Theorem nfcprod1 11495

Description: Bound-variable hypothesis builder for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypothesis

Ref	Expression
nfcprod1.1	⊢ Ⅎ𝑘𝐴

Assertion

Ref	Expression
nfcprod1	⊢ Ⅎ𝑘∏𝑘 ∈ 𝐴 𝐵

Distinct variable group:   𝐴,𝑘

Allowed substitution hint:   𝐵(𝑘)

Proof of Theorem nfcprod1

Dummy variables 𝑓 𝑗 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-proddc 11492	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))
2		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑘ℤ
3		nfcprod1.1	. . . . . . . 8 ⊢ Ⅎ𝑘𝐴
4		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑘(ℤ≥‘𝑚)
5	3, 4	nfss 3135	. . . . . . 7 ⊢ Ⅎ𝑘 𝐴 ⊆ (ℤ≥‘𝑚)
6	3	nfcri 2302	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝐴
7	6	nfdc 1647	. . . . . . . 8 ⊢ Ⅎ𝑘DECID 𝑗 ∈ 𝐴
8	4, 7	nfralxy 2504	. . . . . . 7 ⊢ Ⅎ𝑘∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
9	5, 8	nfan 1553	. . . . . 6 ⊢ Ⅎ𝑘(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
10		nfv 1516	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑦 # 0
11		nfcv 2308	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑛
12		nfcv 2308	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 ·
13		nfmpt1 4075	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
14	11, 12, 13	nfseq 10390	. . . . . . . . . . 11 ⊢ Ⅎ𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
15		nfcv 2308	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 ⇝
16		nfcv 2308	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑦
17	14, 15, 16	nfbr 4028	. . . . . . . . . 10 ⊢ Ⅎ𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
18	10, 17	nfan 1553	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
19	18	nfex 1625	. . . . . . . 8 ⊢ Ⅎ𝑘∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
20	4, 19	nfrexxy 2505	. . . . . . 7 ⊢ Ⅎ𝑘∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
21		nfcv 2308	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑚
22	21, 12, 13	nfseq 10390	. . . . . . . 8 ⊢ Ⅎ𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
23		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑘𝑥
24	22, 15, 23	nfbr 4028	. . . . . . 7 ⊢ Ⅎ𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥
25	20, 24	nfan 1553	. . . . . 6 ⊢ Ⅎ𝑘(∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)
26	9, 25	nfan 1553	. . . . 5 ⊢ Ⅎ𝑘((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
27	2, 26	nfrexxy 2505	. . . 4 ⊢ Ⅎ𝑘∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥))
28		nfcv 2308	. . . . 5 ⊢ Ⅎ𝑘ℕ
29		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑘𝑓
30		nfcv 2308	. . . . . . . 8 ⊢ Ⅎ𝑘(1...𝑚)
31	29, 30, 3	nff1o 5430	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓:(1...𝑚)–1-1-onto→𝐴
32		nfcv 2308	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
33		nfv 1516	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 𝑛 ≤ 𝑚
34		nfcsb1v 3078	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
35	33, 34, 32	nfif 3548	. . . . . . . . . . 11 ⊢ Ⅎ𝑘if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)
36	28, 35	nfmpt 4074	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))
37	32, 12, 36	nfseq 10390	. . . . . . . . 9 ⊢ Ⅎ𝑘seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))
38	37, 21	nffv 5496	. . . . . . . 8 ⊢ Ⅎ𝑘(seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
39	38	nfeq2 2320	. . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
40	31, 39	nfan 1553	. . . . . 6 ⊢ Ⅎ𝑘(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
41	40	nfex 1625	. . . . 5 ⊢ Ⅎ𝑘∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
42	28, 41	nfrexxy 2505	. . . 4 ⊢ Ⅎ𝑘∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))
43	27, 42	nfor 1562	. . 3 ⊢ Ⅎ𝑘(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))
44	43	nfiotaw 5157	. 2 ⊢ Ⅎ𝑘(℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))
45	1, 44	nfcxfr 2305	1 ⊢ Ⅎ𝑘∏𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 103   ∨ wo 698  DECID wdc 824   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Ⅎwnfc 2295  ∀wral 2444  ∃wrex 2445  ⦋csb 3045   ⊆ wss 3116  ifcif 3520   class class class wbr 3982   ↦ cmpt 4043  ℩cio 5151  –1-1-onto→wf1o 5187  ‘cfv 5188  (class class class)co 5842  0cc0 7753  1c1 7754   · cmul 7758   ≤ cle 7934   # cap 8479  ℕcn 8857  ℤcz 9191  ℤ_≥cuz 9466  ...cfz 9944  seqcseq 10380   ⇝ cli 11219  ∏cprod 11491

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-recs 6273  df-frec 6359  df-seqfrec 10381  df-proddc 11492

This theorem is referenced by: (None)