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| Mirrors > Home > MPE Home > Th. List > Mathboxes > measinblem | Structured version Visualization version GIF version | ||
| Description: Lemma for measinb 33514. (Contributed by Thierry Arnoux, 2-Jun-2017.) |
| Ref | Expression |
|---|---|
| measinblem | β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯)) β (πβ(βͺ π΅ β© π΄)) = Ξ£*π₯ β π΅(πβ(π₯ β© π΄))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | iunin1 5076 | . . . 4 β’ βͺ π₯ β π΅ (π₯ β© π΄) = (βͺ π₯ β π΅ π₯ β© π΄) | |
| 2 | uniiun 5062 | . . . . 5 β’ βͺ π΅ = βͺ π₯ β π΅ π₯ | |
| 3 | 2 | ineq1i 4209 | . . . 4 β’ (βͺ π΅ β© π΄) = (βͺ π₯ β π΅ π₯ β© π΄) |
| 4 | 1, 3 | eqtr4i 2762 | . . 3 β’ βͺ π₯ β π΅ (π₯ β© π΄) = (βͺ π΅ β© π΄) |
| 5 | 4 | fveq2i 6895 | . 2 β’ (πββͺ π₯ β π΅ (π₯ β© π΄)) = (πβ(βͺ π΅ β© π΄)) |
| 6 | simplll 772 | . . 3 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯)) β π β (measuresβπ)) | |
| 7 | nfv 1916 | . . . . 5 β’ β²π₯((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) | |
| 8 | nfv 1916 | . . . . . 6 β’ β²π₯ π΅ βΌ Ο | |
| 9 | nfdisj1 5128 | . . . . . 6 β’ β²π₯Disj π₯ β π΅ π₯ | |
| 10 | 8, 9 | nfan 1901 | . . . . 5 β’ β²π₯(π΅ βΌ Ο β§ Disj π₯ β π΅ π₯) |
| 11 | 7, 10 | nfan 1901 | . . . 4 β’ β²π₯(((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯)) |
| 12 | simp1ll 1235 | . . . . . . 7 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯) β§ π₯ β π΅) β π β (measuresβπ)) | |
| 13 | measbase 33490 | . . . . . . 7 β’ (π β (measuresβπ) β π β βͺ ran sigAlgebra) | |
| 14 | 12, 13 | syl 17 | . . . . . 6 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯) β§ π₯ β π΅) β π β βͺ ran sigAlgebra) |
| 15 | simp3 1137 | . . . . . . 7 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯) β§ π₯ β π΅) β π₯ β π΅) | |
| 16 | simp1r 1197 | . . . . . . 7 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯) β§ π₯ β π΅) β π΅ β π« π) | |
| 17 | elelpwi 4613 | . . . . . . 7 β’ ((π₯ β π΅ β§ π΅ β π« π) β π₯ β π) | |
| 18 | 15, 16, 17 | syl2anc 583 | . . . . . 6 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯) β§ π₯ β π΅) β π₯ β π) |
| 19 | simp1lr 1236 | . . . . . 6 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯) β§ π₯ β π΅) β π΄ β π) | |
| 20 | inelsiga 33428 | . . . . . 6 β’ ((π β βͺ ran sigAlgebra β§ π₯ β π β§ π΄ β π) β (π₯ β© π΄) β π) | |
| 21 | 14, 18, 19, 20 | syl3anc 1370 | . . . . 5 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯) β§ π₯ β π΅) β (π₯ β© π΄) β π) |
| 22 | 21 | 3expia 1120 | . . . 4 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯)) β (π₯ β π΅ β (π₯ β© π΄) β π)) |
| 23 | 11, 22 | ralrimi 3253 | . . 3 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯)) β βπ₯ β π΅ (π₯ β© π΄) β π) |
| 24 | simprl 768 | . . 3 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯)) β π΅ βΌ Ο) | |
| 25 | disjin 32081 | . . . 4 β’ (Disj π₯ β π΅ π₯ β Disj π₯ β π΅ (π₯ β© π΄)) | |
| 26 | 25 | ad2antll 726 | . . 3 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯)) β Disj π₯ β π΅ (π₯ β© π΄)) |
| 27 | measvuni 33507 | . . 3 β’ ((π β (measuresβπ) β§ βπ₯ β π΅ (π₯ β© π΄) β π β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ (π₯ β© π΄))) β (πββͺ π₯ β π΅ (π₯ β© π΄)) = Ξ£*π₯ β π΅(πβ(π₯ β© π΄))) | |
| 28 | 6, 23, 24, 26, 27 | syl112anc 1373 | . 2 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯)) β (πββͺ π₯ β π΅ (π₯ β© π΄)) = Ξ£*π₯ β π΅(πβ(π₯ β© π΄))) |
| 29 | 5, 28 | eqtr3id 2785 | 1 β’ ((((π β (measuresβπ) β§ π΄ β π) β§ π΅ β π« π) β§ (π΅ βΌ Ο β§ Disj π₯ β π΅ π₯)) β (πβ(βͺ π΅ β© π΄)) = Ξ£*π₯ β π΅(πβ(π₯ β© π΄))) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βwral 3060 β© cin 3948 π« cpw 4603 βͺ cuni 4909 βͺ ciun 4998 Disj wdisj 5114 class class class wbr 5149 ran crn 5678 βcfv 6544 Οcom 7858 βΌ cdom 8940 Ξ£*cesum 33320 sigAlgebracsiga 33401 measurescmeas 33488 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-ac2 10461 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-dju 9899 df-card 9937 df-acn 9940 df-ac 10114 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-sin 16018 df-cos 16019 df-pi 16021 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-ordt 17452 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-ps 18524 df-tsr 18525 df-plusf 18565 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-mulg 18988 df-subg 19040 df-cntz 19223 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-subrng 20435 df-subrg 20460 df-abv 20569 df-lmod 20617 df-scaf 20618 df-sra 20931 df-rgmod 20932 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-tmd 23797 df-tgp 23798 df-tsms 23852 df-trg 23885 df-xms 24047 df-ms 24048 df-tms 24049 df-nm 24312 df-ngp 24313 df-nrg 24315 df-nlm 24316 df-ii 24618 df-cncf 24619 df-limc 25616 df-dv 25617 df-log 26298 df-esum 33321 df-siga 33402 df-meas 33489 |
| This theorem is referenced by: measinb 33514 |
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