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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 33778 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | nfcv 2891 | . . . 4 ⊢ Ⅎ𝑘(ℝ*𝑠 ↾s (0[,]+∞)) | |
3 | nfcv 2891 | . . . 4 ⊢ Ⅎ𝑘 tsums | |
4 | nfmpt1 5257 | . . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐵) | |
5 | 2, 3, 4 | nfov 7449 | . . 3 ⊢ Ⅎ𝑘((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
6 | 5 | nfuni 4916 | . 2 ⊢ Ⅎ𝑘∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
7 | 1, 6 | nfcxfr 2889 | 1 ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2875 ∪ cuni 4909 ↦ cmpt 5232 (class class class)co 7419 0cc0 11140 +∞cpnf 11277 [,]cicc 13362 ↾s cress 17212 ℝ*𝑠cxrs 17485 tsums ctsu 24074 Σ*cesum 33777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-iota 6501 df-fv 6557 df-ov 7422 df-esum 33778 |
This theorem is referenced by: esumfsup 33820 esum2d 33843 oms0 34048 omssubadd 34051 |
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