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Theorem nfcprod 15794
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 15789 . 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 2907 . . . . 5 𝑥
3 nfcprod.1 . . . . . . 7 𝑥𝐴
4 nfcv 2907 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 3936 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
6 nfv 1917 . . . . . . . . 9 𝑥 𝑧 ≠ 0
7 nfcv 2907 . . . . . . . . . . 11 𝑥𝑛
8 nfcv 2907 . . . . . . . . . . 11 𝑥 ·
93nfcri 2894 . . . . . . . . . . . . 13 𝑥 𝑘𝐴
10 nfcprod.2 . . . . . . . . . . . . 13 𝑥𝐵
11 nfcv 2907 . . . . . . . . . . . . 13 𝑥1
129, 10, 11nfif 4516 . . . . . . . . . . . 12 𝑥if(𝑘𝐴, 𝐵, 1)
132, 12nfmpt 5212 . . . . . . . . . . 11 𝑥(𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
147, 8, 13nfseq 13916 . . . . . . . . . 10 𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
15 nfcv 2907 . . . . . . . . . 10 𝑥
16 nfcv 2907 . . . . . . . . . 10 𝑥𝑧
1714, 15, 16nfbr 5152 . . . . . . . . 9 𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧
186, 17nfan 1902 . . . . . . . 8 𝑥(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧)
1918nfex 2317 . . . . . . 7 𝑥𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧)
204, 19nfrexw 3296 . . . . . 6 𝑥𝑛 ∈ (ℤ𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧)
21 nfcv 2907 . . . . . . . 8 𝑥𝑚
2221, 8, 13nfseq 13916 . . . . . . 7 𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
23 nfcv 2907 . . . . . . 7 𝑥𝑦
2422, 15, 23nfbr 5152 . . . . . 6 𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦
255, 20, 24nf3an 1904 . . . . 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, 25nfrexw 3296 . . . 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 2907 . . . . 5 𝑥
28 nfcv 2907 . . . . . . . 8 𝑥𝑓
29 nfcv 2907 . . . . . . . 8 𝑥(1...𝑚)
3028, 29, 3nff1o 6782 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
31 nfcv 2907 . . . . . . . . . . . 12 𝑥(𝑓𝑛)
3231, 10nfcsbw 3882 . . . . . . . . . . 11 𝑥(𝑓𝑛) / 𝑘𝐵
3327, 32nfmpt 5212 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
3411, 8, 33nfseq 13916 . . . . . . . . 9 𝑥seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
3534, 21nffv 6852 . . . . . . . 8 𝑥(seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3635nfeq2 2924 . . . . . . 7 𝑥 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3730, 36nfan 1902 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3837nfex 2317 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3927, 38nfrexw 3296 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
4026, 39nfor 1907 . . 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 6452 . 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 2905 1 𝑥𝑘𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 396  wo 845  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wnfc 2887  wne 2943  wrex 3073  csb 3855  wss 3910  ifcif 4486   class class class wbr 5105  cmpt 5188  cio 6446  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   · cmul 11056  cn 12153  cz 12499  cuz 12763  ...cfz 13424  seqcseq 13906  cli 15366  cprod 15788
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-seq 13907  df-prod 15789
This theorem is referenced by:  fprod2dlem  15863  fprodcom2  15867  fprodcn  43831  fprodcncf  44131
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