Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfcprod Structured version   Visualization version   GIF version

Theorem nfcprod 15261
 Description: Bound-variable hypothesis builder for product: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in ∏𝑘 ∈ 𝐴𝐵. (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 15256 . 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 2958 . . . . 5 𝑥
3 nfcprod.1 . . . . . . 7 𝑥𝐴
4 nfcv 2958 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 3910 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
6 nfv 1915 . . . . . . . . 9 𝑥 𝑧 ≠ 0
7 nfcv 2958 . . . . . . . . . . 11 𝑥𝑛
8 nfcv 2958 . . . . . . . . . . 11 𝑥 ·
93nfcri 2946 . . . . . . . . . . . . 13 𝑥 𝑘𝐴
10 nfcprod.2 . . . . . . . . . . . . 13 𝑥𝐵
11 nfcv 2958 . . . . . . . . . . . . 13 𝑥1
129, 10, 11nfif 4457 . . . . . . . . . . . 12 𝑥if(𝑘𝐴, 𝐵, 1)
132, 12nfmpt 5130 . . . . . . . . . . 11 𝑥(𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
147, 8, 13nfseq 13378 . . . . . . . . . 10 𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
15 nfcv 2958 . . . . . . . . . 10 𝑥
16 nfcv 2958 . . . . . . . . . 10 𝑥𝑧
1714, 15, 16nfbr 5080 . . . . . . . . 9 𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧
186, 17nfan 1900 . . . . . . . 8 𝑥(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧)
1918nfex 2335 . . . . . . 7 𝑥𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧)
204, 19nfrex 3271 . . . . . 6 𝑥𝑛 ∈ (ℤ𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑧)
21 nfcv 2958 . . . . . . . 8 𝑥𝑚
2221, 8, 13nfseq 13378 . . . . . . 7 𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)))
23 nfcv 2958 . . . . . . 7 𝑥𝑦
2422, 15, 23nfbr 5080 . . . . . 6 𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦
255, 20, 24nf3an 1902 . . . . 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 3271 . . . 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 2958 . . . . 5 𝑥
28 nfcv 2958 . . . . . . . 8 𝑥𝑓
29 nfcv 2958 . . . . . . . 8 𝑥(1...𝑚)
3028, 29, 3nff1o 6592 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
31 nfcv 2958 . . . . . . . . . . . 12 𝑥(𝑓𝑛)
3231, 10nfcsbw 3857 . . . . . . . . . . 11 𝑥(𝑓𝑛) / 𝑘𝐵
3327, 32nfmpt 5130 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
3411, 8, 33nfseq 13378 . . . . . . . . 9 𝑥seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
3534, 21nffv 6659 . . . . . . . 8 𝑥(seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3635nfeq2 2975 . . . . . . 7 𝑥 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3730, 36nfan 1900 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3837nfex 2335 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3927, 38nfrex 3271 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
4026, 39nfor 1905 . . 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 6291 . 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 2956 1 𝑥𝑘𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2112  Ⅎwnfc 2939   ≠ wne 2990  ∃wrex 3110  ⦋csb 3831   ⊆ wss 3884  ifcif 4428   class class class wbr 5033   ↦ cmpt 5113  ℩cio 6285  –1-1-onto→wf1o 6327  ‘cfv 6328  (class class class)co 7139  0cc0 10530  1c1 10531   · cmul 10535  ℕcn 11629  ℤcz 11973  ℤ≥cuz 12235  ...cfz 12889  seqcseq 13368   ⇝ cli 14837  ∏cprod 15255 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seq 13369  df-prod 15256 This theorem is referenced by:  fprod2dlem  15330  fprodcom2  15334  fprodcn  42235  fprodcncf  42535
 Copyright terms: Public domain W3C validator