| 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 2905 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
| 4 | nfre1 3285 | . . . 4 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝑧 = 𝐶 | |
| 5 | 4 | nfab 2911 | . . 3 ⊢ Ⅎ𝑘{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} |
| 6 | esumc.2 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
| 7 | nfmpt1 5250 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶) | |
| 8 | esumc.3 | . . 3 ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵) | |
| 9 | esumc.4 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
| 10 | elex 3501 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
| 12 | 6, 11 | abrexexd 32528 | . . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ∈ V) |
| 13 | esumc.7 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ 𝑊) | |
| 14 | 13 | ex 412 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ 𝑊)) |
| 15 | 1, 14 | ralrimi 3257 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊) |
| 16 | 6 | fnmptf 6704 | . . . . 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 2737 | . . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶) | |
| 20 | 19 | rnmpt 5968 | . . . . 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 6853 | . . . 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 1344 | . . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}) |
| 24 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴) | |
| 25 | 6 | fvmpt2f 7017 | . . . 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 3484 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 28 | eqeq1 2741 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑦 = 𝐶)) | |
| 29 | 28 | rexbidv 3179 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐶 ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶)) |
| 30 | 27, 29 | elab 3679 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶) |
| 31 | 8 | reximi 3084 | . . . . 5 ⊢ (∃𝑘 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) |
| 32 | 30, 31 | sylbi 217 | . . . 4 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) |
| 33 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞) | |
| 34 | 2, 33 | nfel 2920 | . . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞) |
| 35 | esumc.6 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
| 36 | eleq1 2829 | . . . . . . . 8 ⊢ (𝐷 = 𝐵 → (𝐷 ∈ (0[,]+∞) ↔ 𝐵 ∈ (0[,]+∞))) | |
| 37 | 35, 36 | syl5ibrcom 247 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞))) |
| 38 | 37 | ex 412 | . . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞)))) |
| 39 | 1, 34, 38 | rexlimd 3266 | . . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝐴 𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞))) |
| 40 | 39 | imp 406 | . . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) → 𝐷 ∈ (0[,]+∞)) |
| 41 | 32, 40 | sylan2 593 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}) → 𝐷 ∈ (0[,]+∞)) |
| 42 | 1, 2, 3, 5, 6, 7, 8, 12, 23, 26, 41 | esumf1o 34051 | . 2 ⊢ (𝜑 → Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷 = Σ*𝑘 ∈ 𝐴𝐵) |
| 43 | 42 | eqcomd 2743 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐵 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ↦ cmpt 5225 ◡ccnv 5684 ran crn 5686 Fun wfun 6555 Fn wfn 6556 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 0cc0 11155 +∞cpnf 11292 [,]cicc 13390 Σ*cesum 34028 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-xadd 13155 df-icc 13394 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-tset 17316 df-ple 17317 df-ds 17319 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-ordt 17546 df-xrs 17547 df-ps 18611 df-tsr 18612 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-cntz 19335 df-cmn 19800 df-fbas 21361 df-fg 21362 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-ntr 23028 df-nei 23106 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-tsms 24135 df-esum 34029 |
| This theorem is referenced by: esumrnmpt 34053 esum2dlem 34093 measvunilem 34213 omssubadd 34302 |
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