Theorem nfsum 12106

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 12103	. 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 2392	. . . . 5 ⊢ Ⅎ𝑥ℤ
3		nfsum.1	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
4		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑚)
5	3, 4	nfss 3241	. . . . . 6 ⊢ Ⅎ𝑥 𝐴 ⊆ (ℤ≥‘𝑚)
6	3	nfcri 2386	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑗 ∈ 𝐴
7	6	nfdc 1711	. . . . . . 7 ⊢ Ⅎ𝑥DECID 𝑗 ∈ 𝐴
8	4, 7	nfralxy 2588	. . . . . 6 ⊢ Ⅎ𝑥∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴
9		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥𝑚
10		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥 +
11	3	nfcri 2386	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝑛 ∈ 𝐴
12		nfcv 2392	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑛
13		nfsum.2	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐵
14	12, 13	nfcsb 3185	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑛 / 𝑘⦌𝐵
15		nfcv 2392	. . . . . . . . . 10 ⊢ Ⅎ𝑥0
16	11, 14, 15	nfif 3669	. . . . . . . . 9 ⊢ Ⅎ𝑥if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)
17	2, 16	nfmpt 4221	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
18	9, 10, 17	nfseq 10877	. . . . . . 7 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
19		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑥 ⇝
20		nfcv 2392	. . . . . . 7 ⊢ Ⅎ𝑥𝑧
21	18, 19, 20	nfbr 4175	. . . . . 6 ⊢ Ⅎ𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑧
22	5, 8, 21	nf3an 1619	. . . . 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 2589	. . . 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 2392	. . . . 5 ⊢ Ⅎ𝑥ℕ
25		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥𝑓
26		nfcv 2392	. . . . . . . 8 ⊢ Ⅎ𝑥(1...𝑚)
27	25, 26, 3	nff1o 5635	. . . . . . 7 ⊢ Ⅎ𝑥 𝑓:(1...𝑚)–1-1-onto→𝐴
28		nfcv 2392	. . . . . . . . . 10 ⊢ Ⅎ𝑥1
29		nfv 1581	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑛 ≤ 𝑚
30		nfcv 2392	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑓‘𝑛)
31	30, 13	nfcsb 3185	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋(𝑓‘𝑛) / 𝑘⦌𝐵
32	29, 31, 15	nfif 3669	. . . . . . . . . . 11 ⊢ Ⅎ𝑥if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)
33	24, 32	nfmpt 4221	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
34	28, 10, 33	nfseq 10877	. . . . . . . . 9 ⊢ Ⅎ𝑥seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))
35	34, 9	nffv 5703	. . . . . . . 8 ⊢ Ⅎ𝑥(seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)
36	35	nfeq2 2404	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)
37	27, 36	nfan 1618	. . . . . 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 1690	. . . . 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 2589	. . . 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 1627	. . 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 5339	. 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 2389	1 ⊢ Ⅎ𝑥Σ𝑘 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104   ∨ wo 720  DECID wdc 846   ∧ w3a 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Ⅎwnfc 2379  ∀wral 2528  ∃wrex 2529  ⦋csb 3147   ⊆ wss 3220  ifcif 3638   class class class wbr 4128   ↦ cmpt 4190  ℩cio 5333  –1-1-onto→wf1o 5374  ‘cfv 5375  (class class class)co 6079  0cc0 8173  1c1 8174   + caddc 8176   ≤ cle 8355  ℕcn 9287  ℤcz 9627  ℤ_≥cuz 9904  ...cfz 10394  seqcseq 10867   ⇝ cli 12027  Σcsu 12102

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-mpt 4192  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-recs 6570  df-frec 6656  df-seqfrec 10868  df-sumdc 12103

This theorem is referenced by:  fsum2dlemstep  12184  fisumcom2  12188  fsumiun  12227  fsumcncntop  15651  dvmptfsum  15809