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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 23198 | . 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 2751 | . . 3 ⊢ Ⅎ𝑥(0...3) | |
3 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑥(i↑𝑘) | |
4 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑥 · | |
5 | nfcv 2751 | . . . . 5 ⊢ Ⅎ𝑥∫2 | |
6 | nfmpt1 4675 | . . . . 5 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)) | |
7 | 5, 6 | nffv 6110 | . . . 4 ⊢ Ⅎ𝑥(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0))) |
8 | 3, 4, 7 | nfov 6575 | . . 3 ⊢ Ⅎ𝑥((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)))) |
9 | 2, 8 | nfsum 14269 | . 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 2749 | 1 ⊢ Ⅎ𝑥∫𝐴𝐵 d𝑥 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∈ wcel 1977 Ⅎwnfc 2738 ⦋csb 3499 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 ici 9817 · cmul 9820 ≤ cle 9954 / cdiv 10563 3c3 10948 ...cfz 12197 ↑cexp 12722 ℜcre 13685 Σcsu 14264 ∫2citg2 23191 ∫citg 23193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seq 12664 df-sum 14265 df-itg 23198 |
This theorem is referenced by: (None) |
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