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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 2726 | . 2 β’ βͺ dom π = βͺ dom π | |
3 | caragen0.s | . 2 β’ π = (CaraGenβπ) | |
4 | 0elpw 5347 | . . 3 β’ β β π« βͺ dom π | |
5 | 4 | a1i 11 | . 2 β’ (π β β β π« βͺ dom π) |
6 | in0 4386 | . . . . . 6 β’ (π β© β ) = β | |
7 | 6 | fveq2i 6888 | . . . . 5 β’ (πβ(π β© β )) = (πββ ) |
8 | dif0 4367 | . . . . . 6 β’ (π β β ) = π | |
9 | 8 | fveq2i 6888 | . . . . 5 β’ (πβ(π β β )) = (πβπ) |
10 | 7, 9 | oveq12i 7417 | . . . 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 45782 | . . . . 5 β’ (π β (πββ ) = 0) |
13 | 12 | adantr 480 | . . . 4 β’ ((π β§ π β π« βͺ dom π) β (πββ ) = 0) |
14 | 13 | oveq1d 7420 | . . 3 β’ ((π β§ π β π« βͺ dom π) β ((πββ ) +π (πβπ)) = (0 +π (πβπ))) |
15 | iccssxr 13413 | . . . . 5 β’ (0[,]+β) β β* | |
16 | 1 | adantr 480 | . . . . . 6 β’ ((π β§ π β π« βͺ dom π) β π β OutMeas) |
17 | elpwi 4604 | . . . . . . 7 β’ (π β π« βͺ dom π β π β βͺ dom π) | |
18 | 17 | adantl 481 | . . . . . 6 β’ ((π β§ π β π« βͺ dom π) β π β βͺ dom π) |
19 | 16, 2, 18 | omecl 45788 | . . . . 5 β’ ((π β§ π β π« βͺ dom π) β (πβπ) β (0[,]+β)) |
20 | 15, 19 | sselid 3975 | . . . 4 β’ ((π β§ π β π« βͺ dom π) β (πβπ) β β*) |
21 | 20 | xaddlidd 44598 | . . 3 β’ ((π β§ π β π« βͺ dom π) β (0 +π (πβπ)) = (πβπ)) |
22 | 11, 14, 21 | 3eqtrd 2770 | . 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 45787 | 1 β’ (π β β β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 β cdif 3940 β© cin 3942 β wss 3943 β c0 4317 π« cpw 4597 βͺ cuni 4902 dom cdm 5669 βcfv 6537 (class class class)co 7405 0cc0 11112 +βcpnf 11249 β*cxr 11251 +π cxad 13096 [,]cicc 13333 OutMeascome 45774 CaraGenccaragen 45776 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-xadd 13099 df-icc 13337 df-ome 45775 df-caragen 45777 |
This theorem is referenced by: caragenfiiuncl 45800 caragenunicl 45809 caragensal 45810 caratheodory 45813 |
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