Theorem sumeq1 15639

Description: Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Assertion

Ref	Expression
sumeq1	⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Proof of Theorem sumeq1

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

Step	Hyp	Ref	Expression
1		sseq1 4002	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐵 ⊆ (ℤ≥‘𝑚)))
2		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ) → 𝐴 = 𝐵)
3	2	eleq2d 2813	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐵))
4	3	ifbid 4546	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ) → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0) = if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))
5	4	mpteq2dva 5241	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0)))
6	5	seqeq3d 13977	. . . . . . 7 ⊢ (𝐴 = 𝐵 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))))
7	6	breq1d 5151	. . . . . 6 ⊢ (𝐴 = 𝐵 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))
8	1, 7	anbi12d 630	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ↔ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
9	8	rexbidv 3172	. . . 4 ⊢ (𝐴 = 𝐵 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
10		f1oeq3 6816	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑚)–1-1-onto→𝐵))
11	10	anbi1d 629	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
12	11	exbidv 1916	. . . . 5 ⊢ (𝐴 = 𝐵 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
13	12	rexbidv 3172	. . . 4 ⊢ (𝐴 = 𝐵 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
14	9, 13	orbi12d 915	. . 3 ⊢ (𝐴 = 𝐵 → ((∃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))) ↔ (∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))))
15	14	iotabidv 6520	. 2 ⊢ (𝐴 = 𝐵 → (℩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)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))))
16		df-sum 15637	. 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))))
17		df-sum 15637	. 2 ⊢ Σ𝑘 ∈ 𝐵 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
18	15, 16, 17	3eqtr4g 2791	1 ⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃wrex 3064  ⦋csb 3888   ⊆ wss 3943  ifcif 4523   class class class wbr 5141   ↦ cmpt 5224  ℩cio 6486  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7404  0cc0 11109  1c1 11110   + caddc 11112  ℕcn 12213  ℤcz 12559  ℤ_≥cuz 12823  ...cfz 13487  seqcseq 13969   ⇝ cli 15432  Σcsu 15636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-iota 6488  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-seq 13970  df-sum 15637

This theorem is referenced by:  sumeq1i  15648  sumeq1d  15651  sumz  15672  fsumadd  15690  fsum2d  15721  fsumrev2  15732  fsummulc2  15734  fsumconst  15740  modfsummods  15743  modfsummod  15744  fsumabs  15751  fsumrelem  15757  fsumrlim  15761  fsumo1  15762  fsumiun  15771  sumeven  16335  sumodd  16336  bitsinv2  16389  bitsf1ocnv  16390  bitsinv  16394  prmreclem5  16860  gsumfsum  21324  fsumcn  24739  ovolfiniun  25381  volfiniun  25427  itgfsum  25707  dvmptfsum  25858  pntrsumbnd2  27451  finsumvtxdg2size  29312  esumpcvgval  33606  esumcvg  33614  rrnval  37206  mccl  44867  dvmptfprod  45214  dvnprodlem1  45215  dvnprodlem2  45216  dvnprodlem3  45217  dvnprod  45218  sge0rnn0  45637  sge00  45645  fsumlesge0  45646  sge0sn  45648  sge0cl  45650  sge0f1o  45651  sge0resplit  45675  sge0xaddlem1  45702  sge0xaddlem2  45703  sge0reuz  45716