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Mirrors > Home > MPE Home > Th. List > nfitg | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for an integral: if 𝑦 is (effectively) not free in 𝐴 and 𝐵, it is not free in ∫𝐴𝐵 d𝑥. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
nfitg.1 | ⊢ Ⅎ𝑦𝐴 |
nfitg.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfitg | ⊢ Ⅎ𝑦∫𝐴𝐵 d𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))) | |
2 | 1 | dfitg 23536 | . 2 ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) |
3 | nfcv 2764 | . . 3 ⊢ Ⅎ𝑦(0...3) | |
4 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑦(i↑𝑘) | |
5 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑦 · | |
6 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑦∫2 | |
7 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑦ℝ | |
8 | nfitg.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
9 | 8 | nfcri 2758 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
10 | nfcv 2764 | . . . . . . . . 9 ⊢ Ⅎ𝑦0 | |
11 | nfcv 2764 | . . . . . . . . 9 ⊢ Ⅎ𝑦 ≤ | |
12 | nfcv 2764 | . . . . . . . . . 10 ⊢ Ⅎ𝑦ℜ | |
13 | nfitg.2 | . . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵 | |
14 | nfcv 2764 | . . . . . . . . . . 11 ⊢ Ⅎ𝑦 / | |
15 | 13, 14, 4 | nfov 6676 | . . . . . . . . . 10 ⊢ Ⅎ𝑦(𝐵 / (i↑𝑘)) |
16 | 12, 15 | nffv 6198 | . . . . . . . . 9 ⊢ Ⅎ𝑦(ℜ‘(𝐵 / (i↑𝑘))) |
17 | 10, 11, 16 | nfbr 4699 | . . . . . . . 8 ⊢ Ⅎ𝑦0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) |
18 | 9, 17 | nfan 1828 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) |
19 | 18, 16, 10 | nfif 4115 | . . . . . 6 ⊢ Ⅎ𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) |
20 | 7, 19 | nfmpt 4746 | . . . . 5 ⊢ Ⅎ𝑦(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) |
21 | 6, 20 | nffv 6198 | . . . 4 ⊢ Ⅎ𝑦(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) |
22 | 4, 5, 21 | nfov 6676 | . . 3 ⊢ Ⅎ𝑦((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) |
23 | 3, 22 | nfsum 14421 | . 2 ⊢ Ⅎ𝑦Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) |
24 | 2, 23 | nfcxfr 2762 | 1 ⊢ Ⅎ𝑦∫𝐴𝐵 d𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∈ wcel 1990 Ⅎwnfc 2751 ifcif 4086 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 ici 9938 · cmul 9941 ≤ cle 10075 / cdiv 10684 3c3 11071 ...cfz 12326 ↑cexp 12860 ℜcre 13837 Σcsu 14416 ∫2citg2 23385 ∫citg 23387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-seq 12802 df-sum 14417 df-itg 23392 |
This theorem is referenced by: itgfsum 23593 itgulm2 24163 fourierdlem112 40435 |
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