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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 33556 | . 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 5248 | . . . 4 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐴 ↦ 𝐵) |
7 | 2, 3, 6 | nfov 7434 | . . 3 ⊢ Ⅎ𝑥((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
8 | 7 | nfuni 4909 | . 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 4902 ↦ cmpt 5224 (class class class)co 7404 0cc0 11109 +∞cpnf 11246 [,]cicc 13330 ↾s cress 17180 ℝ*𝑠cxrs 17453 tsums ctsu 23981 Σ*cesum 33555 |
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 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-iota 6488 df-fv 6544 df-ov 7407 df-esum 33556 |
This theorem is referenced by: esum2dlem 33620 |
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