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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvmbfm | Structured version Visualization version Unicode version |
Description: A real-valued random variable is a measurable function from its sample space to the Borel sigma-algebra. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
isrrvv.1 | Prob |
Ref | Expression |
---|---|
rrvmbfm | rRndVar MblFnM𝔅ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrrvv.1 | . . 3 Prob | |
2 | dmeq 5324 | . . . . 5 | |
3 | 2 | oveq1d 6665 | . . . 4 MblFnM𝔅ℝ MblFnM𝔅ℝ |
4 | df-rrv 30503 | . . . 4 rRndVar Prob MblFnM𝔅ℝ | |
5 | ovex 6678 | . . . 4 MblFnM𝔅ℝ | |
6 | 3, 4, 5 | fvmpt 6282 | . . 3 Prob rRndVar MblFnM𝔅ℝ |
7 | 1, 6 | syl 17 | . 2 rRndVar MblFnM𝔅ℝ |
8 | 7 | eleq2d 2687 | 1 rRndVar MblFnM𝔅ℝ |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wceq 1483 wcel 1990 cdm 5114 cfv 5888 (class class class)co 6650 𝔅ℝcbrsiga 30244 MblFnMcmbfm 30312 Probcprb 30469 rRndVarcrrv 30502 |
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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
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-eu 2474 df-mo 2475 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-sbc 3436 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-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-rrv 30503 |
This theorem is referenced by: isrrvv 30505 rrvadd 30514 rrvmulc 30515 orrvcval4 30526 orrvcoel 30527 orrvccel 30528 |
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