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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfesum1 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for extended sum. (Contributed by Thierry Arnoux, 19-Oct-2017.) |
Ref | Expression |
---|---|
nfesum1.1 | ⊢ Ⅎ𝑘𝐴 |
Ref | Expression |
---|---|
nfesum1 | ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-esum 32750 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑘(ℝ*𝑠 ↾s (0[,]+∞)) | |
3 | nfcv 2902 | . . . 4 ⊢ Ⅎ𝑘 tsums | |
4 | nfmpt1 5233 | . . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐵) | |
5 | 2, 3, 4 | nfov 7407 | . . 3 ⊢ Ⅎ𝑘((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
6 | 5 | nfuni 4892 | . 2 ⊢ Ⅎ𝑘∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
7 | 1, 6 | nfcxfr 2900 | 1 ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2882 ∪ cuni 4885 ↦ cmpt 5208 (class class class)co 7377 0cc0 11075 +∞cpnf 11210 [,]cicc 13292 ↾s cress 17138 ℝ*𝑠cxrs 17411 tsums ctsu 23529 Σ*cesum 32749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3419 df-v 3461 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-nul 4303 df-if 4507 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-br 5126 df-opab 5188 df-mpt 5209 df-iota 6468 df-fv 6524 df-ov 7380 df-esum 32750 |
This theorem is referenced by: esumfsup 32792 esum2d 32815 oms0 33020 omssubadd 33023 |
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