esumc - Mathbox for Thierry Arnoux

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Theorem esumc 33670

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 2899	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfre1 3279	. . . 4 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝑧 = 𝐶
5	4	nfab 2905	. . 3 ⊢ Ⅎ𝑘{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}
6		esumc.2	. . 3 ⊢ Ⅎ𝑘𝐴
7		nfmpt1 5256	. . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶)
8		esumc.3	. . 3 ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵)
9		esumc.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		elex 3490	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12	6, 11	abrexexd 32317	. . 3 ⊢ (𝜑 → {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ∈ V)
13		esumc.7	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ 𝑊)
14	13	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ 𝑊))
15	1, 14	ralrimi 3251	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊)
16	6	fnmptf 6691	. . . . 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 2728	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
20	19	rnmpt 5957	. . . . 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 6844	. . . 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 1341	. . 3 ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐶):𝐴–1-1-onto→{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶})
24		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
25	6	fvmpt2f 7006	. . . 4 ⊢ ((𝑘 ∈ 𝐴 ∧ 𝐶 ∈ 𝑊) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
26	24, 13, 25	syl2anc 583	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑘) = 𝐶)
27		vex 3475	. . . . . 6 ⊢ 𝑦 ∈ V
28		eqeq1 2732	. . . . . . 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 3667	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶)
31	8	reximi 3081	. . . . 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 2899	. . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞)
34	2, 33	nfel 2914	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞)
35		esumc.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
36		eleq1 2817	. . . . . . . 8 ⊢ (𝐷 = 𝐵 → (𝐷 ∈ (0[,]+∞) ↔ 𝐵 ∈ (0[,]+∞)))
37	35, 36	syl5ibrcom 246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞)))
38	37	ex 412	. . . . . 6 ⊢ (𝜑 → (𝑘 ∈ 𝐴 → (𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞))))
39	1, 34, 38	rexlimd 3260	. . . . 5 ⊢ (𝜑 → (∃𝑘 ∈ 𝐴 𝐷 = 𝐵 → 𝐷 ∈ (0[,]+∞)))
40	39	imp 406	. . . 4 ⊢ ((𝜑 ∧ ∃𝑘 ∈ 𝐴 𝐷 = 𝐵) → 𝐷 ∈ (0[,]+∞))
41	32, 40	sylan2 592	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}) → 𝐷 ∈ (0[,]+∞))
42	1, 2, 3, 5, 6, 7, 8, 12, 23, 26, 41	esumf1o 33669	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷 = Σ^𝑘 ∈ 𝐴𝐵)
43	42	eqcomd 2734	1 ⊢ (𝜑 → Σ^𝑘 ∈ 𝐴𝐵 = Σ^𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 Ⅎwnf 1778 ∈ wcel 2099 {cab 2705 Ⅎwnfc 2879 ∀wral 3058 ∃wrex 3067 Vcvv 3471 ↦ cmpt 5231 ^◡ccnv 5677 ran crn 5679 Fun wfun 6542 Fn wfn 6543 –1-1-onto→wf1o 6547 ‘cfv 6548 (class class class)co 7420 0cc0 11139 +∞cpnf 11276 [,]cicc 13360 Σ^*cesum 33646

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-fi 9435 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-xadd 13126 df-icc 13364 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-tset 17252 df-ple 17253 df-ds 17255 df-rest 17404 df-topn 17405 df-0g 17423 df-gsum 17424 df-topgen 17425 df-ordt 17483 df-xrs 17484 df-ps 18558 df-tsr 18559 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-cntz 19268 df-cmn 19737 df-fbas 21276 df-fg 21277 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-ntr 22937 df-nei 23015 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-tsms 24044 df-esum 33647

This theorem is referenced by: esumrnmpt 33671 esum2dlem 33711 measvunilem 33831 omssubadd 33920

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