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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 23392 | . 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 2764 | . . 3 ⊢ Ⅎ𝑥(0...3) | |
3 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑥(i↑𝑘) | |
4 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑥 · | |
5 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑥∫2 | |
6 | nfmpt1 4747 | . . . . 5 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)) | |
7 | 5, 6 | nffv 6198 | . . . 4 ⊢ Ⅎ𝑥(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0))) |
8 | 3, 4, 7 | nfov 6676 | . . 3 ⊢ Ⅎ𝑥((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)))) |
9 | 2, 8 | nfsum 14421 | . 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 2762 | 1 ⊢ Ⅎ𝑥∫𝐴𝐵 d𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∈ wcel 1990 Ⅎwnfc 2751 ⦋csb 3533 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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 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-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 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-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-seq 12802 df-sum 14417 df-itg 23392 |
This theorem is referenced by: (None) |
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