esumc - Mathbox for Thierry Arnoux

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Theorem esumc 32650

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 2907	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfre1 3268	. . . 4 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝑧 = 𝐶
5	4	nfab 2913	. . 3 ⊢ Ⅎ𝑘{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}
6		esumc.2	. . 3 ⊢ Ⅎ𝑘𝐴
7		nfmpt1 5213	. . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶)
8		esumc.3	. . 3 ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵)
9		esumc.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		elex 3463	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12	6, 11	abrexexd 31435	. . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ∈ V)
13		esumc.7	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ 𝑊)
14	13	ex 413	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ 𝑊))
15	1, 14	ralrimi 3240	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊)
16	6	fnmptf 6637	. . . . 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 2736	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
20	19	rnmpt 5910	. . . . 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 6789	. . . 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 1343	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶})
24		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
25	6	fvmpt2f 6949	. . . 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 3449	. . . . . 6 ⊢ 𝑦 ∈ V
28		eqeq1 2740	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑦 = 𝐶))
29	28	rexbidv 3175	. . . . . 6 ⊢ (𝑧 = 𝑦 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐶 ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶))
30	27, 29	elab 3630	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶)
31	8	reximi 3087	. . . . 5 ⊢ (∃𝑘 ∈ 𝐴 𝑦 = 𝐶 → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵)
32	30, 31	sylbi 216	. . . 4 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} → ∃𝑘 ∈ 𝐴 𝐷 = 𝐵)
33		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞)
34	2, 33	nfel 2921	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞)
35		esumc.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
36		eleq1 2825	. . . . . . . 8 ⊢ (𝐷 = 𝐵 → (𝐷 ∈ (0[,]+∞) ↔ 𝐵 ∈ (0[,]+∞)))
37	35, 36	syl5ibrcom 246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞)))
38	37	ex 413	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞))))
39	1, 34, 38	rexlimd 3249	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝐴 𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞)))
40	39	imp 407	. . . 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 32649	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷 = Σ^𝑘 ∈ 𝐴𝐵)
43	42	eqcomd 2742	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = Σ^𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 {cab 2713 Ⅎwnfc 2887 ∀wral 3064 ∃wrex 3073 Vcvv 3445 ↦ cmpt 5188 ^◡ccnv 5632 ran crn 5634 Fun wfun 6490 Fn wfn 6491 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7357 0cc0 11051 +∞cpnf 11186 [,]cicc 13267 Σ^*cesum 32626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-xadd 13034 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-hash 14231 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-tset 17152 df-ple 17153 df-ds 17155 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-ordt 17383 df-xrs 17384 df-ps 18455 df-tsr 18456 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-cntz 19097 df-cmn 19564 df-fbas 20793 df-fg 20794 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-ntr 22371 df-nei 22449 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-tsms 23478 df-esum 32627

This theorem is referenced by: esumrnmpt 32651 esum2dlem 32691 measvunilem 32811 omssubadd 32900

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