Theorem sumeq1 11520

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 3206	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐵 ⊆ (ℤ≥‘𝑚)))
2		eleq2 2260	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑗 ∈ 𝐴 ↔ 𝑗 ∈ 𝐵))
3	2	dcbid 839	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑗 ∈ 𝐵))
4	3	ralbidv 2497	. . . . . 6 ⊢ (𝐴 = 𝐵 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐵))
5		simpl 109	. . . . . . . . . . 11 ⊢ ((𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ) → 𝐴 = 𝐵)
6	5	eleq2d 2266	. . . . . . . . . 10 ⊢ ((𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ) → (𝑛 ∈ 𝐴 ↔ 𝑛 ∈ 𝐵))
7	6	ifbid 3582	. . . . . . . . 9 ⊢ ((𝐴 = 𝐵 ∧ 𝑛 ∈ ℤ) → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0) = if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0))
8	7	mpteq2dva 4123	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐵, ⦋𝑛 / 𝑘⦌𝐶, 0)))
9	8	seqeq3d 10547	. . . . . . 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 4043	. . . . . 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 1323	. . . . 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 2498	. . . 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 5494	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑚)–1-1-onto→𝐵))
14	13	anbi1d 465	. . . . . 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 1839	. . . . 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 2498	. . . 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 794	. . 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 5241	. 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 11519	. 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 11519	. 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 2254	1 ⊢ (𝐴 = 𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   ∧ w3a 980   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  ⦋csb 3084   ⊆ wss 3157  ifcif 3561   class class class wbr 4033   ↦ cmpt 4094  ℩cio 5217  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922  0cc0 7879  1c1 7880   + caddc 7882   ≤ cle 8062  ℕcn 8990  ℤcz 9326  ℤ_≥cuz 9601  ...cfz 10083  seqcseq 10539   ⇝ cli 11443  Σcsu 11518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-cnv 4671  df-dm 4673  df-rn 4674  df-res 4675  df-iota 5219  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-recs 6363  df-frec 6449  df-seqfrec 10540  df-sumdc 11519

This theorem is referenced by:  sumeq1i  11528  sumeq1d  11531  isumz  11554  fsumadd  11571  fsum2d  11600  fisumrev2  11611  fsummulc2  11613  fsumconst  11619  modfsummod  11623  fsumabs  11630  fsumrelem  11636  fsumiun  11642  fsumcncntop  14803  dvmptfsum  14961