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Mirrors > Home > MPE Home > Th. List > Mathboxes > caragenel2d | Structured version Visualization version GIF version |
Description: Membership in the Caratheodory's construction. Similar to carageneld 45952, but here "less then or equal to" is used, instead of equality. This is Remark 113D of [Fremlin1] p. 21. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
caragenel2d.o | β’ (π β π β OutMeas) |
caragenel2d.x | β’ π = βͺ dom π |
caragenel2d.s | β’ π = (CaraGenβπ) |
caragenel2d.e | β’ (π β πΈ β π« π) |
caragenel2d.a | β’ ((π β§ π β π« π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) β€ (πβπ)) |
Ref | Expression |
---|---|
caragenel2d | β’ (π β πΈ β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caragenel2d.o | . 2 β’ (π β π β OutMeas) | |
2 | caragenel2d.x | . 2 β’ π = βͺ dom π | |
3 | caragenel2d.s | . 2 β’ π = (CaraGenβπ) | |
4 | caragenel2d.e | . 2 β’ (π β πΈ β π« π) | |
5 | 1 | adantr 479 | . . . . 5 β’ ((π β§ π β π« π) β π β OutMeas) |
6 | inss1 4223 | . . . . . . 7 β’ (π β© πΈ) β π | |
7 | elpwi 4605 | . . . . . . 7 β’ (π β π« π β π β π) | |
8 | 6, 7 | sstrid 3984 | . . . . . 6 β’ (π β π« π β (π β© πΈ) β π) |
9 | 8 | adantl 480 | . . . . 5 β’ ((π β§ π β π« π) β (π β© πΈ) β π) |
10 | 5, 2, 9 | omexrcl 45957 | . . . 4 β’ ((π β§ π β π« π) β (πβ(π β© πΈ)) β β*) |
11 | 7 | ssdifssd 4135 | . . . . . 6 β’ (π β π« π β (π β πΈ) β π) |
12 | 11 | adantl 480 | . . . . 5 β’ ((π β§ π β π« π) β (π β πΈ) β π) |
13 | 5, 2, 12 | omexrcl 45957 | . . . 4 β’ ((π β§ π β π« π) β (πβ(π β πΈ)) β β*) |
14 | xaddcl 13248 | . . . 4 β’ (((πβ(π β© πΈ)) β β* β§ (πβ(π β πΈ)) β β*) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) β β*) | |
15 | 10, 13, 14 | syl2anc 582 | . . 3 β’ ((π β§ π β π« π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) β β*) |
16 | 7 | adantl 480 | . . . 4 β’ ((π β§ π β π« π) β π β π) |
17 | 5, 2, 16 | omexrcl 45957 | . . 3 β’ ((π β§ π β π« π) β (πβπ) β β*) |
18 | caragenel2d.a | . . 3 β’ ((π β§ π β π« π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) β€ (πβπ)) | |
19 | 5, 2, 16 | omelesplit 45968 | . . 3 β’ ((π β§ π β π« π) β (πβπ) β€ ((πβ(π β© πΈ)) +π (πβ(π β πΈ)))) |
20 | 15, 17, 18, 19 | xrletrid 13164 | . 2 β’ ((π β§ π β π« π) β ((πβ(π β© πΈ)) +π (πβ(π β πΈ))) = (πβπ)) |
21 | 1, 2, 3, 4, 20 | carageneld 45952 | 1 β’ (π β πΈ β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β cdif 3937 β© cin 3939 β wss 3940 π« cpw 4598 βͺ cuni 4903 class class class wbr 5143 dom cdm 5672 βcfv 6542 (class class class)co 7415 β*cxr 11275 β€ cle 11277 +π cxad 13120 OutMeascome 45939 CaraGenccaragen 45941 |
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 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-xadd 13123 df-ico 13360 df-icc 13361 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-sumge0 45813 df-ome 45940 df-caragen 45942 |
This theorem is referenced by: caragencmpl 45985 hspmbl 46079 |
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