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Mirrors > Home > MPE Home > Th. List > Mathboxes > measvunilem0 | Structured version Visualization version GIF version |
Description: Lemma for measvuni 33886. (Contributed by Thierry Arnoux, 6-Mar-2017.) |
Ref | Expression |
---|---|
measvunilem.0.1 | β’ β²π₯π΄ |
Ref | Expression |
---|---|
measvunilem0 | β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β (πββͺ π₯ β π΄ π΅) = Ξ£*π₯ β π΄(πβπ΅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp3l 1198 | . . 3 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β π΄ βΌ Ο) | |
2 | ctex 8977 | . . 3 β’ (π΄ βΌ Ο β π΄ β V) | |
3 | measvunilem.0.1 | . . . 4 β’ β²π₯π΄ | |
4 | 3 | esum0 33721 | . . 3 β’ (π΄ β V β Ξ£*π₯ β π΄0 = 0) |
5 | 1, 2, 4 | 3syl 18 | . 2 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β Ξ£*π₯ β π΄0 = 0) |
6 | nfv 1909 | . . . 4 β’ β²π₯ π β (measuresβπ) | |
7 | nfra1 3272 | . . . 4 β’ β²π₯βπ₯ β π΄ π΅ β {β } | |
8 | nfcv 2892 | . . . . . 6 β’ β²π₯ βΌ | |
9 | nfcv 2892 | . . . . . 6 β’ β²π₯Ο | |
10 | 3, 8, 9 | nfbr 5191 | . . . . 5 β’ β²π₯ π΄ βΌ Ο |
11 | nfdisj1 5123 | . . . . 5 β’ β²π₯Disj π₯ β π΄ π΅ | |
12 | 10, 11 | nfan 1894 | . . . 4 β’ β²π₯(π΄ βΌ Ο β§ Disj π₯ β π΄ π΅) |
13 | 6, 7, 12 | nf3an 1896 | . . 3 β’ β²π₯(π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) |
14 | eqidd 2726 | . . 3 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β π΄ = π΄) | |
15 | simp2 1134 | . . . . . . 7 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β βπ₯ β π΄ π΅ β {β }) | |
16 | 15 | r19.21bi 3239 | . . . . . 6 β’ (((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β§ π₯ β π΄) β π΅ β {β }) |
17 | elsni 4642 | . . . . . 6 β’ (π΅ β {β } β π΅ = β ) | |
18 | 16, 17 | syl 17 | . . . . 5 β’ (((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β§ π₯ β π΄) β π΅ = β ) |
19 | 18 | fveq2d 6894 | . . . 4 β’ (((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β§ π₯ β π΄) β (πβπ΅) = (πββ )) |
20 | measvnul 33878 | . . . . . 6 β’ (π β (measuresβπ) β (πββ ) = 0) | |
21 | 20 | 3ad2ant1 1130 | . . . . 5 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β (πββ ) = 0) |
22 | 21 | adantr 479 | . . . 4 β’ (((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β§ π₯ β π΄) β (πββ ) = 0) |
23 | 19, 22 | eqtrd 2765 | . . 3 β’ (((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β§ π₯ β π΄) β (πβπ΅) = 0) |
24 | 13, 14, 23 | esumeq12dvaf 33703 | . 2 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β Ξ£*π₯ β π΄(πβπ΅) = Ξ£*π₯ β π΄0) |
25 | 13, 3, 3, 14, 18 | iuneq12daf 32387 | . . . . 5 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β βͺ π₯ β π΄ π΅ = βͺ π₯ β π΄ β ) |
26 | iun0 5061 | . . . . 5 β’ βͺ π₯ β π΄ β = β | |
27 | 25, 26 | eqtrdi 2781 | . . . 4 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β βͺ π₯ β π΄ π΅ = β ) |
28 | 27 | fveq2d 6894 | . . 3 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β (πββͺ π₯ β π΄ π΅) = (πββ )) |
29 | 28, 21 | eqtrd 2765 | . 2 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β (πββͺ π₯ β π΄ π΅) = 0) |
30 | 5, 24, 29 | 3eqtr4rd 2776 | 1 β’ ((π β (measuresβπ) β§ βπ₯ β π΄ π΅ β {β } β§ (π΄ βΌ Ο β§ Disj π₯ β π΄ π΅)) β (πββͺ π₯ β π΄ π΅) = Ξ£*π₯ β π΄(πβπ΅)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1084 = wceq 1533 β wcel 2098 β²wnfc 2875 βwral 3051 Vcvv 3463 β c0 4319 {csn 4625 βͺ ciun 4992 Disj wdisj 5109 class class class wbr 5144 βcfv 6543 Οcom 7865 βΌ cdom 8955 0cc0 11133 Ξ£*cesum 33699 measurescmeas 33867 |
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 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 |
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 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-disj 5110 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-xadd 13120 df-ioo 13355 df-ioc 13356 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-seq 13994 df-hash 14317 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-tset 17246 df-ple 17247 df-ds 17249 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-ordt 17477 df-xrs 17478 df-mre 17560 df-mrc 17561 df-acs 17563 df-ps 18552 df-tsr 18553 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-cntz 19267 df-cmn 19736 df-fbas 21275 df-fg 21276 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22862 df-ntr 22937 df-nei 23015 df-cn 23144 df-haus 23232 df-fil 23763 df-fm 23855 df-flim 23856 df-flf 23857 df-tsms 24044 df-esum 33700 df-meas 33868 |
This theorem is referenced by: measvuni 33886 |
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