esumc - Mathbox for Thierry Arnoux

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Theorem esumc 34616

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 2922	. . 3 ⊢ Ⅎ𝑦𝐵
4		nfre1 3287	. . . 4 ⊢ Ⅎ𝑘∃𝑘 ∈ 𝐴 𝑧 = 𝐶
5	4	nfab 2928	. . 3 ⊢ Ⅎ𝑘{𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}
6		esumc.2	. . 3 ⊢ Ⅎ𝑘𝐴
7		nfmpt1 5203	. . 3 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐶)
8		esumc.3	. . 3 ⊢ (𝑦 = 𝐶 → 𝐷 = 𝐵)
9		esumc.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
10		elex 3471	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝐴 ∈ V)
12	6, 11	abrexexd 33038	. . 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 3260	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ 𝐴 𝐶 ∈ 𝑊)
16	6	fnmptf 6663	. . . . 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 2760	. . . . . 6 ⊢ (𝑘 ∈ 𝐴 ↦ 𝐶) = (𝑘 ∈ 𝐴 ↦ 𝐶)
20	19	rnmpt 5935	. . . . 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 6818	. . . 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 6982	. . . 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 3454	. . . . . 6 ⊢ 𝑦 ∈ V
28		eqeq1 2764	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 = 𝐶 ↔ 𝑦 = 𝐶))
29	28	rexbidv 3186	. . . . . 6 ⊢ (𝑧 = 𝑦 → (∃𝑘 ∈ 𝐴 𝑧 = 𝐶 ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶))
30	27, 29	elab 3632	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶} ↔ ∃𝑘 ∈ 𝐴 𝑦 = 𝐶)
31	8	reximi 3100	. . . . 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 2922	. . . . . . 7 ⊢ Ⅎ𝑘(0[,]+∞)
34	2, 33	nfel 2936	. . . . . 6 ⊢ Ⅎ𝑘 𝐷 ∈ (0[,]+∞)
35		esumc.6	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,]+∞))
36		eleq1 2848	. . . . . . . 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 3269	. . . . 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 34615	. 2 ⊢ (𝜑 → Σ^𝑦 ∈ {𝑧 ∣ ∃𝑘 ∈ 𝐴 𝑧 = 𝐶}𝐷 = Σ^𝑘 ∈ 𝐴𝐵)
43	42	eqcomd 2766	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 2738 Ⅎwnfc 2907 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ↦ cmpt 5185 ^◡ccnv 5646 ran crn 5648 Fun wfun 6521 Fn wfn 6522 –1-1-onto→wf1o 6526 ‘cfv 6527 (class class class)co 7408 0cc0 11171 +∞cpnf 11311 [,]cicc 13448 Σ^*cesum 34592

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 2213 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7734 ax-cnex 11227 ax-resscn 11228 ax-1cn 11229 ax-icn 11230 ax-addcl 11231 ax-addrcl 11232 ax-mulcl 11233 ax-mulrcl 11234 ax-mulcom 11235 ax-addass 11236 ax-mulass 11237 ax-distr 11238 ax-i2m1 11239 ax-1ne0 11240 ax-1rid 11241 ax-rnegex 11242 ax-rrecex 11243 ax-cnre 11244 ax-pre-lttri 11245 ax-pre-lttrn 11246 ax-pre-ltadd 11247 ax-pre-mulgt0 11248

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6293 df-ord 6354 df-on 6355 df-lim 6356 df-suc 6357 df-iota 6483 df-fun 6529 df-fn 6530 df-f 6531 df-f1 6532 df-fo 6533 df-f1o 6534 df-fv 6535 df-isom 6536 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7861 df-1st 7984 df-2nd 7985 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-en 8952 df-dom 8953 df-sdom 8954 df-fin 8955 df-fsupp 9332 df-fi 9381 df-oi 9482 df-card 9991 df-pnf 11316 df-mnf 11317 df-xr 11318 df-ltxr 11319 df-le 11320 df-sub 11514 df-neg 11515 df-nn 12305 df-2 12374 df-3 12375 df-4 12376 df-5 12377 df-6 12378 df-7 12379 df-8 12380 df-9 12381 df-n0 12576 df-z 12663 df-dec 12784 df-uz 12935 df-xadd 13211 df-icc 13452 df-fz 13609 df-fzo 13757 df-seq 14113 df-hash 14442 df-struct 17286 df-sets 17303 df-slot 17321 df-ndx 17333 df-base 17349 df-ress 17370 df-plusg 17402 df-mulr 17403 df-tset 17408 df-ple 17409 df-ds 17411 df-rest 17554 df-topn 17555 df-0g 17573 df-gsum 17574 df-topgen 17575 df-ordt 17634 df-xrs 17635 df-ps 18701 df-tsr 18702 df-mgm 18777 df-sgrp 18869 df-mnd 18885 df-submnd 18940 df-cntz 19492 df-cmn 19957 df-fbas 21636 df-fg 21637 df-top 23173 df-topon 23190 df-topsp 23212 df-bases 23225 df-ntr 23299 df-nei 23377 df-fil 24126 df-fm 24218 df-flim 24219 df-flf 24220 df-tsms 24407 df-esum 34593

This theorem is used by: esumrnmpt 34617 esum2dlem 34657 measvunilem 34778 omssubadd 34866

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