Theorem nfcprod1 11807

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 11804	. 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 2347	. . . . 5 ⊢ Ⅎ𝑘ℤ
3		nfcprod1.1	. . . . . . . 8 ⊢ Ⅎ𝑘𝐴
4		nfcv 2347	. . . . . . . 8 ⊢ Ⅎ𝑘(ℤ≥‘𝑚)
5	3, 4	nfss 3185	. . . . . . 7 ⊢ Ⅎ𝑘 𝐴 ⊆ (ℤ≥‘𝑚)
6	3	nfcri 2341	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝐴
7	6	nfdc 1681	. . . . . . . 8 ⊢ Ⅎ𝑘DECID 𝑗 ∈ 𝐴
8	4, 7	nfralxy 2543	. . . . . . 7 ⊢ Ⅎ𝑘∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
9	5, 8	nfan 1587	. . . . . 6 ⊢ Ⅎ𝑘(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
10		nfv 1550	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑦 # 0
11		nfcv 2347	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑛
12		nfcv 2347	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 ·
13		nfmpt1 4136	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
14	11, 12, 13	nfseq 10600	. . . . . . . . . . 11 ⊢ Ⅎ𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
15		nfcv 2347	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 ⇝
16		nfcv 2347	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑦
17	14, 15, 16	nfbr 4089	. . . . . . . . . 10 ⊢ Ⅎ𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
18	10, 17	nfan 1587	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
19	18	nfex 1659	. . . . . . . 8 ⊢ Ⅎ𝑘∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
20	4, 19	nfrexw 2544	. . . . . . 7 ⊢ Ⅎ𝑘∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
21		nfcv 2347	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑚
22	21, 12, 13	nfseq 10600	. . . . . . . 8 ⊢ Ⅎ𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
23		nfcv 2347	. . . . . . . 8 ⊢ Ⅎ𝑘𝑥
24	22, 15, 23	nfbr 4089	. . . . . . 7 ⊢ Ⅎ𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥
25	20, 24	nfan 1587	. . . . . 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 1587	. . . . 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	nfrexw 2544	. . . 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 2347	. . . . 5 ⊢ Ⅎ𝑘ℕ
29		nfcv 2347	. . . . . . . 8 ⊢ Ⅎ𝑘𝑓
30		nfcv 2347	. . . . . . . 8 ⊢ Ⅎ𝑘(1...𝑚)
31	29, 30, 3	nff1o 5519	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓:(1...𝑚)–1-1-onto→𝐴
32		nfcv 2347	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
33		nfv 1550	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 𝑛 ≤ 𝑚
34		nfcsb1v 3125	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
35	33, 34, 32	nfif 3598	. . . . . . . . . . 11 ⊢ Ⅎ𝑘if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)
36	28, 35	nfmpt 4135	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))
37	32, 12, 36	nfseq 10600	. . . . . . . . 9 ⊢ Ⅎ𝑘seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))
38	37, 21	nffv 5585	. . . . . . . 8 ⊢ Ⅎ𝑘(seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
39	38	nfeq2 2359	. . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
40	31, 39	nfan 1587	. . . . . 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 1659	. . . . 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	nfrexw 2544	. . . 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 1596	. . 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 5235	. 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 2344	1 ⊢ Ⅎ𝑘∏𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1372  ∃wex 1514   ∈ wcel 2175  Ⅎwnfc 2334  ∀wral 2483  ∃wrex 2484  ⦋csb 3092   ⊆ wss 3165  ifcif 3570   class class class wbr 4043   ↦ cmpt 4104  ℩cio 5229  –1-1-onto→wf1o 5269  ‘cfv 5270  (class class class)co 5943  0cc0 7924  1c1 7925   · cmul 7929   ≤ cle 8107   # cap 8653  ℕcn 9035  ℤcz 9371  ℤ_≥cuz 9647  ...cfz 10129  seqcseq 10590   ⇝ cli 11531  ∏cprod 11803

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-if 3571  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-recs 6390  df-frec 6476  df-seqfrec 10591  df-proddc 11804

This theorem is referenced by: (None)