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 30090 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑘(ℝ*𝑠 ↾s (0[,]+∞)) | |
3 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑘 tsums | |
4 | nfmpt1 4747 | . . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐵) | |
5 | 2, 3, 4 | nfov 6676 | . . 3 ⊢ Ⅎ𝑘((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
6 | 5 | nfuni 4442 | . 2 ⊢ Ⅎ𝑘∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
7 | 1, 6 | nfcxfr 2762 | 1 ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2751 ∪ cuni 4436 ↦ cmpt 4729 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 [,]cicc 12178 ↾s cress 15858 ℝ*𝑠cxrs 16160 tsums ctsu 21929 Σ*cesum 30089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-iota 5851 df-fv 5896 df-ov 6653 df-esum 30090 |
This theorem is referenced by: esumfsup 30132 esum2d 30155 oms0 30359 omssubadd 30362 |
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