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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 2903 | . . 3 ⊢ Ⅎ𝑦𝐵 | |
4 | nfre1 3283 | . . . 4 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝑧 = 𝐶 | |
5 | 4 | nfab 2909 | . . 3 ⊢ Ⅎ𝑘{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} |
6 | esumc.2 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
7 | nfmpt1 5256 | . . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶) | |
8 | esumc.3 | . . 3 ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵) | |
9 | esumc.4 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
10 | elex 3499 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ V) |
12 | 6, 11 | abrexexd 32537 | . . 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 3255 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊) |
16 | 6 | fnmptf 6705 | . . . . 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 2735 | . . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶) | |
20 | 19 | rnmpt 5971 | . . . . 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 6854 | . . . 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 1342 | . . 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 3482 | . . . . . 6 ⊢ 𝑦 ∈ V | |
28 | eqeq1 2739 | . . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑦 = 𝐶)) | |
29 | 28 | rexbidv 3177 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐶 ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶)) |
30 | 27, 29 | elab 3681 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶) |
31 | 8 | reximi 3082 | . . . . 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 2903 | . . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞) | |
34 | 2, 33 | nfel 2918 | . . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞) |
35 | esumc.6 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞)) | |
36 | eleq1 2827 | . . . . . . . 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 3264 | . . . . 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 34031 | . 2 ⊢ (𝜑 → Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷 = Σ*𝑘 ∈ 𝐴𝐵) |
43 | 42 | eqcomd 2741 | 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 1537 Ⅎwnf 1780 ∈ wcel 2106 {cab 2712 Ⅎwnfc 2888 ∀wral 3059 ∃wrex 3068 Vcvv 3478 ↦ cmpt 5231 ◡ccnv 5688 ran crn 5690 Fun wfun 6557 Fn wfn 6558 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 [,]cicc 13387 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 |
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-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-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-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-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-xadd 13153 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-ps 18624 df-tsr 18625 df-mgm 18666 df-sgrp 18745 df-mnd 18761 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-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-tsms 24151 df-esum 34009 |
This theorem is referenced by: esumrnmpt 34033 esum2dlem 34073 measvunilem 34193 omssubadd 34282 |
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