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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumrnmpt | Structured version Visualization version GIF version |
Description: Rewrite an extended sum into a sum on the range of a mapping function. (Contributed by Thierry Arnoux, 27-May-2020.) |
Ref | Expression |
---|---|
esumrnmpt.0 | ⊢ Ⅎ𝑘𝐴 |
esumrnmpt.1 | ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷) |
esumrnmpt.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
esumrnmpt.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞)) |
esumrnmpt.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (𝑊 ∖ {∅})) |
esumrnmpt.5 | ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵) |
Ref | Expression |
---|---|
esumrnmpt | ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ*𝑘 ∈ 𝐴𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | rnmpt 5979 | . . 3 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} |
3 | esumeq1 33990 | . . 3 ⊢ (ran (𝑘 ∈ 𝐴 ↦ 𝐵) = {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} → Σ*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵}𝐶) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ Σ*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵}𝐶 |
5 | nfcv 2904 | . . 3 ⊢ Ⅎ𝑘𝐶 | |
6 | nfv 1913 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
7 | esumrnmpt.0 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
8 | esumrnmpt.1 | . . 3 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷) | |
9 | esumrnmpt.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
10 | esumrnmpt.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (𝑊 ∖ {∅})) | |
11 | esumrnmpt.5 | . . . 4 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵) | |
12 | 6, 7, 10, 11 | disjdsct 32706 | . . 3 ⊢ (𝜑 → Fun ◡(𝑘 ∈ 𝐴 ↦ 𝐵)) |
13 | esumrnmpt.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞)) | |
14 | 5, 6, 7, 8, 9, 12, 13, 10 | esumc 34007 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐷 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵}𝐶) |
15 | 4, 14 | eqtr4id 2793 | 1 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ*𝑘 ∈ 𝐴𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2103 {cab 2711 Ⅎwnfc 2888 ∃wrex 3072 ∖ cdif 3967 ∅c0 4347 {csn 4648 Disj wdisj 5136 ↦ cmpt 5252 ran crn 5700 (class class class)co 7445 0cc0 11180 +∞cpnf 11317 [,]cicc 13406 Σ*cesum 33983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-rep 5306 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 ax-un 7766 ax-cnex 11236 ax-resscn 11237 ax-1cn 11238 ax-icn 11239 ax-addcl 11240 ax-addrcl 11241 ax-mulcl 11242 ax-mulrcl 11243 ax-mulcom 11244 ax-addass 11245 ax-mulass 11246 ax-distr 11247 ax-i2m1 11248 ax-1ne0 11249 ax-1rid 11250 ax-rnegex 11251 ax-rrecex 11252 ax-cnre 11253 ax-pre-lttri 11254 ax-pre-lttrn 11255 ax-pre-ltadd 11256 ax-pre-mulgt0 11257 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3383 df-reu 3384 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-pss 3990 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4973 df-iun 5021 df-disj 5137 df-br 5170 df-opab 5232 df-mpt 5253 df-tr 5287 df-id 5597 df-eprel 5603 df-po 5611 df-so 5612 df-fr 5654 df-se 5655 df-we 5656 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-pred 6331 df-ord 6397 df-on 6398 df-lim 6399 df-suc 6400 df-iota 6524 df-fun 6574 df-fn 6575 df-f 6576 df-f1 6577 df-fo 6578 df-f1o 6579 df-fv 6580 df-isom 6581 df-riota 7401 df-ov 7448 df-oprab 7449 df-mpo 7450 df-om 7900 df-1st 8026 df-2nd 8027 df-supp 8198 df-frecs 8318 df-wrecs 8349 df-recs 8423 df-rdg 8462 df-1o 8518 df-2o 8519 df-er 8759 df-map 8882 df-en 9000 df-dom 9001 df-sdom 9002 df-fin 9003 df-fsupp 9428 df-fi 9476 df-oi 9575 df-card 10004 df-pnf 11322 df-mnf 11323 df-xr 11324 df-ltxr 11325 df-le 11326 df-sub 11518 df-neg 11519 df-nn 12290 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12550 df-z 12636 df-dec 12755 df-uz 12900 df-xadd 13172 df-icc 13410 df-fz 13564 df-fzo 13708 df-seq 14049 df-hash 14376 df-struct 17189 df-sets 17206 df-slot 17224 df-ndx 17236 df-base 17254 df-ress 17283 df-plusg 17319 df-mulr 17320 df-tset 17325 df-ple 17326 df-ds 17328 df-rest 17477 df-topn 17478 df-0g 17496 df-gsum 17497 df-topgen 17498 df-ordt 17556 df-xrs 17557 df-ps 18631 df-tsr 18632 df-mgm 18673 df-sgrp 18752 df-mnd 18768 df-submnd 18814 df-cntz 19352 df-cmn 19819 df-fbas 21379 df-fg 21380 df-top 22914 df-topon 22931 df-topsp 22953 df-bases 22967 df-ntr 23042 df-nei 23120 df-fil 23868 df-fm 23960 df-flim 23961 df-flf 23962 df-tsms 24149 df-esum 33984 |
This theorem is referenced by: esumrnmpt2 34024 |
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