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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 2824 | . . 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 2980 | . . 3 ⊢ Ⅎ𝑦(0...3) | |
4 | nfcv 2980 | . . . 4 ⊢ Ⅎ𝑦(i↑𝑘) | |
5 | nfcv 2980 | . . . 4 ⊢ Ⅎ𝑦 · | |
6 | nfcv 2980 | . . . . 5 ⊢ Ⅎ𝑦∫2 | |
7 | nfcv 2980 | . . . . . 6 ⊢ Ⅎ𝑦ℝ | |
8 | nfitg.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
9 | 8 | nfcri 2974 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
10 | nfcv 2980 | . . . . . . . . 9 ⊢ Ⅎ𝑦0 | |
11 | nfcv 2980 | . . . . . . . . 9 ⊢ Ⅎ𝑦 ≤ | |
12 | nfcv 2980 | . . . . . . . . . 10 ⊢ Ⅎ𝑦ℜ | |
13 | nfitg.2 | . . . . . . . . . . 11 ⊢ Ⅎ𝑦𝐵 | |
14 | nfcv 2980 | . . . . . . . . . . 11 ⊢ Ⅎ𝑦 / | |
15 | 13, 14, 4 | nfov 7189 | . . . . . . . . . 10 ⊢ Ⅎ𝑦(𝐵 / (i↑𝑘)) |
16 | 12, 15 | nffv 6683 | . . . . . . . . 9 ⊢ Ⅎ𝑦(ℜ‘(𝐵 / (i↑𝑘))) |
17 | 10, 11, 16 | nfbr 5116 | . . . . . . . 8 ⊢ Ⅎ𝑦0 ≤ (ℜ‘(𝐵 / (i↑𝑘))) |
18 | 9, 17 | nfan 1899 | . . . . . . 7 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))) |
19 | 18, 16, 10 | nfif 4499 | . . . . . 6 ⊢ Ⅎ𝑦if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0) |
20 | 7, 19 | nfmpt 5166 | . . . . 5 ⊢ Ⅎ𝑦(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)) |
21 | 6, 20 | nffv 6683 | . . . 4 ⊢ Ⅎ𝑦(∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0))) |
22 | 4, 5, 21 | nfov 7189 | . . 3 ⊢ Ⅎ𝑦((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ if((𝑥 ∈ 𝐴 ∧ 0 ≤ (ℜ‘(𝐵 / (i↑𝑘)))), (ℜ‘(𝐵 / (i↑𝑘))), 0)))) |
23 | 3, 22 | nfsumw 15050 | . 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 2978 | 1 ⊢ Ⅎ𝑦∫𝐴𝐵 d𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2113 Ⅎwnfc 2964 ifcif 4470 class class class wbr 5069 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 ℝcr 10539 0cc0 10540 ici 10542 · cmul 10545 ≤ cle 10679 / cdiv 11300 3c3 11696 ...cfz 12895 ↑cexp 13432 ℜcre 14459 Σcsu 15045 ∫2citg2 24220 ∫citg 24222 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-seq 13373 df-sum 15046 df-itg 24227 |
This theorem is referenced by: itgfsum 24430 itgulm2 25000 fourierdlem112 42510 |
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