Theorem sumeq1 11124

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 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3120	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐵 ⊆ (ℤ≥‘𝑚)))
2		eleq2 2203	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑗 ∈ 𝐴 ↔ 𝑗 ∈ 𝐵))
3	2	dcbid 823	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑗 ∈ 𝐵))
4	3	ralbidv 2437	. . . . . 6 ⊢ (𝐴 = 𝐵 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐵))
5		simpl 108	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ) → 𝐴 = 𝐵)
6	5	eleq2d 2209	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐵))
7	6	ifbid 3493	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ) → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0) = if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))
8	7	mpteq2dva 4018	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0)))
9	8	seqeq3d 10226	. . . . . . 7 ⊢ (𝐴 = 𝐵 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))))
10	9	breq1d 3939	. . . . . 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))
11	1, 4, 10	3anbi123d 1290	. . . . 5 ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ↔ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
12	11	rexbidv 2438	. . . 4 ⊢ (𝐴 = 𝐵 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
13		f1oeq3 5358	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑚)–1-1-onto→𝐵))
14	13	anbi1d 460	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))))
15	14	exbidv 1797	. . . . 5 ⊢ (𝐴 = 𝐵 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))))
16	15	rexbidv 2438	. . . 4 ⊢ (𝐴 = 𝐵 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))))
17	12, 16	orbi12d 782	. . 3 ⊢ (𝐴 = 𝐵 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ 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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐵 ∧ 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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚)))))
18	17	iotabidv 5109	. 2 ⊢ (𝐴 = 𝐵 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ 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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐵 ∧ 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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚)))))
19		df-sumdc 11123	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ 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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))))
20		df-sumdc 11123	. 2 ⊢ Σ𝑘 ∈ 𝐵 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐵 ∧ 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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))))
21	18, 19, 20	3eqtr4g 2197	1 ⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∀wral 2416  ∃wrex 2417  ⦋csb 3003   ⊆ wss 3071  ifcif 3474   class class class wbr 3929   ↦ cmpt 3989  ℩cio 5086  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774  0cc0 7620  1c1 7621   + caddc 7623   ≤ cle 7801  ℕcn 8720  ℤcz 9054  ℤ_≥cuz 9326  ...cfz 9790  seqcseq 10218   ⇝ cli 11047  Σcsu 11122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-seqfrec 10219  df-sumdc 11123

This theorem is referenced by:  sumeq1i  11132  sumeq1d  11135  isumz  11158  fsumadd  11175  fsum2d  11204  fisumrev2  11215  fsummulc2  11217  fsumconst  11223  modfsummod  11227  fsumabs  11234  fsumrelem  11240  fsumiun  11246  fsumcncntop  12725