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Mirrors > Home > MPE Home > Th. List > Mathboxes > measvunilem0 | Structured version Visualization version GIF version |
Description: Lemma for measvuni 33756. (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 1199 | . . 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 8973 | . . 3 β’ (π΄ βΌ Ο β π΄ β V) | |
3 | measvunilem.0.1 | . . . 4 β’ β²π₯π΄ | |
4 | 3 | esum0 33591 | . . 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 1910 | . . . 4 β’ β²π₯ π β (measuresβπ) | |
7 | nfra1 3276 | . . . 4 β’ β²π₯βπ₯ β π΄ π΅ β {β } | |
8 | nfcv 2898 | . . . . . 6 β’ β²π₯ βΌ | |
9 | nfcv 2898 | . . . . . 6 β’ β²π₯Ο | |
10 | 3, 8, 9 | nfbr 5189 | . . . . 5 β’ β²π₯ π΄ βΌ Ο |
11 | nfdisj1 5121 | . . . . 5 β’ β²π₯Disj π₯ β π΄ π΅ | |
12 | 10, 11 | nfan 1895 | . . . 4 β’ β²π₯(π΄ βΌ Ο β§ Disj π₯ β π΄ π΅) |
13 | 6, 7, 12 | nf3an 1897 | . . 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 2728 | . . 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 1135 | . . . . . . 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 3243 | . . . . . 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 4641 | . . . . . 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 6895 | . . . 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 33748 | . . . . . 6 β’ (π β (measuresβπ) β (πββ ) = 0) | |
21 | 20 | 3ad2ant1 1131 | . . . . 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 480 | . . . 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 2767 | . . 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 33573 | . 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 32319 | . . . . 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 5059 | . . . . 5 β’ βͺ π₯ β π΄ β = β | |
27 | 25, 26 | eqtrdi 2783 | . . . 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 6895 | . . 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 2767 | . 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 2778 | 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 395 β§ w3a 1085 = wceq 1534 β wcel 2099 β²wnfc 2878 βwral 3056 Vcvv 3469 β c0 4318 {csn 4624 βͺ ciun 4991 Disj wdisj 5107 class class class wbr 5142 βcfv 6542 Οcom 7862 βΌ cdom 8951 0cc0 11124 Ξ£*cesum 33569 measurescmeas 33737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-xadd 13111 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-seq 13985 df-hash 14308 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-tset 17237 df-ple 17238 df-ds 17240 df-rest 17389 df-topn 17390 df-0g 17408 df-gsum 17409 df-topgen 17410 df-ordt 17468 df-xrs 17469 df-mre 17551 df-mrc 17552 df-acs 17554 df-ps 18543 df-tsr 18544 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-submnd 18726 df-cntz 19252 df-cmn 19721 df-fbas 21256 df-fg 21257 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-ntr 22898 df-nei 22976 df-cn 23105 df-haus 23193 df-fil 23724 df-fm 23816 df-flim 23817 df-flf 23818 df-tsms 24005 df-esum 33570 df-meas 33738 |
This theorem is referenced by: measvuni 33756 |
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