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Theorem nfsum 14541
 Description: Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Hypotheses
Ref Expression
nfsum.1 𝑥𝐴
nfsum.2 𝑥𝐵
Assertion
Ref Expression
nfsum 𝑥Σ𝑘𝐴 𝐵

Proof of Theorem nfsum
Dummy variables 𝑓 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 14537 . 2 Σ𝑘𝐴 𝐵 = (℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
2 nfcv 2866 . . . . 5 𝑥
3 nfsum.1 . . . . . . 7 𝑥𝐴
4 nfcv 2866 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 3702 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2866 . . . . . . . 8 𝑥𝑚
7 nfcv 2866 . . . . . . . 8 𝑥 +
83nfcri 2860 . . . . . . . . . 10 𝑥 𝑛𝐴
9 nfcv 2866 . . . . . . . . . . 11 𝑥𝑛
10 nfsum.2 . . . . . . . . . . 11 𝑥𝐵
119, 10nfcsb 3657 . . . . . . . . . 10 𝑥𝑛 / 𝑘𝐵
12 nfcv 2866 . . . . . . . . . 10 𝑥0
138, 11, 12nfif 4223 . . . . . . . . 9 𝑥if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
142, 13nfmpt 4854 . . . . . . . 8 𝑥(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
156, 7, 14nfseq 12926 . . . . . . 7 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
16 nfcv 2866 . . . . . . 7 𝑥
17 nfcv 2866 . . . . . . 7 𝑥𝑧
1815, 16, 17nfbr 4807 . . . . . 6 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧
195, 18nfan 1941 . . . . 5 𝑥(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
202, 19nfrex 3109 . . . 4 𝑥𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
21 nfcv 2866 . . . . 5 𝑥
22 nfcv 2866 . . . . . . . 8 𝑥𝑓
23 nfcv 2866 . . . . . . . 8 𝑥(1...𝑚)
2422, 23, 3nff1o 6248 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
25 nfcv 2866 . . . . . . . . . 10 𝑥1
26 nfcv 2866 . . . . . . . . . . . 12 𝑥(𝑓𝑛)
2726, 10nfcsb 3657 . . . . . . . . . . 11 𝑥(𝑓𝑛) / 𝑘𝐵
2821, 27nfmpt 4854 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
2925, 7, 28nfseq 12926 . . . . . . . . 9 𝑥seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
3029, 6nffv 6311 . . . . . . . 8 𝑥(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3130nfeq2 2882 . . . . . . 7 𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3224, 31nfan 1941 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3332nfex 2265 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3421, 33nfrex 3109 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3520, 34nfor 1947 . . 3 𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
3635nfiota 5968 . 2 𝑥(℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
371, 36nfcxfr 2864 1 𝑥Σ𝑘𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 382   ∧ wa 383   = wceq 1596  ∃wex 1817   ∈ wcel 2103  Ⅎwnfc 2853  ∃wrex 3015  ⦋csb 3639   ⊆ wss 3680  ifcif 4194   class class class wbr 4760   ↦ cmpt 4837  ℩cio 5962  –1-1-onto→wf1o 6000  ‘cfv 6001  (class class class)co 6765  0cc0 10049  1c1 10050   + caddc 10052  ℕcn 11133  ℤcz 11490  ℤ≥cuz 11800  ...cfz 12440  seqcseq 12916   ⇝ cli 14335  Σcsu 14536 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-seq 12917  df-sum 14537 This theorem is referenced by:  fsum2dlem  14621  fsumcom2  14625  fsumcom2OLD  14626  fsumrlim  14663  fsumiun  14673  fsumcn  22795  fsum2cn  22796  nfitg1  23660  nfitg  23661  dvmptfsum  23858  fsumdvdscom  25031  binomcxplemdvsum  38973  binomcxplemnotnn0  38974  fsumcnf  39596  fsumiunss  40227  dvmptfprod  40580  sge0iunmptlemre  41052
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