Theorem nfcprod 15857

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 𝑚 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prod 15852	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑦(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
2		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfcprod.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3974	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ≠ 0
7		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
8		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ·
9	3	nfcri 2890	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑘 ∈ 𝐴
10		nfcprod.2	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐵
11		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥1
12	9, 10, 11	nfif 4558	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥if(𝑘 ∈ 𝐴, 𝐵, 1)
13	2, 12	nfmpt 5255	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
14	7, 8, 13	nfseq 13978	. . . . . . . . . 10 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
15		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑥 ⇝
16		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
17	14, 15, 16	nfbr 5195	. . . . . . . . 9 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧
18	6, 17	nfan 1902	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
19	18	nfex 2317	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
20	4, 19	nfrexw 3310	. . . . . 6 ⊢ Ⅎ𝑥∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
21		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
22	21, 8, 13	nfseq 13978	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
23		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
24	22, 15, 23	nfbr 5195	. . . . . 6 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
25	5, 20, 24	nf3an 1904	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
26	2, 25	nfrexw 3310	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦)
27		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑥ℕ
28		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
29		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
30	28, 29, 3	nff1o 6831	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
31		nfcv 2903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑓‘𝑛)
32	31, 10	nfcsbw 3920	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
33	27, 32	nfmpt 5255	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
34	11, 8, 33	nfseq 13978	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
35	34, 21	nffv 6901	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
36	35	nfeq2 2920	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
37	30, 36	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑥(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
38	37	nfex 2317	. . . . 5 ⊢ Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
39	27, 38	nfrexw 3310	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
40	26, 39	nfor 1907	. . 3 ⊢ Ⅎ𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))
41	40	nfiotaw 6499	. 2 ⊢ Ⅎ𝑥(℩𝑦(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
42	1, 41	nfcxfr 2901	1 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Ⅎwnfc 2883   ≠ wne 2940  ∃wrex 3070  ⦋csb 3893   ⊆ wss 3948  ifcif 4528   class class class wbr 5148   ↦ cmpt 5231  ℩cio 6493  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7411  0cc0 11112  1c1 11113   · cmul 11117  ℕcn 12214  ℤcz 12560  ℤ_≥cuz 12824  ...cfz 13486  seqcseq 13968   ⇝ cli 15430  ∏cprod 15851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-seq 13969  df-prod 15852

This theorem is referenced by:  fprod2dlem  15926  fprodcom2  15930  fprodcn  44395  fprodcncf  44695