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 29417 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑘(ℝ*𝑠 ↾s (0[,]+∞)) | |
3 | nfcv 2751 | . . . 4 ⊢ Ⅎ𝑘 tsums | |
4 | nfmpt1 4675 | . . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐵) | |
5 | 2, 3, 4 | nfov 6575 | . . 3 ⊢ Ⅎ𝑘((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
6 | 5 | nfuni 4378 | . 2 ⊢ Ⅎ𝑘∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
7 | 1, 6 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2738 ∪ cuni 4372 ↦ cmpt 4643 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 [,]cicc 12049 ↾s cress 15696 ℝ*𝑠cxrs 15983 tsums ctsu 21739 Σ*cesum 29416 |
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-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-iota 5768 df-fv 5812 df-ov 6552 df-esum 29417 |
This theorem is referenced by: esumfsup 29459 esum2d 29482 oms0 29686 omssubadd 29689 |
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