Theorem cbvsum 15657

Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Hypotheses

Ref	Expression
cbvsum.1	⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
cbvsum.2	⊢ Ⅎ𝑘𝐵
cbvsum.3	⊢ Ⅎ𝑗𝐶

Assertion

Ref	Expression
cbvsum	⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶

Distinct variable group:   𝑗,𝑘

Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐵(𝑗,𝑘)   𝐶(𝑗,𝑘)

Proof of Theorem cbvsum

Dummy variables 𝑓 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvsum.2	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐵
2		cbvsum.3	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐶
3		cbvsum.1	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
4	1, 2, 3	cbvcsbw 3847	. . . . . . . . . . . 12 ⊢ ⦋𝑛 / 𝑗⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶
5	4	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ⦋𝑛 / 𝑗⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶)
6	5	ifeq1d 4486	. . . . . . . . . 10 ⊢ (⊤ → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))
7	6	mpteq2dv 5179	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))
8	7	seqeq3d 13971	. . . . . . . 8 ⊢ (⊤ → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))))
9	8	mptru 1549	. . . . . . 7 ⊢ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))
10	9	breq1i 5092	. . . . . 6 ⊢ (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)
11	10	anbi2i 624	. . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))
12	11	rexbii 3084	. . . 4 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))
13	1, 2, 3	cbvcsbw 3847	. . . . . . . . . . . . 13 ⊢ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶
14	13	a1i 11	. . . . . . . . . . . 12 ⊢ (⊤ → ⦋(𝑓‘𝑛) / 𝑗⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
15	14	mpteq2dv 5179	. . . . . . . . . . 11 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
16	15	seqeq3d 13971	. . . . . . . . . 10 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)))
17	16	mptru 1549	. . . . . . . . 9 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
18	17	fveq1i 6841	. . . . . . . 8 ⊢ (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)
19	18	eqeq2i 2749	. . . . . . 7 ⊢ (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))
20	19	anbi2i 624	. . . . . 6 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))
21	20	exbii 1850	. . . . 5 ⊢ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))
22	21	rexbii 3084	. . . 4 ⊢ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))
23	12, 22	orbi12i 915	. . 3 ⊢ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
24	23	iotabii 6483	. 2 ⊢ (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
25		df-sum 15649	. 2 ⊢ Σ𝑗 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚))))
26		df-sum 15649	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
27	24, 25, 26	3eqtr4i 2769	1 ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542  ⊤wtru 1543  ∃wex 1781   ∈ wcel 2114  Ⅎwnfc 2883  ∃wrex 3061  ⦋csb 3837   ⊆ wss 3889  ifcif 4466   class class class wbr 5085   ↦ cmpt 5166  ℩cio 6452  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  0cc0 11038  1c1 11039   + caddc 11041  ℕcn 12174  ℤcz 12524  ℤ_≥cuz 12788  ...cfz 13461  seqcseq 13963   ⇝ cli 15446  Σcsu 15648

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-12 2185  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-xp 5637  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-iota 6454  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-seq 13964  df-sum 15649

This theorem is referenced by:  sumfc  15671  sumss2  15688  fsumclf  15700  fsumzcl2  15701  fsumsplitf  15704  sumsnf  15705  sumsns  15712  fsummsnunz  15716  fsumsplitsnun  15717  fsum2dlem  15732  fsumcom2  15736  fsumshftm  15743  fsumrlim  15774  fsumo1  15775  o1fsum  15776  fsumiun  15784  ovolfiniun  25468  ovoliun2  25473  volfiniun  25514  itgfsum  25794  elplyd  26167  coeeq2  26207  fsumdvdscom  27148  fsumdvdsmul  27158  fsumvma  27176  fsumiunle  32902  esumpfinvalf  34220  fsumshftd  39398  sticksstones8  42592  sticksstones10  42594  sticksstones12a  42596  sticksstones12  42597  aks6d1c7lem3  42621  binomcxplemdvsum  44782  sumsnd  45457  fsummulc1f  46001  fsumf1of  46004  fsumiunss  46005  fsumreclf  46006  fsumlessf  46007  fsumsermpt  46009  dvnmul  46371  fourierdlem115  46649  sge0revalmpt  46806  sge0fsummpt  46818  sge0iunmptlemfi  46841  sge0iunmptlemre  46843  sge0ltfirpmpt2  46854  sge0isummpt2  46860  sge0xaddlem2  46862  sge0fsummptf  46864  fsummsndifre  47828  fsumsplitsndif  47829  fsummmodsndifre  47830  fsummmodsnunz  47831