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Theorem sumeq2ii 14271
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 476 . . . . . . . 8 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
2 simpr 476 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) ∧ 𝑛𝐴) → 𝑛𝐴)
3 simplll 794 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) ∧ 𝑛𝐴) → ∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶))
4 nfcv 2751 . . . . . . . . . . . . . . . . 17 𝑘 I
5 nfcsb1v 3515 . . . . . . . . . . . . . . . . 17 𝑘𝑛 / 𝑘𝐵
64, 5nffv 6110 . . . . . . . . . . . . . . . 16 𝑘( I ‘𝑛 / 𝑘𝐵)
7 nfcsb1v 3515 . . . . . . . . . . . . . . . . 17 𝑘𝑛 / 𝑘𝐶
84, 7nffv 6110 . . . . . . . . . . . . . . . 16 𝑘( I ‘𝑛 / 𝑘𝐶)
96, 8nfeq 2762 . . . . . . . . . . . . . . 15 𝑘( I ‘𝑛 / 𝑘𝐵) = ( I ‘𝑛 / 𝑘𝐶)
10 csbeq1a 3508 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
1110fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ( I ‘𝐵) = ( I ‘𝑛 / 𝑘𝐵))
12 csbeq1a 3508 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑛𝐶 = 𝑛 / 𝑘𝐶)
1312fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑛 → ( I ‘𝐶) = ( I ‘𝑛 / 𝑘𝐶))
1411, 13eqeq12d 2625 . . . . . . . . . . . . . . 15 (𝑘 = 𝑛 → (( I ‘𝐵) = ( I ‘𝐶) ↔ ( I ‘𝑛 / 𝑘𝐵) = ( I ‘𝑛 / 𝑘𝐶)))
159, 14rspc 3276 . . . . . . . . . . . . . 14 (𝑛𝐴 → (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ( I ‘𝑛 / 𝑘𝐵) = ( I ‘𝑛 / 𝑘𝐶)))
162, 3, 15sylc 63 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) ∧ 𝑛𝐴) → ( I ‘𝑛 / 𝑘𝐵) = ( I ‘𝑛 / 𝑘𝐶))
1716ifeq1da 4066 . . . . . . . . . . . 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 6114 . . . . . . . . . . . 12 ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = if(𝑛𝐴, ( I ‘𝑛 / 𝑘𝐵), ( I ‘0))
19 fvif 6114 . . . . . . . . . . . 12 ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)) = if(𝑛𝐴, ( I ‘𝑛 / 𝑘𝐶), ( I ‘0))
2017, 18, 193eqtr4g 2669 . . . . . . . . . . 11 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) → ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
2120mpteq2dv 4673 . . . . . . . . . 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 6105 . . . . . . . . 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 2610 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
24 eqid 2610 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
2523, 24fvmptex 6203 . . . . . . . . 9 ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))‘𝑥) = ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))‘𝑥)
26 eqid 2610 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))
27 eqid 2610 . . . . . . . . . 10 (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) = (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
2826, 27fvmptex 6203 . . . . . . . . 9 ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))‘𝑥) = ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))‘𝑥)
2922, 25, 283eqtr4g 2669 . . . . . . . 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 12688 . . . . . . 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 4593 . . . . . 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 736 . . . . 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 3031 . . . 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 788 . . . . . . . . . 10 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ)
35 nnuz 11599 . . . . . . . . . 10 ℕ = (ℤ‘1)
3634, 35syl6eleq 2698 . . . . . . . . 9 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ (ℤ‘1))
37 f1of 6050 . . . . . . . . . . . . . 14 (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)⟶𝐴)
3837ad2antlr 759 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑓:(1...𝑚)⟶𝐴)
39 ffvelrn 6265 . . . . . . . . . . . . 13 ((𝑓:(1...𝑚)⟶𝐴𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
4038, 39sylancom 698 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
41 simplll 794 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶))
42 nfcsb1v 3515 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵)
43 nfcsb1v 3515 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐶)
4442, 43nfeq 2762 . . . . . . . . . . . . 13 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)
45 csbeq1a 3508 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐵))
46 csbeq1a 3508 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐶) = (𝑓𝑥) / 𝑘( I ‘𝐶))
4745, 46eqeq12d 2625 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)))
4844, 47rspc 3276 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ 𝐴 → (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)))
4940, 41, 48sylc 63 . . . . . . . . . . 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 6113 . . . . . . . . . . . 12 (𝑓𝑥) ∈ V
51 csbfv2g 6142 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ V → (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵))
5250, 51ax-mp 5 . . . . . . . . . . 11 (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵)
53 csbfv2g 6142 . . . . . . . . . . . 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 2667 . . . . . . . . . 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 12241 . . . . . . . . . . . 12 (𝑥 ∈ (1...𝑚) → 𝑥 ∈ ℕ)
5756adantl 481 . . . . . . . . . . 11 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑥 ∈ ℕ)
58 fveq2 6103 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → (𝑓𝑛) = (𝑓𝑥))
5958csbeq1d 3506 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑥) / 𝑘𝐵)
60 eqid 2610 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
6159, 60fvmpti 6190 . . . . . . . . . . 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 3506 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐶 = (𝑓𝑥) / 𝑘𝐶)
64 eqid 2610 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)
6563, 64fvmpti 6190 . . . . . . . . . . 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 2654 . . . . . . . . 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 12687 . . . . . . . 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 2620 . . . . . . 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 671 . . . . . 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 1837 . . . . 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 3031 . . . 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 742 . . 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 5789 . 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 14265 . 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 14265 . 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 2669 1 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1475  wex 1695  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  csb 3499  wss 3540  ifcif 4036   class class class wbr 4583  cmpt 4643   I cid 4948  cio 5766  wf 5800  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  cn 10897  cz 11254  cuz 11563  ...cfz 12197  seqcseq 12663  cli 14063  Σcsu 14264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-sum 14265
This theorem is referenced by:  sumeq2  14272  sum2id  14286
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