Theorem nfsum 12035

Description: Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)

Hypotheses

Ref	Expression
nfsum.1	⊢ Ⅎ𝑥𝐴
nfsum.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfsum	⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

Proof of Theorem nfsum

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

Step	Hyp	Ref	Expression
1		df-sumdc 12032	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))))
2		nfcv 2384	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfsum.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4		nfcv 2384	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3230	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6	3	nfcri 2378	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑗 ∈ 𝐴
7	6	nfdc 1707	. . . . . . 7 ⊢ Ⅎ𝑥DECID 𝑗 ∈ 𝐴
8	4, 7	nfralxy 2580	. . . . . 6 ⊢ Ⅎ𝑥∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
9		nfcv 2384	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
10		nfcv 2384	. . . . . . . 8 ⊢ Ⅎ𝑥 +
11	3	nfcri 2378	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑛 ∈ 𝐴
12		nfcv 2384	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
13		nfsum.2	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐵
14	12, 13	nfcsb 3175	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑛 / 𝑘⦌𝐵
15		nfcv 2384	. . . . . . . . . 10 ⊢ Ⅎ𝑥0
16	11, 14, 15	nfif 3650	. . . . . . . . 9 ⊢ Ⅎ𝑥if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
17	2, 16	nfmpt 4201	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
18	9, 10, 17	nfseq 10815	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
19		nfcv 2384	. . . . . . 7 ⊢ Ⅎ𝑥 ⇝
20		nfcv 2384	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
21	18, 19, 20	nfbr 4155	. . . . . 6 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧
22	5, 8, 21	nf3an 1615	. . . . 5 ⊢ Ⅎ𝑥(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧)
23	2, 22	nfrexw 2581	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧)
24		nfcv 2384	. . . . 5 ⊢ Ⅎ𝑥ℕ
25		nfcv 2384	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
26		nfcv 2384	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
27	25, 26, 3	nff1o 5611	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
28		nfcv 2384	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
29		nfv 1577	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑛 ≤ 𝑚
30		nfcv 2384	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑓‘𝑛)
31	30, 13	nfcsb 3175	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
32	29, 31, 15	nfif 3650	. . . . . . . . . . 11 ⊢ Ⅎ𝑥if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)
33	24, 32	nfmpt 4201	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
34	28, 10, 33	nfseq 10815	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))
35	34, 9	nffv 5679	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)
36	35	nfeq2 2396	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)
37	27, 36	nfan 1614	. . . . . 6 ⊢ Ⅎ𝑥(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))
38	37	nfex 1686	. . . . 5 ⊢ Ⅎ𝑥∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))
39	24, 38	nfrexw 2581	. . . 4 ⊢ Ⅎ𝑥∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))
40	23, 39	nfor 1623	. . 3 ⊢ Ⅎ𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)))
41	40	nfiotaw 5315	. 2 ⊢ Ⅎ𝑥(℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))))
42	1, 41	nfcxfr 2381	1 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104   ∨ wo 716  DECID wdc 842   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2203  Ⅎwnfc 2371  ∀wral 2520  ∃wrex 2521  ⦋csb 3137   ⊆ wss 3210  ifcif 3619   class class class wbr 4108   ↦ cmpt 4170  ℩cio 5309  –1-1-onto→wf1o 5350  ‘cfv 5351  (class class class)co 6049  0cc0 8123  1c1 8124   + caddc 8126   ≤ cle 8305  ℕcn 9233  ℤcz 9573  ℤ_≥cuz 9849  ...cfz 10338  seqcseq 10805   ⇝ cli 11956  Σcsu 12031

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 2214

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 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-if 3620  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-oprab 6053  df-mpo 6054  df-recs 6535  df-frec 6621  df-seqfrec 10806  df-sumdc 12032

This theorem is referenced by:  fsum2dlemstep  12113  fisumcom2  12117  fsumiun  12156  fsumcncntop  15419  dvmptfsum  15577