esumc - Mathbox for Thierry Arnoux

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Theorem esumc 34750

Description: Convert from the collection form to the class-variable form of a sum. (Contributed by Thierry Arnoux, 10-May-2017.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
esumc	⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = Σ^𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷)

Distinct variable groups: 𝑦,𝑘,𝑧 𝑦,𝐴,𝑧 𝑦,𝐵 𝑦,𝐶,𝑧 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑧, 𝑘) 𝐴(𝑘) 𝐵(𝑧, 𝑘) 𝐶(𝑘) 𝐷(𝑦, 𝑧, 𝑘) 𝑉(𝑦, 𝑧, 𝑘) 𝑊(𝑦, 𝑧, 𝑘)

**Proof of Theorem esumc**
Step	Hyp	Ref	Expression
1		esumc.1	. . 3 ⊢ Ⅎ𝑘𝜑
2		esumc.0	. . 3 ⊢ Ⅎ𝑘𝐷
3		nfcv 2926	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfre1 3291	. . . 4 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝑧 = 𝐶
5	4	nfab 2932	. . 3 ⊢ Ⅎ𝑘{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}
6		esumc.2	. . 3 ⊢ Ⅎ𝑘𝐴
7		nfmpt1 5207	. . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶)
8		esumc.3	. . 3 ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵)
9		esumc.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		elex 3475	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12	6, 11	abrexexd 33174	. . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ∈ V)
13		esumc.7	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ 𝑊)
14	13	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ 𝑊))
15	1, 14	ralrimi 3264	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊)
16	6	fnmptf 6694	. . . . 5 ⊢ (∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊 → (𝑘 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
17	15, 16	syl 18	. . . 4 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶) Fn 𝐴)
18		esumc.5	. . . 4 ⊢ (𝜑 → Fun ^◡(𝑘 ∈ 𝐴 ↦ 𝐶))
19		eqid 2763	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
20	19	rnmpt 5945	. . . . 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 6849	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶1-1onto{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 1362	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴⟶1-1onto{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶})
24		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
25	6	fvmpt2f 7014	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐶 ∈ 𝑊) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
26	24, 13, 25	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
27		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
28		eqeq1 2767	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑦 = 𝐶))
29	28	rexbidv 3190	. . . . . 6 ⊢ (𝑧 = 𝑦 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐶 ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶))
30	27, 29	elab 3636	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶)
31	8	reximi 3104	. . . . 5 ⊢ (∃𝑘 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵)
32	30, 31	sylbi 220	. . . 4 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵)
33		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞)
34	2, 33	nfel 2940	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞)
35		esumc.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
36		eleq1 2852	. . . . . . . 8 ⊢ (𝐷 = 𝐵 → (𝐷 ∈ (0[,]+∞) ↔ 𝐵 ∈ (0[,]+∞)))
37	35, 36	syl5ibrcom 250	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞)))
38	37	ex 418	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞))))
39	1, 34, 38	rexlimd 3273	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝐴 𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞)))
40	39	imp 412	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) → 𝐷 ∈ (0[,]+∞))
41	32, 40	sylan2 605	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}) → 𝐷 ∈ (0[,]+∞))
42	1, 2, 3, 5, 6, 7, 8, 12, 23, 26, 41	esumf1o 34749	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷 = Σ^𝑘 ∈ 𝐴𝐵)
43	42	eqcomd 2769	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = Σ^𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2147 {cab 2741 Ⅎwnfc 2911 ∀wral 3080 ∃wrex 3090 Vcvv 3454 ↦ cmpt 5189 ^◡ccnv 5654 ran crn 5656 Fun wfun 6549 Fn wfn 6550 ⟶1-1ontowf1o 6554 ‘cfv 6555 (class class class)co 7440 0cc0 11224 +∞cpnf 11364 [,]cicc 13503 Σ^*cesum 34726

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-isom 6564 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-supp 8202 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-1o 8501 df-2o 8502 df-er 8742 df-map 8874 df-en 8999 df-dom 9000 df-sdom 9001 df-fin 9002 df-fsupp 9379 df-fi 9427 df-oi 9528 df-card 10044 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-2 12429 df-3 12430 df-4 12431 df-5 12432 df-6 12433 df-7 12434 df-8 12435 df-9 12436 df-n0 12631 df-z 12718 df-dec 12839 df-uz 12990 df-xadd 13266 df-icc 13507 df-fz 13664 df-fzo 13813 df-seq 14169 df-hash 14499 df-struct 17351 df-sets 17368 df-slot 17386 df-ndx 17398 df-base 17414 df-ress 17435 df-plusg 17467 df-mulr 17468 df-tset 17473 df-ple 17474 df-ds 17476 df-rest 17619 df-topn 17620 df-0g 17638 df-gsum 17639 df-topgen 17640 df-ordt 17699 df-xrs 17700 df-ps 18766 df-tsr 18767 df-mgm 18842 df-sgrp 18934 df-mnd 18950 df-submnd 19005 df-cntz 19558 df-cmn 20023 df-fbas 21702 df-fg 21703 df-top 23239 df-topon 23256 df-topsp 23278 df-bases 23291 df-ntr 23365 df-nei 23443 df-fil 24192 df-fm 24284 df-flim 24285 df-flf 24286 df-tsms 24473 df-esum 34727

This theorem is used by: esumrnmpt 34751 esum2dlem 34791 measvunilem 34912 omssubadd 34999

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