Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 31291 | . 2 ⊢ Σ*𝑘 ∈ 𝐴𝐵 = ∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) | |
2 | nfcv 2980 | . . . 4 ⊢ Ⅎ𝑥(ℝ*𝑠 ↾s (0[,]+∞)) | |
3 | nfcv 2980 | . . . 4 ⊢ Ⅎ𝑥 tsums | |
4 | nfesum2.1 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | nfesum2.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
6 | 4, 5 | nfmpt 5166 | . . . 4 ⊢ Ⅎ𝑥(𝑘 ∈ 𝐴 ↦ 𝐵) |
7 | 2, 3, 6 | nfov 7189 | . . 3 ⊢ Ⅎ𝑥((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
8 | 7 | nfuni 4848 | . 2 ⊢ Ⅎ𝑥∪ ((ℝ*𝑠 ↾s (0[,]+∞)) tsums (𝑘 ∈ 𝐴 ↦ 𝐵)) |
9 | 1, 8 | nfcxfr 2978 | 1 ⊢ Ⅎ𝑥Σ*𝑘 ∈ 𝐴𝐵 |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2964 ∪ cuni 4841 ↦ cmpt 5149 (class class class)co 7159 0cc0 10540 +∞cpnf 10675 [,]cicc 12744 ↾s cress 16487 ℝ*𝑠cxrs 16776 tsums ctsu 22737 Σ*cesum 31290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-iota 6317 df-fv 6366 df-ov 7162 df-esum 31291 |
This theorem is referenced by: esum2dlem 31355 |
Copyright terms: Public domain | W3C validator |