Theorem nfcprod 11566

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 11562	. 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 2319	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfcprod.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
4		nfcv 2319	. . . . . . . 8 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3150	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6	3	nfcri 2313	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑗 ∈ 𝐴
7	6	nfdc 1659	. . . . . . . 8 ⊢ Ⅎ𝑥DECID 𝑗 ∈ 𝐴
8	4, 7	nfralxy 2515	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
9	5, 8	nfan 1565	. . . . . 6 ⊢ Ⅎ𝑥(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
10		nfv 1528	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑧 # 0
11		nfcv 2319	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑛
12		nfcv 2319	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 ·
13	3	nfcri 2313	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 𝑘 ∈ 𝐴
14		nfcprod.2	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐵
15		nfcv 2319	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥1
16	13, 14, 15	nfif 3564	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥if(𝑘 ∈ 𝐴, 𝐵, 1)
17	2, 16	nfmpt 4097	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
18	11, 12, 17	nfseq 10458	. . . . . . . . . . 11 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
19		nfcv 2319	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ⇝
20		nfcv 2319	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑧
21	18, 19, 20	nfbr 4051	. . . . . . . . . 10 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧
22	10, 21	nfan 1565	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
23	22	nfex 1637	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑧(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
24	4, 23	nfrexxy 2516	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
25		nfcv 2319	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑚
26	25, 12, 17	nfseq 10458	. . . . . . . 8 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
27		nfcv 2319	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
28	26, 19, 27	nfbr 4051	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
29	24, 28	nfan 1565	. . . . . 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 1565	. . . . 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 2516	. . . 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 2319	. . . . 5 ⊢ Ⅎ𝑥ℕ
33		nfcv 2319	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
34		nfcv 2319	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
35	33, 34, 3	nff1o 5461	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
36		nfv 1528	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑛 ≤ 𝑚
37		nfcv 2319	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑓‘𝑛)
38	37, 14	nfcsb 3096	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
39	36, 38, 15	nfif 3564	. . . . . . . . . . 11 ⊢ Ⅎ𝑥if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)
40	32, 39	nfmpt 4097	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))
41	15, 12, 40	nfseq 10458	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))
42	41, 25	nffv 5527	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
43	42	nfeq2 2331	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
44	35, 43	nfan 1565	. . . . . 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 1637	. . . . 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 2516	. . . 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 1574	. . 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 5184	. 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 2316	1 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104   ∨ wo 708  DECID wdc 834   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Ⅎwnfc 2306  ∀wral 2455  ∃wrex 2456  ⦋csb 3059   ⊆ wss 3131  ifcif 3536   class class class wbr 4005   ↦ cmpt 4066  ℩cio 5178  –1-1-onto→wf1o 5217  ‘cfv 5218  (class class class)co 5878  0cc0 7814  1c1 7815   · cmul 7819   ≤ cle 7996   # cap 8541  ℕcn 8922  ℤcz 9256  ℤ_≥cuz 9531  ...cfz 10011  seqcseq 10448   ⇝ cli 11289  ∏cprod 11561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-if 3537  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5881  df-oprab 5882  df-mpo 5883  df-recs 6309  df-frec 6395  df-seqfrec 10449  df-proddc 11562

This theorem is referenced by:  fprod2dlemstep  11633  fprodcom2fi  11637