Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfesum2 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 2-May-2020.) |
Ref | Expression |
---|---|
nfesum2.1 | ⊢ Ⅎ𝑥𝐴 |
nfesum2.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfesum2 | ⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-esum 31662 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑥(ℝ*𝑠 ↾s (0[,]+∞)) | |
3 | nfcv 2897 | . . . 4 ⊢ Ⅎ𝑥 tsums | |
4 | nfesum2.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | nfesum2.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | nfmpt 5137 | . . . 4 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐴 ↦ 𝐵) |
7 | 2, 3, 6 | nfov 7221 | . . 3 ⊢ Ⅎ𝑥((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
8 | 7 | nfuni 4812 | . 2 ⊢ Ⅎ𝑥∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
9 | 1, 8 | nfcxfr 2895 | 1 ⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2877 ∪ cuni 4805 ↦ cmpt 5120 (class class class)co 7191 0cc0 10694 +∞cpnf 10829 [,]cicc 12903 ↾s cress 16667 ℝ*𝑠cxrs 16959 tsums ctsu 22977 Σ*cesum 31661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-iota 6316 df-fv 6366 df-ov 7194 df-esum 31662 |
This theorem is referenced by: esum2dlem 31726 |
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