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Theorem nfsum1 10964
 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 𝑗 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sumdc 10962 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
2 nfcv 2240 . . . . 5 𝑘
3 nfsum1.1 . . . . . . 7 𝑘𝐴
4 nfcv 2240 . . . . . . 7 𝑘(ℤ𝑚)
53, 4nfss 3040 . . . . . 6 𝑘 𝐴 ⊆ (ℤ𝑚)
63nfcri 2234 . . . . . . . 8 𝑘 𝑗𝐴
76nfdc 1605 . . . . . . 7 𝑘DECID 𝑗𝐴
84, 7nfralxy 2430 . . . . . 6 𝑘𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴
9 nfcv 2240 . . . . . . . 8 𝑘𝑚
10 nfcv 2240 . . . . . . . 8 𝑘 +
113nfcri 2234 . . . . . . . . . 10 𝑘 𝑛𝐴
12 nfcsb1v 2985 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐵
13 nfcv 2240 . . . . . . . . . 10 𝑘0
1411, 12, 13nfif 3447 . . . . . . . . 9 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
152, 14nfmpt 3960 . . . . . . . 8 𝑘(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
169, 10, 15nfseq 10069 . . . . . . 7 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
17 nfcv 2240 . . . . . . 7 𝑘
18 nfcv 2240 . . . . . . 7 𝑘𝑥
1916, 17, 18nfbr 3919 . . . . . 6 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥
205, 8, 19nf3an 1513 . . . . 5 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
212, 20nfrexya 2433 . . . 4 𝑘𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
22 nfcv 2240 . . . . 5 𝑘
23 nfcv 2240 . . . . . . . 8 𝑘𝑓
24 nfcv 2240 . . . . . . . 8 𝑘(1...𝑚)
2523, 24, 3nff1o 5299 . . . . . . 7 𝑘 𝑓:(1...𝑚)–1-1-onto𝐴
26 nfcv 2240 . . . . . . . . . 10 𝑘1
27 nfv 1476 . . . . . . . . . . . 12 𝑘 𝑛𝑚
28 nfcsb1v 2985 . . . . . . . . . . . 12 𝑘(𝑓𝑛) / 𝑘𝐵
2927, 28, 13nfif 3447 . . . . . . . . . . 11 𝑘if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)
3022, 29nfmpt 3960 . . . . . . . . . 10 𝑘(𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
3126, 10, 30nfseq 10069 . . . . . . . . 9 𝑘seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))
3231, 9nffv 5363 . . . . . . . 8 𝑘(seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)
3332nfeq2 2252 . . . . . . 7 𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)
3425, 33nfan 1512 . . . . . 6 𝑘(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
3534nfex 1584 . . . . 5 𝑘𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
3622, 35nfrexya 2433 . . . 4 𝑘𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
3721, 36nfor 1521 . . 3 𝑘(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
3837nfiotaxy 5028 . 2 𝑘(℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
391, 38nfcxfr 2237 1 𝑘Σ𝑘𝐴 𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ∨ wo 670  DECID wdc 786   ∧ w3a 930   = wceq 1299  ∃wex 1436   ∈ wcel 1448  Ⅎwnfc 2227  ∀wral 2375  ∃wrex 2376  ⦋csb 2955   ⊆ wss 3021  ifcif 3421   class class class wbr 3875   ↦ cmpt 3929  ℩cio 5022  –1-1-onto→wf1o 5058  ‘cfv 5059  (class class class)co 5706  0cc0 7500  1c1 7501   + caddc 7503   ≤ cle 7673  ℕcn 8578  ℤcz 8906  ℤ≥cuz 9176  ...cfz 9631  seqcseq 10059   ⇝ cli 10886  Σcsu 10961 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082 This theorem depends on definitions:  df-bi 116  df-dc 787  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-recs 6132  df-frec 6218  df-seqfrec 10060  df-sumdc 10962 This theorem is referenced by:  mertenslem2  11144
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