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Theorem nfcprod 15253
Description: Bound-variable hypothesis builder for product: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in 𝑘𝐴𝐵. (Contributed by Scott Fenton, 1-Dec-2017.)
Hypotheses
Ref Expression
nfcprod.1 𝑥𝐴
nfcprod.2 𝑥𝐵
Assertion
Ref Expression
nfcprod 𝑥𝑘𝐴 𝐵
Distinct variable group:   𝑥,𝑘
Allowed substitution hints:   𝐴(𝑥,𝑘)   𝐵(𝑥,𝑘)

Proof of Theorem nfcprod
Dummy variables 𝑓 𝑚 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prod 15248 . 2 𝑘𝐴 𝐵 = (℩𝑦(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
2 nfcv 2974 . . . . 5 𝑥
3 nfcprod.1 . . . . . . 7 𝑥𝐴
4 nfcv 2974 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 3957 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
6 nfv 1906 . . . . . . . . 9 𝑥 𝑧 ≠ 0
7 nfcv 2974 . . . . . . . . . . 11 𝑥𝑛
8 nfcv 2974 . . . . . . . . . . 11 𝑥 ·
93nfcri 2968 . . . . . . . . . . . . 13 𝑥 𝑘𝐴
10 nfcprod.2 . . . . . . . . . . . . 13 𝑥𝐵
11 nfcv 2974 . . . . . . . . . . . . 13 𝑥1
129, 10, 11nfif 4492 . . . . . . . . . . . 12 𝑥if(𝑘𝐴, 𝐵, 1)
132, 12nfmpt 5154 . . . . . . . . . . 11 𝑥(𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
147, 8, 13nfseq 13367 . . . . . . . . . 10 𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
15 nfcv 2974 . . . . . . . . . 10 𝑥
16 nfcv 2974 . . . . . . . . . 10 𝑥𝑧
1714, 15, 16nfbr 5104 . . . . . . . . 9 𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧
186, 17nfan 1891 . . . . . . . 8 𝑥(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧)
1918nfex 2334 . . . . . . 7 𝑥𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧)
204, 19nfrex 3306 . . . . . 6 𝑥𝑛 ∈ (ℤ𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧)
21 nfcv 2974 . . . . . . . 8 𝑥𝑚
2221, 8, 13nfseq 13367 . . . . . . 7 𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
23 nfcv 2974 . . . . . . 7 𝑥𝑦
2422, 15, 23nfbr 5104 . . . . . 6 𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦
255, 20, 24nf3an 1893 . . . . 5 𝑥(𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)
262, 25nfrex 3306 . . . 4 𝑥𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦)
27 nfcv 2974 . . . . 5 𝑥
28 nfcv 2974 . . . . . . . 8 𝑥𝑓
29 nfcv 2974 . . . . . . . 8 𝑥(1...𝑚)
3028, 29, 3nff1o 6606 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
31 nfcv 2974 . . . . . . . . . . . 12 𝑥(𝑓𝑛)
3231, 10nfcsbw 3906 . . . . . . . . . . 11 𝑥(𝑓𝑛) / 𝑘𝐵
3327, 32nfmpt 5154 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
3411, 8, 33nfseq 13367 . . . . . . . . 9 𝑥seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
3534, 21nffv 6673 . . . . . . . 8 𝑥(seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3635nfeq2 2992 . . . . . . 7 𝑥 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3730, 36nfan 1891 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3837nfex 2334 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3927, 38nfrex 3306 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
4026, 39nfor 1896 . . 3 𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
4140nfiotaw 6311 . 2 𝑥(℩𝑦(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
421, 41nfcxfr 2972 1 𝑥𝑘𝐴 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:  fprod2dlem  15322  fprodcom2  15326  fprodcn  41757  fprodcncf  42060
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