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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 34009 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥(ℝ*𝑠 ↾s (0[,]+∞)) | |
3 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑥 tsums | |
4 | nfesum2.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | nfesum2.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | nfmpt 5255 | . . . 4 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐴 ↦ 𝐵) |
7 | 2, 3, 6 | nfov 7461 | . . 3 ⊢ Ⅎ𝑥((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
8 | 7 | nfuni 4919 | . 2 ⊢ Ⅎ𝑥∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
9 | 1, 8 | nfcxfr 2901 | 1 ⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2888 ∪ cuni 4912 ↦ cmpt 5231 (class class class)co 7431 0cc0 11153 +∞cpnf 11290 [,]cicc 13387 ↾s cress 17274 ℝ*𝑠cxrs 17547 tsums ctsu 24150 Σ*cesum 34008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-iota 6516 df-fv 6571 df-ov 7434 df-esum 34009 |
This theorem is referenced by: esum2dlem 34073 |
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