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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragen0 | Structured version Visualization version GIF version |
Description: The empty set belongs to any Caratheodory's construction. First part of Step (b) in the proof of Theorem 113C of [Fremlin1] p. 19. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
caragen0.o | β’ (π β π β OutMeas) |
caragen0.s | β’ π = (CaraGenβπ) |
Ref | Expression |
---|---|
caragen0 | β’ (π β β β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragen0.o | . 2 β’ (π β π β OutMeas) | |
2 | eqid 2728 | . 2 β’ βͺ dom π = βͺ dom π | |
3 | caragen0.s | . 2 β’ π = (CaraGenβπ) | |
4 | 0elpw 5360 | . . 3 β’ β β π« βͺ dom π | |
5 | 4 | a1i 11 | . 2 β’ (π β β β π« βͺ dom π) |
6 | in0 4395 | . . . . . 6 β’ (π β© β ) = β | |
7 | 6 | fveq2i 6905 | . . . . 5 β’ (πβ(π β© β )) = (πββ ) |
8 | dif0 4376 | . . . . . 6 β’ (π β β ) = π | |
9 | 8 | fveq2i 6905 | . . . . 5 β’ (πβ(π β β )) = (πβπ) |
10 | 7, 9 | oveq12i 7438 | . . . 4 β’ ((πβ(π β© β )) +π (πβ(π β β ))) = ((πββ ) +π (πβπ)) |
11 | 10 | a1i 11 | . . 3 β’ ((π β§ π β π« βͺ dom π) β ((πβ(π β© β )) +π (πβ(π β β ))) = ((πββ ) +π (πβπ))) |
12 | 1 | ome0 45914 | . . . . 5 β’ (π β (πββ ) = 0) |
13 | 12 | adantr 479 | . . . 4 β’ ((π β§ π β π« βͺ dom π) β (πββ ) = 0) |
14 | 13 | oveq1d 7441 | . . 3 β’ ((π β§ π β π« βͺ dom π) β ((πββ ) +π (πβπ)) = (0 +π (πβπ))) |
15 | iccssxr 13447 | . . . . 5 β’ (0[,]+β) β β* | |
16 | 1 | adantr 479 | . . . . . 6 β’ ((π β§ π β π« βͺ dom π) β π β OutMeas) |
17 | elpwi 4613 | . . . . . . 7 β’ (π β π« βͺ dom π β π β βͺ dom π) | |
18 | 17 | adantl 480 | . . . . . 6 β’ ((π β§ π β π« βͺ dom π) β π β βͺ dom π) |
19 | 16, 2, 18 | omecl 45920 | . . . . 5 β’ ((π β§ π β π« βͺ dom π) β (πβπ) β (0[,]+β)) |
20 | 15, 19 | sselid 3980 | . . . 4 β’ ((π β§ π β π« βͺ dom π) β (πβπ) β β*) |
21 | 20 | xaddlidd 44730 | . . 3 β’ ((π β§ π β π« βͺ dom π) β (0 +π (πβπ)) = (πβπ)) |
22 | 11, 14, 21 | 3eqtrd 2772 | . 2 β’ ((π β§ π β π« βͺ dom π) β ((πβ(π β© β )) +π (πβ(π β β ))) = (πβπ)) |
23 | 1, 2, 3, 5, 22 | carageneld 45919 | 1 β’ (π β β β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β cdif 3946 β© cin 3948 β wss 3949 β c0 4326 π« cpw 4606 βͺ cuni 4912 dom cdm 5682 βcfv 6553 (class class class)co 7426 0cc0 11146 +βcpnf 11283 β*cxr 11285 +π cxad 13130 [,]cicc 13367 OutMeascome 45906 CaraGenccaragen 45908 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-xadd 13133 df-icc 13371 df-ome 45907 df-caragen 45909 |
This theorem is referenced by: caragenfiiuncl 45932 caragenunicl 45941 caragensal 45942 caratheodory 45945 |
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