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Theorem nfsum1 10106
Description: Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Hypothesis
Ref Expression
nfsum1.1 𝑘𝐴
Assertion
Ref Expression
nfsum1 𝑘Σ𝑘𝐴 𝐵

Proof of Theorem nfsum1
Dummy variables 𝑓 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 10104 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))))
2 nfcv 2194 . . . . 5 𝑘
3 nfsum1.1 . . . . . . 7 𝑘𝐴
4 nfcv 2194 . . . . . . 7 𝑘(ℤ𝑚)
53, 4nfss 2966 . . . . . 6 𝑘 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2194 . . . . . . . 8 𝑘𝑚
7 nfcv 2194 . . . . . . . 8 𝑘 +
83nfcri 2188 . . . . . . . . . 10 𝑘 𝑛𝐴
9 nfcsb1v 2910 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐵
10 nfcv 2194 . . . . . . . . . 10 𝑘0
118, 9, 10nfif 3384 . . . . . . . . 9 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
122, 11nfmpt 3877 . . . . . . . 8 𝑘(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
13 nfcv 2194 . . . . . . . 8 𝑘
146, 7, 12, 13nfiseq 9382 . . . . . . 7 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ)
15 nfcv 2194 . . . . . . 7 𝑘
16 nfcv 2194 . . . . . . 7 𝑘𝑥
1714, 15, 16nfbr 3836 . . . . . 6 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥
185, 17nfan 1473 . . . . 5 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥)
192, 18nfrexya 2380 . . . 4 𝑘𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥)
20 nfcv 2194 . . . . 5 𝑘
21 nfcv 2194 . . . . . . . 8 𝑘𝑓
22 nfcv 2194 . . . . . . . 8 𝑘(1...𝑚)
2321, 22, 3nff1o 5152 . . . . . . 7 𝑘 𝑓:(1...𝑚)–1-1-onto𝐴
24 nfcv 2194 . . . . . . . . . 10 𝑘1
25 nfcsb1v 2910 . . . . . . . . . . 11 𝑘(𝑓𝑛) / 𝑘𝐵
2620, 25nfmpt 3877 . . . . . . . . . 10 𝑘(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
2724, 7, 26, 13nfiseq 9382 . . . . . . . . 9 𝑘seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)
2827, 6nffv 5213 . . . . . . . 8 𝑘(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)
2928nfeq2 2205 . . . . . . 7 𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)
3023, 29nfan 1473 . . . . . 6 𝑘(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
3130nfex 1544 . . . . 5 𝑘𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
3220, 31nfrexya 2380 . . . 4 𝑘𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
3319, 32nfor 1482 . . 3 𝑘(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)))
3433nfiotaxy 4899 . 2 𝑘(℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))))
351, 34nfcxfr 2191 1 𝑘Σ𝑘𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wa 101  wo 639   = wceq 1259  wex 1397  wcel 1409  wnfc 2181  wrex 2324  csb 2880  wss 2945  ifcif 3359   class class class wbr 3792  cmpt 3846  cio 4893  1-1-ontowf1o 4929  cfv 4930  (class class class)co 5540  cc 6945  0cc0 6947  1c1 6948   + caddc 6950  cn 7990  cz 8302  cuz 8569  ...cfz 8976  seqcseq 9375  cli 10030  Σcsu 10103
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-un 2950  df-in 2952  df-ss 2959  df-if 3360  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-recs 5951  df-frec 6009  df-iseq 9376  df-sum 10104
This theorem is referenced by: (None)
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