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Theorem nfsum1 15048
 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 15045 . 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))))
2 nfcv 2982 . . . . 5 𝑘
3 nfsum1.1 . . . . . . 7 𝑘𝐴
4 nfcv 2982 . . . . . . 7 𝑘(ℤ𝑚)
53, 4nfss 3945 . . . . . 6 𝑘 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2982 . . . . . . . 8 𝑘𝑚
7 nfcv 2982 . . . . . . . 8 𝑘 +
83nfcri 2969 . . . . . . . . . 10 𝑘 𝑛𝐴
9 nfcsb1v 3890 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐵
10 nfcv 2982 . . . . . . . . . 10 𝑘0
118, 9, 10nfif 4479 . . . . . . . . 9 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
122, 11nfmpt 5150 . . . . . . . 8 𝑘(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
136, 7, 12nfseq 13385 . . . . . . 7 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
14 nfcv 2982 . . . . . . 7 𝑘
15 nfcv 2982 . . . . . . 7 𝑘𝑥
1613, 14, 15nfbr 5100 . . . . . 6 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥
175, 16nfan 1901 . . . . 5 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
182, 17nfrex 3301 . . . 4 𝑘𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
19 nfcv 2982 . . . . 5 𝑘
20 nfcv 2982 . . . . . . . 8 𝑘𝑓
21 nfcv 2982 . . . . . . . 8 𝑘(1...𝑚)
2220, 21, 3nff1o 6606 . . . . . . 7 𝑘 𝑓:(1...𝑚)–1-1-onto𝐴
23 nfcv 2982 . . . . . . . . . 10 𝑘1
24 nfcsb1v 3890 . . . . . . . . . . 11 𝑘(𝑓𝑛) / 𝑘𝐵
2519, 24nfmpt 5150 . . . . . . . . . 10 𝑘(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
2623, 7, 25nfseq 13385 . . . . . . . . 9 𝑘seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
2726, 6nffv 6673 . . . . . . . 8 𝑘(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
2827nfeq2 2999 . . . . . . 7 𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
2922, 28nfan 1901 . . . . . 6 𝑘(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3029nfex 2345 . . . . 5 𝑘𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3119, 30nfrex 3301 . . . 4 𝑘𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3218, 31nfor 1906 . . 3 𝑘(∃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)))
3332nfiotaw 6308 . 2 𝑘(℩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))))
341, 33nfcxfr 2980 1 𝑘Σ𝑘𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2115  Ⅎwnfc 2962  ∃wrex 3134  ⦋csb 3866   ⊆ wss 3919  ifcif 4450   class class class wbr 5053   ↦ cmpt 5133  ℩cio 6302  –1-1-onto→wf1o 6344  ‘cfv 6345  (class class class)co 7151  0cc0 10537  1c1 10538   + caddc 10540  ℕcn 11636  ℤcz 11980  ℤ≥cuz 12242  ...cfz 12896  seqcseq 13375   ⇝ cli 14843  Σcsu 15044 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-mpt 5134  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-seq 13376  df-sum 15045 This theorem is referenced by:  dvmptfprod  42540  dvnprodlem1  42541  fourierdlem112  42813  etransclem32  42861  sge0reuz  43039
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