Theorem cbvsum 15786

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 3860	. . . . . . . . . . . 12 ⊢ ⦋𝑛 / 𝑗⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶
5	4	a1i 11	. . . . . . . . . . 11 ⊢ (⊤ → ⦋𝑛 / 𝑗⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶)
6	5	ifeq1d 4505	. . . . . . . . . 10 ⊢ (⊤ → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))
7	6	mpteq2dv 5203	. . . . . . . . 9 ⊢ (⊤ → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))
8	7	seqeq3d 14077	. . . . . . . 8 ⊢ (⊤ → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))))
9	8	mptru 1577	. . . . . . 7 ⊢ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑗⦌𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))
10	9	breq1i 5114	. . . . . 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 635	. . . . 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 3111	. . . 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 3860	. . . . . . . . . . . . 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 5203	. . . . . . . . . . 11 ⊢ (⊤ → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
16	15	seqeq3d 14077	. . . . . . . . . 10 ⊢ (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)))
17	16	mptru 1577	. . . . . . . . 9 ⊢ seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
18	17	fveq1i 6883	. . . . . . . 8 ⊢ (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)
19	18	eqeq2i 2775	. . . . . . 7 ⊢ (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))
20	19	anbi2i 635	. . . . . 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 1881	. . . . 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 3111	. . . 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 928	. . 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 6522	. 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 15778	. 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 15778	. 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 2795	1 ⊢ Σ𝑗 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570  ⊤wtru 1571  ∃wex 1812   ∈ wcel 2145  Ⅎwnfc 2909  ∃wrex 3088  ⦋csb 3850   ⊆ wss 3902  ifcif 4485   class class class wbr 5107   ↦ cmpt 5190  ℩cio 6491  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7417  0cc0 11128  1c1 11129   + caddc 11131  ℕcn 12261  ℤcz 12619  ℤ_≥cuz 12891  ...cfz 13565  seqcseq 14069   ⇝ cli 15575  Σcsu 15777

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-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rex 3089  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-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-xp 5665  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-iota 6493  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-seq 14070  df-sum 15778

This theorem is used by:  sumfc  15799  sumss2  15816  fsumclf  15828  fsumzcl2  15829  fsumsplitf  15832  sumsnf  15833  sumsns  15840  fsummsnunz  15844  fsumsplitsnun  15845  fsum2dlem  15860  fsumcom2  15864  fsumshftm  15871  fsumrlim  15902  fsumo1  15903  o1fsum  15904  fsumiun  15912  ovolfiniun  25735  ovoliun2  25740  volfiniun  25781  itgfsum  26061  elplyd  26434  coeeq2  26475  fsumdvdscom  27429  fsumdvdsmul  27439  fsumvma  27457  fsumiunle  33307  esumpfinvalf  34594  fsumshftd  39833  sticksstones8  43027  sticksstones10  43029  sticksstones12a  43031  sticksstones12  43032  aks6d1c7lem3  43056  binomcxplemdvsum  45187  sumsnd  45868  fsummulc1f  46409  fsumf1of  46412  fsumiunss  46413  fsumreclf  46414  fsumlessf  46415  fsumsermpt  46417  dvnmul  46779  fourierdlem115  47057  sge0revalmpt  47214  sge0fsummpt  47226  sge0iunmptlemfi  47249  sge0iunmptlemre  47251  sge0ltfirpmpt2  47262  sge0isummpt2  47268  sge0xaddlem2  47270  sge0fsummptf  47272  fsummsndifre  48276  fsumsplitsndif  48277  fsummmodsndifre  48278  fsummmodsnunz  48279