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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 24226 | . 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 2979 | . . 3 ⊢ Ⅎ𝑥(0...3) | |
3 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑥(i↑𝑘) | |
4 | nfcv 2979 | . . . 4 ⊢ Ⅎ𝑥 · | |
5 | nfcv 2979 | . . . . 5 ⊢ Ⅎ𝑥∫2 | |
6 | nfmpt1 5166 | . . . . 5 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)) | |
7 | 5, 6 | nffv 6682 | . . . 4 ⊢ Ⅎ𝑥(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0))) |
8 | 3, 4, 7 | nfov 7188 | . . 3 ⊢ Ⅎ𝑥((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)))) |
9 | 2, 8 | nfsumw 15049 | . 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 2977 | 1 ⊢ Ⅎ𝑥∫𝐴𝐵 d𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2114 Ⅎwnfc 2963 ⦋csb 3885 ifcif 4469 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 0cc0 10539 ici 10541 · cmul 10544 ≤ cle 10678 / cdiv 11299 3c3 11696 ...cfz 12895 ↑cexp 13432 ℜcre 14458 Σcsu 15044 ∫2citg2 24219 ∫citg 24221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-seq 13373 df-sum 15045 df-itg 24226 |
This theorem is referenced by: (None) |
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