8 ⊢ (𝐷 = 𝐵 → (𝐷 ∈ (0[,]+∞) ↔ 𝐵 ∈ (0[,]+∞))) | |
37 | 35, 36 | syl5ibrcom 239 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞))) |
38 | 37 | ex 403 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞)))) |
39 | 1, 34, 38 | rexlimd 3234 | . . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝐴 𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞))) |
40 | 39 | imp 397 | . . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
41 | 32, 40 | sylan2 588 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}) → 𝐷 ∈ (0[,]+∞)) |
42 | 1, 2, 3, 5, 6, 7, 8, 12, 23, 26, 41 | esumf1o 30656 | . 2 ⊢ (𝜑 → Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷 = Σ*𝑘 ∈ 𝐴𝐵) |
43 | 42 | eqcomd 2830 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 Ⅎwnf 1884 ∈ wcel 2166 {cab 2810 Ⅎwnfc 2955 ∀wral 3116 ∃wrex 3117 Vcvv 3413 ↦ cmpt 4951 ◡ccnv 5340 ran crn 5342 Fun wfun 6116 Fn wfn 6117 –1-1-onto→wf1o 6121 ‘cfv 6122 (class class class)co 6904 0cc0 10251 +∞cpnf 10387 [,]cicc 12465 Σ*cesum 30633 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-se 5301 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-isom 6131 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-om 7326 df-1st 7427 df-2nd 7428 df-supp 7559 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-fsupp 8544 df-fi 8585 df-oi 8683 df-card 9077 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-7 11418 df-8 11419 df-9 11420 df-n0 11618 df-z 11704 df-dec 11821 df-uz 11968 df-xadd 12232 df-icc 12469 df-fz 12619 df-fzo 12760 df-seq 13095 df-hash 13410 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-tset 16323 df-ple 16324 df-ds 16326 df-rest 16435 df-topn 16436 df-0g 16454 df-gsum 16455 df-topgen 16456 df-ordt 16513 df-xrs 16514 df-ps 17552 df-tsr 17553 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-cntz 18099 df-cmn 18547 df-fbas 20102 df-fg 20103 df-top 21068 df-topon 21085 df-topsp 21107 df-bases 21120 df-ntr 21194 df-nei 21272 df-fil 22019 df-fm 22111 df-flim 22112 df-flf 22113 df-tsms 22299 df-esum 30634 |
This theorem is referenced by: esumrnmpt 30658 esum2dlem 30698 measvunilem 30819 omssubadd 30906 |
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