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Theorem nfcprod1 15941
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 15937 . 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 2903 . . . . 5 𝑘
3 nfcprod1.1 . . . . . . 7 𝑘𝐴
4 nfcv 2903 . . . . . . 7 𝑘(ℤ𝑚)
53, 4nfss 3988 . . . . . 6 𝑘 𝐴 ⊆ (ℤ𝑚)
6 nfv 1912 . . . . . . . . 9 𝑘 𝑦 ≠ 0
7 nfcv 2903 . . . . . . . . . . 11 𝑘𝑛
8 nfcv 2903 . . . . . . . . . . 11 𝑘 ·
9 nfmpt1 5256 . . . . . . . . . . 11 𝑘(𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
107, 8, 9nfseq 14049 . . . . . . . . . 10 𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
11 nfcv 2903 . . . . . . . . . 10 𝑘
12 nfcv 2903 . . . . . . . . . 10 𝑘𝑦
1310, 11, 12nfbr 5195 . . . . . . . . 9 𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦
146, 13nfan 1897 . . . . . . . 8 𝑘(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)
1514nfex 2323 . . . . . . 7 𝑘𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)
164, 15nfrexw 3311 . . . . . 6 𝑘𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)
17 nfcv 2903 . . . . . . . 8 𝑘𝑚
1817, 8, 9nfseq 14049 . . . . . . 7 𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
19 nfcv 2903 . . . . . . 7 𝑘𝑥
2018, 11, 19nfbr 5195 . . . . . 6 𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥
215, 16, 20nf3an 1899 . . . . 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, 21nfrexw 3311 . . . 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 2903 . . . . 5 𝑘
24 nfcv 2903 . . . . . . . 8 𝑘𝑓
25 nfcv 2903 . . . . . . . 8 𝑘(1...𝑚)
2624, 25, 3nff1o 6847 . . . . . . 7 𝑘 𝑓:(1...𝑚)–1-1-onto𝐴
27 nfcv 2903 . . . . . . . . . 10 𝑘1
28 nfcsb1v 3933 . . . . . . . . . . 11 𝑘(𝑓𝑛) / 𝑘𝐵
2923, 28nfmpt 5255 . . . . . . . . . 10 𝑘(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
3027, 8, 29nfseq 14049 . . . . . . . . 9 𝑘seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
3130, 17nffv 6917 . . . . . . . 8 𝑘(seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3231nfeq2 2921 . . . . . . 7 𝑘 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3326, 32nfan 1897 . . . . . 6 𝑘(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3433nfex 2323 . . . . 5 𝑘𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3523, 34nfrexw 3311 . . . 4 𝑘𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3622, 35nfor 1902 . . 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 6520 . 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 2901 1 𝑘𝑘𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395  wo 847  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wnfc 2888  wne 2938  wrex 3068  csb 3908  wss 3963  ifcif 4531   class class class wbr 5148  cmpt 5231  cio 6514  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   · cmul 11158  cn 12264  cz 12611  cuz 12876  ...cfz 13544  seqcseq 14039  cli 15517  cprod 15936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-seq 14040  df-prod 15937
This theorem is referenced by:  fprodcn  45556  dvmptfprod  45901  vonicc  46641
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