esumc - Mathbox for Thierry Arnoux

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Theorem esumc 34548

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 2924	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfre1 3289	. . . 4 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝑧 = 𝐶
5	4	nfab 2930	. . 3 ⊢ Ⅎ𝑘{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}
6		esumc.2	. . 3 ⊢ Ⅎ𝑘𝐴
7		nfmpt1 5208	. . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶)
8		esumc.3	. . 3 ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵)
9		esumc.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		elex 3474	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12	6, 11	abrexexd 32970	. . 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 3262	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊)
16	6	fnmptf 6672	. . . . 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 2762	. . . . . 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 6827	. . . 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 1362	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶})
24		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
25	6	fvmpt2f 6991	. . . 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 3457	. . . . . 6 ⊢ 𝑦 ∈ V
28		eqeq1 2766	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑦 = 𝐶))
29	28	rexbidv 3188	. . . . . 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 3102	. . . . 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 2924	. . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞)
34	2, 33	nfel 2938	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞)
35		esumc.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
36		eleq1 2850	. . . . . . . 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 3271	. . . . 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 34547	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷 = Σ^𝑘 ∈ 𝐴𝐵)
43	42	eqcomd 2768	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 2145 {cab 2740 Ⅎwnfc 2909 ∀wral 3078 ∃wrex 3088 Vcvv 3453 ↦ cmpt 5190 ^◡ccnv 5658 ran crn 5660 Fun wfun 6531 Fn wfn 6532 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7416 0cc0 11127 +∞cpnf 11267 [,]cicc 13403 Σ^*cesum 34524

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

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 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-supp 8162 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-2o 8459 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fsupp 9335 df-fi 9384 df-oi 9485 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12532 df-z 12619 df-dec 12740 df-uz 12891 df-xadd 13166 df-icc 13407 df-fz 13564 df-fzo 13712 df-seq 14068 df-hash 14397 df-struct 17243 df-sets 17260 df-slot 17278 df-ndx 17290 df-base 17306 df-ress 17327 df-plusg 17359 df-mulr 17360 df-tset 17365 df-ple 17366 df-ds 17368 df-rest 17511 df-topn 17512 df-0g 17530 df-gsum 17531 df-topgen 17532 df-ordt 17591 df-xrs 17592 df-ps 18658 df-tsr 18659 df-mgm 18734 df-sgrp 18823 df-mnd 18839 df-submnd 18893 df-cntz 19445 df-cmn 19910 df-fbas 21583 df-fg 21584 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-ntr 23246 df-nei 23324 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-tsms 24354 df-esum 34525

This theorem is used by: esumrnmpt 34549 esum2dlem 34589 measvunilem 34710 omssubadd 34798

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