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 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑘𝐶 | |
2 | nfv 1915 | . . 3 ⊢ Ⅎ𝑘𝜑 | |
3 | esumrnmpt.0 | . . 3 ⊢ Ⅎ𝑘𝐴 | |
4 | esumrnmpt.1 | . . 3 ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐷) | |
5 | esumrnmpt.2 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
6 | esumrnmpt.4 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (𝑊 ∖ {∅})) | |
7 | esumrnmpt.5 | . . . 4 ⊢ (𝜑 → Disj 𝑘 ∈ 𝐴 𝐵) | |
8 | 2, 3, 6, 7 | disjdsct 30438 | . . 3 ⊢ (𝜑 → Fun ◡(𝑘 ∈ 𝐴 ↦ 𝐵)) |
9 | esumrnmpt.3 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐷 ∈ (0[,]+∞)) | |
10 | 1, 2, 3, 4, 5, 8, 9, 6 | esumc 31310 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ 𝐴𝐷 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵}𝐶) |
11 | eqid 2821 | . . . 4 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) | |
12 | 11 | rnmpt 5827 | . . 3 ⊢ ran (𝑘 ∈ 𝐴 ↦ 𝐵) = {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵} |
13 | esumeq1 31293 | . . 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) | |
14 | 12, 13 | ax-mp 5 | . 2 ⊢ Σ*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ*𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐵}𝐶 |
15 | 10, 14 | syl6reqr 2875 | 1 ⊢ (𝜑 → Σ*𝑦 ∈ ran (𝑘 ∈ 𝐴 ↦ 𝐵)𝐶 = Σ*𝑘 ∈ 𝐴𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2799 Ⅎwnfc 2961 ∃wrex 3139 ∖ cdif 3933 ∅c0 4291 {csn 4567 Disj wdisj 5031 ↦ cmpt 5146 ran crn 5556 (class class class)co 7156 0cc0 10537 +∞cpnf 10672 [,]cicc 12742 Σ*cesum 31286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-disj 5032 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-xadd 12509 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-hash 13692 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-tset 16584 df-ple 16585 df-ds 16587 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-ordt 16774 df-xrs 16775 df-ps 17810 df-tsr 17811 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-cntz 18447 df-cmn 18908 df-fbas 20542 df-fg 20543 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-ntr 21628 df-nei 21706 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-tsms 22735 df-esum 31287 |
This theorem is referenced by: esumrnmpt2 31327 |
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