Theorem nfcprod 11941

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 11937	. 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 2349	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfcprod.1	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
4		nfcv 2349	. . . . . . . 8 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3190	. . . . . . 7 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6	3	nfcri 2343	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑗 ∈ 𝐴
7	6	nfdc 1683	. . . . . . . 8 ⊢ Ⅎ𝑥DECID 𝑗 ∈ 𝐴
8	4, 7	nfralxy 2545	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
9	5, 8	nfan 1589	. . . . . 6 ⊢ Ⅎ𝑥(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
10		nfv 1552	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑧 # 0
11		nfcv 2349	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥𝑛
12		nfcv 2349	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 ·
13	3	nfcri 2343	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥 𝑘 ∈ 𝐴
14		nfcprod.2	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥𝐵
15		nfcv 2349	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥1
16	13, 14, 15	nfif 3604	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥if(𝑘 ∈ 𝐴, 𝐵, 1)
17	2, 16	nfmpt 4144	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
18	11, 12, 17	nfseq 10624	. . . . . . . . . . 11 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
19		nfcv 2349	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ⇝
20		nfcv 2349	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑧
21	18, 19, 20	nfbr 4098	. . . . . . . . . 10 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧
22	10, 21	nfan 1589	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
23	22	nfex 1661	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑧(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
24	4, 23	nfrexw 2546	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
25		nfcv 2349	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑚
26	25, 12, 17	nfseq 10624	. . . . . . . 8 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
27		nfcv 2349	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
28	26, 19, 27	nfbr 4098	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
29	24, 28	nfan 1589	. . . . . 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 1589	. . . . 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	nfrexw 2546	. . . 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 2349	. . . . 5 ⊢ Ⅎ𝑥ℕ
33		nfcv 2349	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
34		nfcv 2349	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
35	33, 34, 3	nff1o 5532	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
36		nfv 1552	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑛 ≤ 𝑚
37		nfcv 2349	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑓‘𝑛)
38	37, 14	nfcsb 3135	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
39	36, 38, 15	nfif 3604	. . . . . . . . . . 11 ⊢ Ⅎ𝑥if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)
40	32, 39	nfmpt 4144	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))
41	15, 12, 40	nfseq 10624	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))
42	41, 25	nffv 5599	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
43	42	nfeq2 2361	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
44	35, 43	nfan 1589	. . . . . 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 1661	. . . . 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	nfrexw 2546	. . . 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 1598	. . 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 5245	. 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 2346	1 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104   ∨ wo 710  DECID wdc 836   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Ⅎwnfc 2336  ∀wral 2485  ∃wrex 2486  ⦋csb 3097   ⊆ wss 3170  ifcif 3575   class class class wbr 4051   ↦ cmpt 4113  ℩cio 5239  –1-1-onto→wf1o 5279  ‘cfv 5280  (class class class)co 5957  0cc0 7945  1c1 7946   · cmul 7950   ≤ cle 8128   # cap 8674  ℕcn 9056  ℤcz 9392  ℤ_≥cuz 9668  ...cfz 10150  seqcseq 10614   ⇝ cli 11664  ∏cprod 11936

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-if 3576  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-recs 6404  df-frec 6490  df-seqfrec 10615  df-proddc 11937

This theorem is referenced by:  fprod2dlemstep  12008  fprodcom2fi  12012