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Mirrors > Home > MPE Home > Th. List > volres | Structured version Visualization version GIF version |
Description: A self-referencing abbreviated definition of the Lebesgue measure. (Contributed by Mario Carneiro, 19-Mar-2014.) |
Ref | Expression |
---|---|
volres | ⊢ vol = (vol* ↾ dom vol) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resdmres 6188 | . 2 ⊢ (vol* ↾ dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})) = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))}) | |
2 | df-vol 24852 | . . . 4 ⊢ vol = (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))}) | |
3 | 2 | dmeqi 5864 | . . 3 ⊢ dom vol = dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))}) |
4 | 3 | reseq2i 5938 | . 2 ⊢ (vol* ↾ dom vol) = (vol* ↾ dom (vol* ↾ {𝑥 ∣ ∀𝑦 ∈ (◡vol* “ ℝ)(vol*‘𝑦) = ((vol*‘(𝑦 ∩ 𝑥)) + (vol*‘(𝑦 ∖ 𝑥)))})) |
5 | 1, 4, 2 | 3eqtr4ri 2772 | 1 ⊢ vol = (vol* ↾ dom vol) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 {cab 2710 ∀wral 3061 ∖ cdif 3911 ∩ cin 3913 ◡ccnv 5636 dom cdm 5637 ↾ cres 5639 “ cima 5640 ‘cfv 6500 (class class class)co 7361 ℝcr 11058 + caddc 11062 vol*covol 24849 volcvol 24850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 df-vol 24852 |
This theorem is referenced by: volf 24916 mblvol 24917 |
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