Theorem sumeq2sdv 15885

Description: Equality deduction for sum. (Contributed by NM, 3-Jan-2006.) (Proof shortened by Glauco Siliprandi, 5-Apr-2020.) Avoid axioms. (Revised by GG, 14-Aug-2025.)

Hypothesis

Ref	Expression
sumeq2sdv.1	⊢ (𝜑 → 𝐵 = 𝐶)

Assertion

Ref	Expression
sumeq2sdv	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)

Distinct variable group:   𝜑,𝑘

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

Proof of Theorem sumeq2sdv

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

Step	Hyp	Ref	Expression
1		sumeq2sdv.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = 𝐶)
2	1	csbeq2dv 3858	. . . . . . . . . 10 ⊢ (𝜑 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶)
3	2	ifeq1d 4505	. . . . . . . . 9 ⊢ (𝜑 → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))
4	3	mpteq2dv 5202	. . . . . . . 8 ⊢ (𝜑 → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))
5	4	seqeq3d 14165	. . . . . . 7 ⊢ (𝜑 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))))
6	5	breq1d 5116	. . . . . 6 ⊢ (𝜑 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))
7	6	anbi2d 642	. . . . 5 ⊢ (𝜑 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
8	7	rexbidv 3190	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
9	1	csbeq2dv 3858	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
10	9	mpteq2dv 5202	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
11	10	seqeq3d 14165	. . . . . . . . 9 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)))
12	11	fveq1d 6898	. . . . . . . 8 ⊢ (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))
13	12	eqeq2d 2774	. . . . . . 7 ⊢ (𝜑 → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))
14	13	anbi2d 642	. . . . . 6 ⊢ (𝜑 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
15	14	exbidv 1955	. . . . 5 ⊢ (𝜑 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
16	15	rexbidv 3190	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
17	8, 16	orbi12d 932	. . 3 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌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)))))
18	17	iotabidv 6530	. 2 ⊢ (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌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)))))
19		df-sum 15869	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
20		df-sum 15869	. 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))))
21	18, 19, 20	3eqtr4g 2823	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2146  ∃wrex 3090  ⦋csb 3851   ⊆ wss 3900  ifcif 4485   class class class wbr 5106   ↦ cmpt 5189  ℩cio 6499  –1-1-onto→wf1o 6546  ‘cfv 6547  (class class class)co 7430  0cc0 11213  1c1 11214   + caddc 11216  ℕcn 12348  ℤcz 12706  ℤ_≥cuz 12978  ...cfz 13652  seqcseq 14157   ⇝ cli 15666  Σcsu 15868

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-xp 5661  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-iota 6501  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-seq 14158  df-sum 15869

This theorem is used by:  sumsplit  15949  fsumrlim  15993  hash2iun1dif1  16006  incexclem  16020  bpolylem  16229  bpolyval  16230  efval  16260  rpnnen2lem12  16408  pcfac  17092  ramcl  17222  cshwshashnsame  17296  fsumcn  25206  fsum2cn  25207  lebnumlem3  25299  rrxdsfival  25749  uniioombllem6  25924  itg1climres  26050  itgeq1f  26107  itgeq1  26108  cbvitgv  26112  itgeq2  26113  dvmptfsum  26310  elplyr  26534  plyeq0lem  26544  plyadd  26551  plymul  26552  coeeu  26559  coelem  26560  coeeq  26561  coeidlem  26571  coeid  26572  coeid2  26573  plyco  26575  plycjlem  26610  aareccl  26668  taylply2  26710  pserdvlem2  26770  pserdv  26771  abelthlem6  26778  abelthlem9  26782  logtayl  27003  leibpi  27285  basellem3  27425  dchrvmasum2if  27839  dchrvmaeq0  27846  rpvmasum2  27854  dchrisum0re  27855  brcgr  29493  axsegcon  29520  dipfval  31319  ipval  31320  fsumiunle  33476  itgeq12dv  35014  eulerpartleme  35051  eulerpartlemr  35062  eulerpartlemn  35069  reprsum  35298  reprsuc  35300  reprpmtf1o  35311  vtsval  35322  iprodgam  36637  fwddifnval  37058  sumeq12sdv  37136  itgeq12sdv  37138  cbvitgdavw  37200  cbvitgdavw2  37216  knoppndvlem6  37513  knoppf  37531  rrnmval  38905  fsumshftd  40152  fsumcnf  46180  mccl  46752  dvnmul  47095  dvmptfprod  47097  dvnprodlem1  47098  dvnprodlem3  47100  dvnprod  47101  stoweidlem17  47169  stoweidlem26  47178  stoweidlem30  47182  stoweidlem32  47184  dirkertrigeq  47253  dirkeritg  47254  fourierdlem83  47341  fourierdlem103  47361  etransclem11  47397  etransclem24  47410  etransclem26  47412  etransclem27  47413  etransclem28  47414  etransclem31  47417  etransclem35  47421  etransclem46  47432  etransclem47  47433  rrndistlt  47442  ioorrnopn  47457  sge0val  47518  hoiqssbllem2  47775  nnsum3primes4  49028  nnsum4primesodd  49036  nnsum4primesoddALTV  49037  nnsum4primesevenALTV  49041  nn0sumshdiglemB  49874  nn0sumshdiglem1  49875  aacllem  51156