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Theorem sumeq2ii 14631
 Description: Equality theorem for sum, with the class expressions 𝐵 and 𝐶 guarded by I to be always sets. (Contributed by Mario Carneiro, 13-Jun-2019.)
Assertion
Ref Expression
sumeq2ii (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
Distinct variable group:   𝐴,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem sumeq2ii
Dummy variables 𝑓 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 471 . . . . . . . 8 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
2 simpr 471 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) ∧ 𝑛𝐴) → 𝑛𝐴)
3 simplll 758 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) ∧ 𝑛𝐴) → ∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶))
4 nfcv 2913 . . . . . . . . . . . . . . . . 17 𝑘 I
5 nfcsb1v 3698 . . . . . . . . . . . . . . . . 17 𝑘𝑛 / 𝑘𝐵
64, 5nffv 6339 . . . . . . . . . . . . . . . 16 𝑘( I ‘𝑛 / 𝑘𝐵)
7 nfcsb1v 3698 . . . . . . . . . . . . . . . . 17 𝑘𝑛 / 𝑘𝐶
84, 7nffv 6339 . . . . . . . . . . . . . . . 16 𝑘( I ‘𝑛 / 𝑘𝐶)
96, 8nfeq 2925 . . . . . . . . . . . . . . 15 𝑘( I ‘𝑛 / 𝑘𝐵) = ( I ‘𝑛 / 𝑘𝐶)
10 csbeq1a 3691 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
1110fveq2d 6336 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ( I ‘𝐵) = ( I ‘𝑛 / 𝑘𝐵))
12 csbeq1a 3691 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛𝐶 = 𝑛 / 𝑘𝐶)
1312fveq2d 6336 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ( I ‘𝐶) = ( I ‘𝑛 / 𝑘𝐶))
1411, 13eqeq12d 2786 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (( I ‘𝐵) = ( I ‘𝐶) ↔ ( I ‘𝑛 / 𝑘𝐵) = ( I ‘𝑛 / 𝑘𝐶)))
159, 14rspc 3454 . . . . . . . . . . . . . 14 (𝑛𝐴 → (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ( I ‘𝑛 / 𝑘𝐵) = ( I ‘𝑛 / 𝑘𝐶)))
162, 3, 15sylc 65 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) ∧ 𝑛𝐴) → ( I ‘𝑛 / 𝑘𝐵) = ( I ‘𝑛 / 𝑘𝐶))
1716ifeq1da 4255 . . . . . . . . . . . 12 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) → if(𝑛𝐴, ( I ‘𝑛 / 𝑘𝐵), ( I ‘0)) = if(𝑛𝐴, ( I ‘𝑛 / 𝑘𝐶), ( I ‘0)))
18 fvif 6345 . . . . . . . . . . . 12 ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = if(𝑛𝐴, ( I ‘𝑛 / 𝑘𝐵), ( I ‘0))
19 fvif 6345 . . . . . . . . . . . 12 ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)) = if(𝑛𝐴, ( I ‘𝑛 / 𝑘𝐶), ( I ‘0))
2017, 18, 193eqtr4g 2830 . . . . . . . . . . 11 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) → ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
2120mpteq2dv 4879 . . . . . . . . . 10 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) → (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))))
2221fveq1d 6334 . . . . . . . . 9 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) → ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))‘𝑥) = ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))‘𝑥))
23 eqid 2771 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
24 eqid 2771 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
2523, 24fvmptex 6436 . . . . . . . . 9 ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑥) = ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))‘𝑥)
26 eqid 2771 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))
27 eqid 2771 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) = (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
2826, 27fvmptex 6436 . . . . . . . . 9 ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))‘𝑥) = ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))‘𝑥)
2922, 25, 283eqtr4g 2830 . . . . . . . 8 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) → ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑥) = ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))‘𝑥))
301, 29seqfeq 13033 . . . . . . 7 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))))
3130breq1d 4796 . . . . . 6 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥))
3231anbi2d 614 . . . . 5 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥)))
3332rexbidva 3197 . . . 4 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥)))
34 simplr 752 . . . . . . . . . 10 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ)
35 nnuz 11925 . . . . . . . . . 10 ℕ = (ℤ‘1)
3634, 35syl6eleq 2860 . . . . . . . . 9 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ (ℤ‘1))
37 f1of 6278 . . . . . . . . . . . . . 14 (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)⟶𝐴)
3837ad2antlr 706 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑓:(1...𝑚)⟶𝐴)
39 ffvelrn 6500 . . . . . . . . . . . . 13 ((𝑓:(1...𝑚)⟶𝐴𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
4038, 39sylancom 576 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
41 simplll 758 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶))
42 nfcsb1v 3698 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵)
43 nfcsb1v 3698 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐶)
4442, 43nfeq 2925 . . . . . . . . . . . . 13 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)
45 csbeq1a 3691 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐵))
46 csbeq1a 3691 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐶) = (𝑓𝑥) / 𝑘( I ‘𝐶))
4745, 46eqeq12d 2786 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)))
4844, 47rspc 3454 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ 𝐴 → (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)))
4940, 41, 48sylc 65 . . . . . . . . . . 11 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶))
50 fvex 6342 . . . . . . . . . . . 12 (𝑓𝑥) ∈ V
51 csbfv2g 6373 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ V → (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵))
5250, 51ax-mp 5 . . . . . . . . . . 11 (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵)
53 csbfv2g 6373 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ V → (𝑓𝑥) / 𝑘( I ‘𝐶) = ( I ‘(𝑓𝑥) / 𝑘𝐶))
5450, 53ax-mp 5 . . . . . . . . . . 11 (𝑓𝑥) / 𝑘( I ‘𝐶) = ( I ‘(𝑓𝑥) / 𝑘𝐶)
5549, 52, 543eqtr3g 2828 . . . . . . . . . 10 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ( I ‘(𝑓𝑥) / 𝑘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐶))
56 elfznn 12577 . . . . . . . . . . . 12 (𝑥 ∈ (1...𝑚) → 𝑥 ∈ ℕ)
5756adantl 467 . . . . . . . . . . 11 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑥 ∈ ℕ)
58 fveq2 6332 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → (𝑓𝑛) = (𝑓𝑥))
5958csbeq1d 3689 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑥) / 𝑘𝐵)
60 eqid 2771 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
6159, 60fvmpti 6423 . . . . . . . . . . 11 (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)‘𝑥) = ( I ‘(𝑓𝑥) / 𝑘𝐵))
6257, 61syl 17 . . . . . . . . . 10 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)‘𝑥) = ( I ‘(𝑓𝑥) / 𝑘𝐵))
6358csbeq1d 3689 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐶 = (𝑓𝑥) / 𝑘𝐶)
64 eqid 2771 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)
6563, 64fvmpti 6423 . . . . . . . . . . 11 (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)‘𝑥) = ( I ‘(𝑓𝑥) / 𝑘𝐶))
6657, 65syl 17 . . . . . . . . . 10 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)‘𝑥) = ( I ‘(𝑓𝑥) / 𝑘𝐶))
6755, 62, 663eqtr4d 2815 . . . . . . . . 9 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)‘𝑥) = ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)‘𝑥))
6836, 67seqfveq 13032 . . . . . . . 8 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))
6968eqeq2d 2781 . . . . . . 7 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)))
7069pm5.32da 568 . . . . . 6 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
7170exbidv 2002 . . . . 5 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
7271rexbidva 3197 . . . 4 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
7333, 72orbi12d 902 . . 3 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)))))
7473iotabidv 6015 . 2 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)))))
75 df-sum 14625 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
76 df-sum 14625 . 2 Σ𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
7774, 75, 763eqtr4g 2830 1 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   ∨ wo 834   = wceq 1631  ∃wex 1852   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  Vcvv 3351  ⦋csb 3682   ⊆ wss 3723  ifcif 4225   class class class wbr 4786   ↦ cmpt 4863   I cid 5156  ℩cio 5992  ⟶wf 6027  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6793  0cc0 10138  1c1 10139   + caddc 10141  ℕcn 11222  ℤcz 11579  ℤ≥cuz 11888  ...cfz 12533  seqcseq 13008   ⇝ cli 14423  Σcsu 14624 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-seq 13009  df-sum 14625 This theorem is referenced by:  sumeq2  14632  sum2id  14647
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