Theorem nfcprod1 12178

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 12175	. 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 2375	. . . . 5 ⊢ Ⅎ𝑘ℤ
3		nfcprod1.1	. . . . . . . 8 ⊢ Ⅎ𝑘𝐴
4		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑘(ℤ≥‘𝑚)
5	3, 4	nfss 3221	. . . . . . 7 ⊢ Ⅎ𝑘 𝐴 ⊆ (ℤ≥‘𝑚)
6	3	nfcri 2369	. . . . . . . . 9 ⊢ Ⅎ𝑘 𝑗 ∈ 𝐴
7	6	nfdc 1707	. . . . . . . 8 ⊢ Ⅎ𝑘DECID 𝑗 ∈ 𝐴
8	4, 7	nfralxy 2571	. . . . . . 7 ⊢ Ⅎ𝑘∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
9	5, 8	nfan 1614	. . . . . 6 ⊢ Ⅎ𝑘(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
10		nfv 1577	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑦 # 0
11		nfcv 2375	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘𝑛
12		nfcv 2375	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 ·
13		nfmpt1 4187	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘(𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
14	11, 12, 13	nfseq 10765	. . . . . . . . . . 11 ⊢ Ⅎ𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
15		nfcv 2375	. . . . . . . . . . 11 ⊢ Ⅎ𝑘 ⇝
16		nfcv 2375	. . . . . . . . . . 11 ⊢ Ⅎ𝑘𝑦
17	14, 15, 16	nfbr 4140	. . . . . . . . . 10 ⊢ Ⅎ𝑘seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
18	10, 17	nfan 1614	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
19	18	nfex 1686	. . . . . . . 8 ⊢ Ⅎ𝑘∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
20	4, 19	nfrexw 2572	. . . . . . 7 ⊢ Ⅎ𝑘∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
21		nfcv 2375	. . . . . . . . 9 ⊢ Ⅎ𝑘𝑚
22	21, 12, 13	nfseq 10765	. . . . . . . 8 ⊢ Ⅎ𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
23		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑘𝑥
24	22, 15, 23	nfbr 4140	. . . . . . 7 ⊢ Ⅎ𝑘seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥
25	20, 24	nfan 1614	. . . . . 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 1614	. . . . 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 2572	. . . 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 2375	. . . . 5 ⊢ Ⅎ𝑘ℕ
29		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑘𝑓
30		nfcv 2375	. . . . . . . 8 ⊢ Ⅎ𝑘(1...𝑚)
31	29, 30, 3	nff1o 5590	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓:(1...𝑚)–1-1-onto→𝐴
32		nfcv 2375	. . . . . . . . . 10 ⊢ Ⅎ𝑘1
33		nfv 1577	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘 𝑛 ≤ 𝑚
34		nfcsb1v 3161	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
35	33, 34, 32	nfif 3638	. . . . . . . . . . 11 ⊢ Ⅎ𝑘if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)
36	28, 35	nfmpt 4186	. . . . . . . . . 10 ⊢ Ⅎ𝑘(𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))
37	32, 12, 36	nfseq 10765	. . . . . . . . 9 ⊢ Ⅎ𝑘seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))
38	37, 21	nffv 5658	. . . . . . . 8 ⊢ Ⅎ𝑘(seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
39	38	nfeq2 2387	. . . . . . 7 ⊢ Ⅎ𝑘 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)
40	31, 39	nfan 1614	. . . . . 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 1686	. . . . 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 2572	. . . 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 1623	. . 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 5297	. 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 2372	1 ⊢ Ⅎ𝑘∏𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104   ∨ wo 716  DECID wdc 842   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Ⅎwnfc 2362  ∀wral 2511  ∃wrex 2512  ⦋csb 3128   ⊆ wss 3201  ifcif 3607   class class class wbr 4093   ↦ cmpt 4155  ℩cio 5291  –1-1-onto→wf1o 5332  ‘cfv 5333  (class class class)co 6028  0cc0 8075  1c1 8076   · cmul 8080   ≤ cle 8257   # cap 8803  ℕcn 9185  ℤcz 9523  ℤ_≥cuz 9799  ...cfz 10288  seqcseq 10755   ⇝ cli 11901  ∏cprod 12174

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-recs 6514  df-frec 6600  df-seqfrec 10756  df-proddc 12175

This theorem is referenced by: (None)