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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 2798 | . . 3 ⊢ (ℜ‘(𝐵 / (i↑𝑘))) = (ℜ‘(𝐵 / (i↑𝑘))) | |
2 | 1 | dfitg 24373 | . 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 2955 | . . 3 ⊢ Ⅎ𝑦(0...3) | |
4 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑦(i↑𝑘) | |
5 | nfcv 2955 | . . . 4 ⊢ Ⅎ𝑦 · | |
6 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑦∫2 | |
7 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑦ℝ | |
8 | nfitg.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
9 | 8 | nfcri 2943 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
10 | nfcv 2955 | . . . . . . . . 9 ⊢ Ⅎ𝑦0 | |
11 | nfcv 2955 | . . . . . . . . 9 ⊢ Ⅎ𝑦 ≤ | |
12 | nfcv 2955 | . . . . . . . . . 10 ⊢ Ⅎ𝑦ℜ | |
13 | nfitg.2 | . . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵 | |
14 | nfcv 2955 | . . . . . . . . . . 11 ⊢ Ⅎ𝑦 / | |
15 | 13, 14, 4 | nfov 7165 | . . . . . . . . . 10 ⊢ Ⅎ𝑦(𝐵 / (i↑𝑘)) |
16 | 12, 15 | nffv 6655 | . . . . . . . . 9 ⊢ Ⅎ𝑦(ℜ‘(𝐵 / (i↑𝑘))) |
17 | 10, 11, 16 | nfbr 5077 | . . . . . . . 8 ⊢ Ⅎ𝑦0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) |
18 | 9, 17 | nfan 1900 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) |
19 | 18, 16, 10 | nfif 4454 | . . . . . 6 ⊢ Ⅎ𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) |
20 | 7, 19 | nfmpt 5127 | . . . . 5 ⊢ Ⅎ𝑦(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) |
21 | 6, 20 | nffv 6655 | . . . 4 ⊢ Ⅎ𝑦(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) |
22 | 4, 5, 21 | nfov 7165 | . . 3 ⊢ Ⅎ𝑦((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) |
23 | 3, 22 | nfsum 15039 | . 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 2953 | 1 ⊢ Ⅎ𝑦∫𝐴𝐵 d𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∈ wcel 2111 Ⅎwnfc 2936 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 0cc0 10526 ici 10528 · cmul 10531 ≤ cle 10665 / cdiv 11286 3c3 11681 ...cfz 12885 ↑cexp 13425 ℜcre 14448 Σcsu 15034 ∫2citg2 24220 ∫citg 24222 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-seq 13365 df-sum 15035 df-itg 24227 |
This theorem is referenced by: itgfsum 24430 itgulm2 25004 fourierdlem112 42860 |
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