Theorem nfcprod 15882

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 15877	. 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 2892	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfcprod.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3942	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑧 ≠ 0
7		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
8		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 ·
9	3	nfcri 2884	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 𝑘 ∈ 𝐴
10		nfcprod.2	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐵
11		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥1
12	9, 10, 11	nfif 4522	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥if(𝑘 ∈ 𝐴, 𝐵, 1)
13	2, 12	nfmpt 5208	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
14	7, 8, 13	nfseq 13983	. . . . . . . . . 10 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
15		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑥 ⇝
16		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
17	14, 15, 16	nfbr 5157	. . . . . . . . 9 ⊢ Ⅎ𝑥seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧
18	6, 17	nfan 1899	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
19	18	nfex 2323	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
20	4, 19	nfrexw 3289	. . . . . 6 ⊢ Ⅎ𝑥∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑧(𝑧 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑧)
21		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
22	21, 8, 13	nfseq 13983	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)))
23		nfcv 2892	. . . . . . 7 ⊢ Ⅎ𝑥𝑦
24	22, 15, 23	nfbr 5157	. . . . . 6 ⊢ Ⅎ𝑥seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦
25	5, 20, 24	nf3an 1901	. . . . 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 3289	. . . 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 2892	. . . . 5 ⊢ Ⅎ𝑥ℕ
28		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
29		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
30	28, 29, 3	nff1o 6801	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
31		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑓‘𝑛)
32	31, 10	nfcsbw 3891	. . . . . . . . . . 11 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
33	27, 32	nfmpt 5208	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
34	11, 8, 33	nfseq 13983	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))
35	34, 21	nffv 6871	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
36	35	nfeq2 2910	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)
37	30, 36	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑥(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
38	37	nfex 2323	. . . . 5 ⊢ Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))
39	27, 38	nfrexw 3289	. . . 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 1904	. . 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 6471	. 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 2890	1 ⊢ Ⅎ𝑥∏𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Ⅎwnfc 2877   ≠ wne 2926  ∃wrex 3054  ⦋csb 3865   ⊆ wss 3917  ifcif 4491   class class class wbr 5110   ↦ cmpt 5191  ℩cio 6465  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390  0cc0 11075  1c1 11076   · cmul 11080  ℕcn 12193  ℤcz 12536  ℤ_≥cuz 12800  ...cfz 13475  seqcseq 13973   ⇝ cli 15457  ∏cprod 15876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-seq 13974  df-prod 15877

This theorem is referenced by:  fprod2dlem  15953  fprodcom2  15957  fprodcn  45605  fprodcncf  45905