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| 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 25659 | . 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 5249 | . . . . 5 ⊢ Ⅎ𝑥(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)) | |
| 7 | 5, 6 | nffv 6915 | . . . 4 ⊢ Ⅎ𝑥(∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0))) | 
| 8 | 3, 4, 7 | nfov 7462 | . . 3 ⊢ Ⅎ𝑥((i↑𝑘) · (∫2‘(𝑥 ∈ ℝ ↦ ⦋(ℜ‘(𝐵 / (i↑𝑘))) / 𝑧⦌if((𝑥 ∈ 𝐴 ∧ 0 ≤ 𝑧), 𝑧, 0)))) | 
| 9 | 2, 8 | nfsum 15728 | . 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 395 ∈ wcel 2107 Ⅎwnfc 2889 ⦋csb 3898 ifcif 4524 class class class wbr 5142 ↦ cmpt 5224 ‘cfv 6560 (class class class)co 7432 ℝcr 11155 0cc0 11156 ici 11158 · cmul 11161 ≤ cle 11297 / cdiv 11921 3c3 12323 ...cfz 13548 ↑cexp 14103 ℜcre 15137 Σcsu 15723 ∫2citg2 25652 ∫citg 25654 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-seq 14044 df-sum 15724 df-itg 25659 | 
| This theorem is referenced by: (None) | 
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