Theorem nfcprod 11518

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 𝑗 𝑚 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-proddc 11514	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑦(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))
2		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfcprod.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
4		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3140	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6	3	nfcri 2306	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑗 ∈ 𝐴
7	6	nfdc 1652	. . . . . . . 8 ⊢ Ⅎ𝑥DECID 𝑗 ∈ 𝐴
8	4, 7	nfralxy 2508	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
9	5, 8	nfan 1558	. . . . . 6 ⊢ Ⅎ𝑥(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
10		nfv 1521	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑧 # 0
11		nfcv 2312	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑛
12		nfcv 2312	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 ·
13	3	nfcri 2306	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 𝑘 ∈ 𝐴
14		nfcprod.2	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐵
15		nfcv 2312	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥1
16	13, 14, 15	nfif 3554	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥if(𝑘 ∈ 𝐴, 𝐵, 1)
17	2, 16	nfmpt 4081	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
18	11, 12, 17	nfseq 10411	. . . . . . . . . . 11 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
19		nfcv 2312	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ⇝
20		nfcv 2312	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑧
21	18, 19, 20	nfbr 4035	. . . . . . . . . 10 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧
22	10, 21	nfan 1558	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
23	22	nfex 1630	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑧(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
24	4, 23	nfrexxy 2509	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
25		nfcv 2312	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑚
26	25, 12, 17	nfseq 10411	. . . . . . . 8 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
27		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
28	26, 19, 27	nfbr 4035	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
29	24, 28	nfan 1558	. . . . . 6 ⊢ Ⅎ𝑥(∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
30	9, 29	nfan 1558	. . . . 5 ⊢ Ⅎ𝑥((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
31	2, 30	nfrexxy 2509	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦))
32		nfcv 2312	. . . . 5 ⊢ Ⅎ𝑥ℕ
33		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
34		nfcv 2312	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
35	33, 34, 3	nff1o 5440	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
36		nfv 1521	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑛 ≤ 𝑚
37		nfcv 2312	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑓‘𝑛)
38	37, 14	nfcsb 3086	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
39	36, 38, 15	nfif 3554	. . . . . . . . . . 11 ⊢ Ⅎ𝑥if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)
40	32, 39	nfmpt 4081	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))
41	15, 12, 40	nfseq 10411	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))
42	41, 25	nffv 5506	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
43	42	nfeq2 2324	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
44	35, 43	nfan 1558	. . . . . 6 ⊢ Ⅎ𝑥(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
45	44	nfex 1630	. . . . 5 ⊢ Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
46	32, 45	nfrexxy 2509	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
47	31, 46	nfor 1567	. . 3 ⊢ Ⅎ𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))
48	47	nfiotaw 5164	. 2 ⊢ Ⅎ𝑥(℩𝑦(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃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 ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))
49	1, 48	nfcxfr 2309	1 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 103   ∨ wo 703  DECID wdc 829   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Ⅎwnfc 2299  ∀wral 2448  ∃wrex 2449  ⦋csb 3049   ⊆ wss 3121  ifcif 3526   class class class wbr 3989   ↦ cmpt 4050  ℩cio 5158  –1-1-onto→wf1o 5197  ‘cfv 5198  (class class class)co 5853  0cc0 7774  1c1 7775   · cmul 7779   ≤ cle 7955   # cap 8500  ℕcn 8878  ℤcz 9212  ℤ_≥cuz 9487  ...cfz 9965  seqcseq 10401   ⇝ cli 11241  ∏cprod 11513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-if 3527  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-recs 6284  df-frec 6370  df-seqfrec 10402  df-proddc 11514

This theorem is referenced by:  fprod2dlemstep  11585  fprodcom2fi  11589