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Theorem nfsum1 9875
 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 9873 . 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 2178 . . . . 5 𝑘
3 nfsum1.1 . . . . . . 7 𝑘𝐴
4 nfcv 2178 . . . . . . 7 𝑘(ℤ𝑚)
53, 4nfss 2938 . . . . . 6 𝑘 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2178 . . . . . . . 8 𝑘𝑚
7 nfcv 2178 . . . . . . . 8 𝑘 +
83nfcri 2172 . . . . . . . . . 10 𝑘 𝑛𝐴
9 nfcsb1v 2882 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐵
10 nfcv 2178 . . . . . . . . . 10 𝑘0
118, 9, 10nfif 3356 . . . . . . . . 9 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
122, 11nfmpt 3849 . . . . . . . 8 𝑘(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
13 nfcv 2178 . . . . . . . 8 𝑘
146, 7, 12, 13nfiseq 9218 . . . . . . 7 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ)
15 nfcv 2178 . . . . . . 7 𝑘
16 nfcv 2178 . . . . . . 7 𝑘𝑥
1714, 15, 16nfbr 3808 . . . . . 6 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥
185, 17nfan 1457 . . . . 5 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥)
192, 18nfrexya 2363 . . . 4 𝑘𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥)
20 nfcv 2178 . . . . 5 𝑘
21 nfcv 2178 . . . . . . . 8 𝑘𝑓
22 nfcv 2178 . . . . . . . 8 𝑘(1...𝑚)
2321, 22, 3nff1o 5124 . . . . . . 7 𝑘 𝑓:(1...𝑚)–1-1-onto𝐴
24 nfcv 2178 . . . . . . . . . 10 𝑘1
25 nfcsb1v 2882 . . . . . . . . . . 11 𝑘(𝑓𝑛) / 𝑘𝐵
2620, 25nfmpt 3849 . . . . . . . . . 10 𝑘(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
2724, 7, 26, 13nfiseq 9218 . . . . . . . . 9 𝑘seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)
2827, 6nffv 5185 . . . . . . . 8 𝑘(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)
2928nfeq2 2189 . . . . . . 7 𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚)
3023, 29nfan 1457 . . . . . 6 𝑘(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
3130nfex 1528 . . . . 5 𝑘𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
3220, 31nfrexya 2363 . . . 4 𝑘𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵), ℂ)‘𝑚))
3319, 32nfor 1466 . . 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 4871 . 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 2175 1 𝑘Σ𝑘𝐴 𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ∨ wo 629   = wceq 1243  ∃wex 1381   ∈ wcel 1393  Ⅎwnfc 2165  ∃wrex 2307  ⦋csb 2852   ⊆ wss 2917  ifcif 3331   class class class wbr 3764   ↦ cmpt 3818  ℩cio 4865  –1-1-onto→wf1o 4901  ‘cfv 4902  (class class class)co 5512  ℂcc 6887  0cc0 6889  1c1 6890   + caddc 6892  ℕcn 7914  ℤcz 8245  ℤ≥cuz 8473  ...cfz 8874  seqcseq 9211   ⇝ cli 9799  Σcsu 9872 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-if 3332  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-frec 5978  df-iseq 9212  df-sum 9873 This theorem is referenced by: (None)
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