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Theorem nfcprod1 15252
Description: Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypothesis
Ref Expression
nfcprod1.1 𝑘𝐴
Assertion
Ref Expression
nfcprod1 𝑘𝑘𝐴 𝐵
Distinct variable group:   𝐴,𝑘
Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem nfcprod1
Dummy variables 𝑓 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prod 15248 . 2 𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
2 nfcv 2974 . . . . 5 𝑘
3 nfcprod1.1 . . . . . . 7 𝑘𝐴
4 nfcv 2974 . . . . . . 7 𝑘(ℤ𝑚)
53, 4nfss 3957 . . . . . 6 𝑘 𝐴 ⊆ (ℤ𝑚)
6 nfv 1906 . . . . . . . . 9 𝑘 𝑦 ≠ 0
7 nfcv 2974 . . . . . . . . . . 11 𝑘𝑛
8 nfcv 2974 . . . . . . . . . . 11 𝑘 ·
9 nfmpt1 5155 . . . . . . . . . . 11 𝑘(𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
107, 8, 9nfseq 13367 . . . . . . . . . 10 𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
11 nfcv 2974 . . . . . . . . . 10 𝑘
12 nfcv 2974 . . . . . . . . . 10 𝑘𝑦
1310, 11, 12nfbr 5104 . . . . . . . . 9 𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦
146, 13nfan 1891 . . . . . . . 8 𝑘(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)
1514nfex 2334 . . . . . . 7 𝑘𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)
164, 15nfrex 3306 . . . . . 6 𝑘𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)
17 nfcv 2974 . . . . . . . 8 𝑘𝑚
1817, 8, 9nfseq 13367 . . . . . . 7 𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
19 nfcv 2974 . . . . . . 7 𝑘𝑥
2018, 11, 19nfbr 5104 . . . . . 6 𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥
215, 16, 20nf3an 1893 . . . . 5 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)
222, 21nfrex 3306 . . . 4 𝑘𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)
23 nfcv 2974 . . . . 5 𝑘
24 nfcv 2974 . . . . . . . 8 𝑘𝑓
25 nfcv 2974 . . . . . . . 8 𝑘(1...𝑚)
2624, 25, 3nff1o 6606 . . . . . . 7 𝑘 𝑓:(1...𝑚)–1-1-onto𝐴
27 nfcv 2974 . . . . . . . . . 10 𝑘1
28 nfcsb1v 3904 . . . . . . . . . . 11 𝑘(𝑓𝑛) / 𝑘𝐵
2923, 28nfmpt 5154 . . . . . . . . . 10 𝑘(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
3027, 8, 29nfseq 13367 . . . . . . . . 9 𝑘seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
3130, 17nffv 6673 . . . . . . . 8 𝑘(seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3231nfeq2 2992 . . . . . . 7 𝑘 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3326, 32nfan 1891 . . . . . 6 𝑘(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3433nfex 2334 . . . . 5 𝑘𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3523, 34nfrex 3306 . . . 4 𝑘𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3622, 35nfor 1896 . . 3 𝑘(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
3736nfiotaw 6311 . 2 𝑘(℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
381, 37nfcxfr 2972 1 𝑘𝑘𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 396  wo 841  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wnfc 2958  wne 3013  wrex 3136  csb 3880  wss 3933  ifcif 4463   class class class wbr 5057  cmpt 5137  cio 6305  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526   · cmul 10530  cn 11626  cz 11969  cuz 12231  ...cfz 12880  seqcseq 13357  cli 14829  cprod 15247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-seq 13358  df-prod 15248
This theorem is referenced by:  fprodcn  41757  dvmptfprod  42106  vonicc  42844
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