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Mirrors > Home > MPE Home > Th. List > nfitg1 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for an integral. (Contributed by Mario Carneiro, 28-Jun-2014.) |
Ref | Expression |
---|---|
nfitg1 | ⊢ Ⅎ𝑥∫𝐴𝐵 d𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-itg 25132 | . 2 ⊢ ∫𝐴𝐵 d𝑥 = Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)))) | |
2 | nfcv 2904 | . . 3 ⊢ Ⅎ𝑥(0...3) | |
3 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑥(i↑𝑘) | |
4 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑥 · | |
5 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥∫2 | |
6 | nfmpt1 5256 | . . . . 5 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)) | |
7 | 5, 6 | nffv 6899 | . . . 4 ⊢ Ⅎ𝑥(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0))) |
8 | 3, 4, 7 | nfov 7436 | . . 3 ⊢ Ⅎ𝑥((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)))) |
9 | 2, 8 | nfsum 15634 | . 2 ⊢ Ⅎ𝑥Σ𝑘 ∈ (0...3)((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)))) |
10 | 1, 9 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑥∫𝐴𝐵 d𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 Ⅎwnfc 2884 ⦋csb 3893 ifcif 4528 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6541 (class class class)co 7406 ℝcr 11106 0cc0 11107 ici 11109 · cmul 11112 ≤ cle 11246 / cdiv 11868 3c3 12265 ...cfz 13481 ↑cexp 14024 ℜcre 15041 Σcsu 15629 ∫2citg2 25125 ∫citg 25127 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-seq 13964 df-sum 15630 df-itg 25132 |
This theorem is referenced by: (None) |
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