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 31287 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑘(ℝ*𝑠 ↾s (0[,]+∞)) | |
3 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑘 tsums | |
4 | nfmpt1 5164 | . . . 4 ⊢ Ⅎ𝑘(𝑘 ∈ 𝐴 ↦ 𝐵) | |
5 | 2, 3, 4 | nfov 7186 | . . 3 ⊢ Ⅎ𝑘((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
6 | 5 | nfuni 4845 | . 2 ⊢ Ⅎ𝑘∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
7 | 1, 6 | nfcxfr 2975 | 1 ⊢ Ⅎ𝑘Σ*𝑘 ∈ 𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2961 ∪ cuni 4838 ↦ cmpt 5146 (class class class)co 7156 0cc0 10537 +∞cpnf 10672 [,]cicc 12742 ↾s cress 16484 ℝ*𝑠cxrs 16773 tsums ctsu 22734 Σ*cesum 31286 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-iota 6314 df-fv 6363 df-ov 7159 df-esum 31287 |
This theorem is referenced by: esumfsup 31329 esum2d 31352 oms0 31555 omssubadd 31558 |
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