Theorem nfcprod 15959

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	⊢ Ⅎ𝑥𝐴
nfcprod.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfcprod	⊢ Ⅎ𝑥∏𝑘 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		df-prod 15954	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑦(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
2		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfcprod.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3930	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfv 1944	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ≠ 0
7		nfcv 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
8		nfcv 2925	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ·
9	3	nfcri 2917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑘 ∈ 𝐴
10		nfcprod.2	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐵
11		nfcv 2925	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥1
12	9, 10, 11	nfif 4518	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥if(𝑘 ∈ 𝐴, 𝐵, 1)
13	2, 12	nfmpt 5209	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
14	7, 8, 13	nfseq 14043	. . . . . . . . . 10 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
15		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥 ⇝
16		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
17	14, 15, 16	nfbr 5158	. . . . . . . . 9 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧
18	6, 17	nfan 1929	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
19	18	nfex 2357	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
20	4, 19	nfrexw 3313	. . . . . 6 ⊢ Ⅎ𝑥∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
21		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
22	21, 8, 13	nfseq 14043	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
23		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
24	22, 15, 23	nfbr 5158	. . . . . 6 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
25	5, 20, 24	nf3an 1931	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
26	2, 25	nfrexw 3313	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
27		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥ℕ
28		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
29		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
30	28, 29, 3	nff1o 6818	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
31		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑓‘𝑛)
32	31, 10	nfcsbw 3879	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
33	27, 32	nfmpt 5209	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
34	11, 8, 33	nfseq 14043	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
35	34, 21	nffv 6891	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
36	35	nfeq2 2942	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
37	30, 36	nfan 1929	. . . . . 6 ⊢ Ⅎ𝑥(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
38	37	nfex 2357	. . . . 5 ⊢ Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
39	27, 38	nfrexw 3313	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
40	26, 39	nfor 1934	. . 3 ⊢ Ⅎ𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
41	40	nfiotaw 6496	. 2 ⊢ Ⅎ𝑥(℩𝑦(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
42	1, 41	nfcxfr 2923	1 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Ⅎwnfc 2910   ≠ wne 2958  ∃wrex 3089  ⦋csb 3853   ⊆ wss 3905  ifcif 4487   class class class wbr 5109   ↦ cmpt 5192  ℩cio 6490  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  0cc0 11095  1c1 11096   · cmul 11100  ℕcn 12228  ℤcz 12586  ℤ_≥cuz 12857  ...cfz 13530  seqcseq 14033   ⇝ cli 15531  ∏cprod 15953

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-seq 14034  df-prod 15954

This theorem is referenced by:  fprod2dlem  16030  fprodcom2  16034  fprodcn  46316  fprodcncf  46614