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Mirrors > Home > MPE Home > Th. List > Mathboxes > 3factsumint | Structured version Visualization version GIF version |
Description: Helpful equation for lcm inequality proof. (Contributed by metakunt, 26-Apr-2024.) |
Ref | Expression |
---|---|
3factsumint.1 | β’ π΄ = (πΏ[,]π) |
3factsumint.2 | β’ (π β π΅ β Fin) |
3factsumint.3 | β’ (π β πΏ β β) |
3factsumint.4 | β’ (π β π β β) |
3factsumint.5 | β’ (π β (π₯ β π΄ β¦ πΉ) β (π΄βcnββ)) |
3factsumint.6 | β’ ((π β§ π β π΅) β πΊ β β) |
3factsumint.7 | β’ ((π β§ π β π΅) β (π₯ β π΄ β¦ π») β (π΄βcnββ)) |
Ref | Expression |
---|---|
3factsumint | β’ (π β β«π΄(πΉ Β· Ξ£π β π΅ (πΊ Β· π»)) dπ₯ = Ξ£π β π΅ (πΊ Β· β«π΄(πΉ Β· π») dπ₯)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3factsumint.2 | . . . 4 β’ (π β π΅ β Fin) | |
2 | 3factsumint.5 | . . . . . . 7 β’ (π β (π₯ β π΄ β¦ πΉ) β (π΄βcnββ)) | |
3 | cncff 24272 | . . . . . . 7 β’ ((π₯ β π΄ β¦ πΉ) β (π΄βcnββ) β (π₯ β π΄ β¦ πΉ):π΄βΆβ) | |
4 | 2, 3 | syl 17 | . . . . . 6 β’ (π β (π₯ β π΄ β¦ πΉ):π΄βΆβ) |
5 | eqid 2737 | . . . . . . 7 β’ (π₯ β π΄ β¦ πΉ) = (π₯ β π΄ β¦ πΉ) | |
6 | 5 | fmpt 7063 | . . . . . 6 β’ (βπ₯ β π΄ πΉ β β β (π₯ β π΄ β¦ πΉ):π΄βΆβ) |
7 | 4, 6 | sylibr 233 | . . . . 5 β’ (π β βπ₯ β π΄ πΉ β β) |
8 | 7 | r19.21bi 3237 | . . . 4 β’ ((π β§ π₯ β π΄) β πΉ β β) |
9 | 3factsumint.6 | . . . 4 β’ ((π β§ π β π΅) β πΊ β β) | |
10 | 3factsumint.7 | . . . . . . . 8 β’ ((π β§ π β π΅) β (π₯ β π΄ β¦ π») β (π΄βcnββ)) | |
11 | cncff 24272 | . . . . . . . 8 β’ ((π₯ β π΄ β¦ π») β (π΄βcnββ) β (π₯ β π΄ β¦ π»):π΄βΆβ) | |
12 | 10, 11 | syl 17 | . . . . . . 7 β’ ((π β§ π β π΅) β (π₯ β π΄ β¦ π»):π΄βΆβ) |
13 | eqid 2737 | . . . . . . . 8 β’ (π₯ β π΄ β¦ π») = (π₯ β π΄ β¦ π») | |
14 | 13 | fmpt 7063 | . . . . . . 7 β’ (βπ₯ β π΄ π» β β β (π₯ β π΄ β¦ π»):π΄βΆβ) |
15 | 12, 14 | sylibr 233 | . . . . . 6 β’ ((π β§ π β π΅) β βπ₯ β π΄ π» β β) |
16 | 15 | r19.21bi 3237 | . . . . 5 β’ (((π β§ π β π΅) β§ π₯ β π΄) β π» β β) |
17 | anass 470 | . . . . . . 7 β’ (((π β§ π β π΅) β§ π₯ β π΄) β (π β§ (π β π΅ β§ π₯ β π΄))) | |
18 | ancom 462 | . . . . . . . 8 β’ ((π β π΅ β§ π₯ β π΄) β (π₯ β π΄ β§ π β π΅)) | |
19 | 18 | anbi2i 624 | . . . . . . 7 β’ ((π β§ (π β π΅ β§ π₯ β π΄)) β (π β§ (π₯ β π΄ β§ π β π΅))) |
20 | 17, 19 | bitri 275 | . . . . . 6 β’ (((π β§ π β π΅) β§ π₯ β π΄) β (π β§ (π₯ β π΄ β§ π β π΅))) |
21 | 20 | imbi1i 350 | . . . . 5 β’ ((((π β§ π β π΅) β§ π₯ β π΄) β π» β β) β ((π β§ (π₯ β π΄ β§ π β π΅)) β π» β β)) |
22 | 16, 21 | mpbi 229 | . . . 4 β’ ((π β§ (π₯ β π΄ β§ π β π΅)) β π» β β) |
23 | 1, 8, 9, 22 | 3factsumint4 40510 | . . 3 β’ (π β β«π΄Ξ£π β π΅ (πΉ Β· (πΊ Β· π»)) dπ₯ = β«π΄(πΉ Β· Ξ£π β π΅ (πΊ Β· π»)) dπ₯) |
24 | 3factsumint.1 | . . . 4 β’ π΄ = (πΏ[,]π) | |
25 | 3factsumint.3 | . . . 4 β’ (π β πΏ β β) | |
26 | 3factsumint.4 | . . . 4 β’ (π β π β β) | |
27 | 24, 1, 25, 26, 8, 2, 9, 22, 10 | 3factsumint1 40507 | . . 3 β’ (π β β«π΄Ξ£π β π΅ (πΉ Β· (πΊ Β· π»)) dπ₯ = Ξ£π β π΅ β«π΄(πΉ Β· (πΊ Β· π»)) dπ₯) |
28 | 23, 27 | eqtr3d 2779 | . 2 β’ (π β β«π΄(πΉ Β· Ξ£π β π΅ (πΊ Β· π»)) dπ₯ = Ξ£π β π΅ β«π΄(πΉ Β· (πΊ Β· π»)) dπ₯) |
29 | 8, 9, 22 | 3factsumint2 40508 | . 2 β’ (π β Ξ£π β π΅ β«π΄(πΉ Β· (πΊ Β· π»)) dπ₯ = Ξ£π β π΅ β«π΄(πΊ Β· (πΉ Β· π»)) dπ₯) |
30 | 24, 25, 26, 8, 2, 9, 22, 10 | 3factsumint3 40509 | . 2 β’ (π β Ξ£π β π΅ β«π΄(πΊ Β· (πΉ Β· π»)) dπ₯ = Ξ£π β π΅ (πΊ Β· β«π΄(πΉ Β· π») dπ₯)) |
31 | 28, 29, 30 | 3eqtrd 2781 | 1 β’ (π β β«π΄(πΉ Β· Ξ£π β π΅ (πΊ Β· π»)) dπ₯ = Ξ£π β π΅ (πΊ Β· β«π΄(πΉ Β· π») dπ₯)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 βwral 3065 β¦ cmpt 5193 βΆwf 6497 (class class class)co 7362 Fincfn 8890 βcc 11056 βcr 11057 Β· cmul 11063 [,]cicc 13274 Ξ£csu 15577 βcnβccncf 24255 β«citg 24998 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-inf2 9584 ax-cc 10378 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-iin 4962 df-disj 5076 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-se 5594 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-of 7622 df-ofr 7623 df-om 7808 df-1st 7926 df-2nd 7927 df-supp 8098 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-oadd 8421 df-omul 8422 df-er 8655 df-map 8774 df-pm 8775 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9313 df-fi 9354 df-sup 9385 df-inf 9386 df-oi 9453 df-dju 9844 df-card 9882 df-acn 9885 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-q 12881 df-rp 12923 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13275 df-ioc 13276 df-ico 13277 df-icc 13278 df-fz 13432 df-fzo 13575 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-limsup 15360 df-clim 15377 df-rlim 15378 df-sum 15578 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-hom 17164 df-cco 17165 df-rest 17311 df-topn 17312 df-0g 17330 df-gsum 17331 df-topgen 17332 df-pt 17333 df-prds 17336 df-xrs 17391 df-qtop 17396 df-imas 17397 df-xps 17399 df-mre 17473 df-mrc 17474 df-acs 17476 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-submnd 18609 df-mulg 18880 df-cntz 19104 df-cmn 19571 df-psmet 20804 df-xmet 20805 df-met 20806 df-bl 20807 df-mopn 20808 df-cnfld 20813 df-top 22259 df-topon 22276 df-topsp 22298 df-bases 22312 df-cn 22594 df-cnp 22595 df-cmp 22754 df-tx 22929 df-hmeo 23122 df-xms 23689 df-ms 23690 df-tms 23691 df-cncf 24257 df-ovol 24844 df-vol 24845 df-mbf 24999 df-itg1 25000 df-itg2 25001 df-ibl 25002 df-itg 25003 df-0p 25050 |
This theorem is referenced by: lcmineqlem2 40516 |
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