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Mirrors > Home > MPE Home > Th. List > ftc1lem2 | Structured version Visualization version GIF version |
Description: Lemma for ftc1 25927. (Contributed by Mario Carneiro, 12-Aug-2014.) |
Ref | Expression |
---|---|
ftc1.g | β’ πΊ = (π₯ β (π΄[,]π΅) β¦ β«(π΄(,)π₯)(πΉβπ‘) dπ‘) |
ftc1.a | β’ (π β π΄ β β) |
ftc1.b | β’ (π β π΅ β β) |
ftc1.le | β’ (π β π΄ β€ π΅) |
ftc1.s | β’ (π β (π΄(,)π΅) β π·) |
ftc1.d | β’ (π β π· β β) |
ftc1.i | β’ (π β πΉ β πΏ1) |
ftc1a.f | β’ (π β πΉ:π·βΆβ) |
Ref | Expression |
---|---|
ftc1lem2 | β’ (π β πΊ:(π΄[,]π΅)βΆβ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvexd 6899 | . . 3 β’ (((π β§ π₯ β (π΄[,]π΅)) β§ π‘ β (π΄(,)π₯)) β (πΉβπ‘) β V) | |
2 | ftc1.b | . . . . . . . 8 β’ (π β π΅ β β) | |
3 | 2 | adantr 480 | . . . . . . 7 β’ ((π β§ π₯ β (π΄[,]π΅)) β π΅ β β) |
4 | 3 | rexrd 11265 | . . . . . 6 β’ ((π β§ π₯ β (π΄[,]π΅)) β π΅ β β*) |
5 | ftc1.a | . . . . . . . . 9 β’ (π β π΄ β β) | |
6 | elicc2 13392 | . . . . . . . . 9 β’ ((π΄ β β β§ π΅ β β) β (π₯ β (π΄[,]π΅) β (π₯ β β β§ π΄ β€ π₯ β§ π₯ β€ π΅))) | |
7 | 5, 2, 6 | syl2anc 583 | . . . . . . . 8 β’ (π β (π₯ β (π΄[,]π΅) β (π₯ β β β§ π΄ β€ π₯ β§ π₯ β€ π΅))) |
8 | 7 | biimpa 476 | . . . . . . 7 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π₯ β β β§ π΄ β€ π₯ β§ π₯ β€ π΅)) |
9 | 8 | simp3d 1141 | . . . . . 6 β’ ((π β§ π₯ β (π΄[,]π΅)) β π₯ β€ π΅) |
10 | iooss2 13363 | . . . . . 6 β’ ((π΅ β β* β§ π₯ β€ π΅) β (π΄(,)π₯) β (π΄(,)π΅)) | |
11 | 4, 9, 10 | syl2anc 583 | . . . . 5 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π΄(,)π₯) β (π΄(,)π΅)) |
12 | ftc1.s | . . . . . 6 β’ (π β (π΄(,)π΅) β π·) | |
13 | 12 | adantr 480 | . . . . 5 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π΄(,)π΅) β π·) |
14 | 11, 13 | sstrd 3987 | . . . 4 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π΄(,)π₯) β π·) |
15 | ioombl 25444 | . . . . 5 β’ (π΄(,)π₯) β dom vol | |
16 | 15 | a1i 11 | . . . 4 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π΄(,)π₯) β dom vol) |
17 | fvexd 6899 | . . . 4 β’ (((π β§ π₯ β (π΄[,]π΅)) β§ π‘ β π·) β (πΉβπ‘) β V) | |
18 | ftc1a.f | . . . . . . 7 β’ (π β πΉ:π·βΆβ) | |
19 | 18 | feqmptd 6953 | . . . . . 6 β’ (π β πΉ = (π‘ β π· β¦ (πΉβπ‘))) |
20 | ftc1.i | . . . . . 6 β’ (π β πΉ β πΏ1) | |
21 | 19, 20 | eqeltrrd 2828 | . . . . 5 β’ (π β (π‘ β π· β¦ (πΉβπ‘)) β πΏ1) |
22 | 21 | adantr 480 | . . . 4 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π‘ β π· β¦ (πΉβπ‘)) β πΏ1) |
23 | 14, 16, 17, 22 | iblss 25684 | . . 3 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π‘ β (π΄(,)π₯) β¦ (πΉβπ‘)) β πΏ1) |
24 | 1, 23 | itgcl 25663 | . 2 β’ ((π β§ π₯ β (π΄[,]π΅)) β β«(π΄(,)π₯)(πΉβπ‘) dπ‘ β β) |
25 | ftc1.g | . 2 β’ πΊ = (π₯ β (π΄[,]π΅) β¦ β«(π΄(,)π₯)(πΉβπ‘) dπ‘) | |
26 | 24, 25 | fmptd 7108 | 1 β’ (π β πΊ:(π΄[,]π΅)βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 class class class wbr 5141 β¦ cmpt 5224 dom cdm 5669 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcc 11107 βcr 11108 β*cxr 11248 β€ cle 11250 (,)cioo 13327 [,]cicc 13330 volcvol 25342 πΏ1cibl 25496 β«citg 25497 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-er 8702 df-map 8821 df-pm 8822 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-n0 12474 df-z 12560 df-uz 12824 df-q 12934 df-rp 12978 df-xadd 13096 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-rlim 15436 df-sum 15636 df-xmet 21228 df-met 21229 df-ovol 25343 df-vol 25344 df-mbf 25498 df-itg1 25499 df-itg2 25500 df-ibl 25501 df-itg 25502 |
This theorem is referenced by: ftc1a 25922 ftc1lem5 25925 ftc1lem6 25926 ftc1 25927 ftc1cn 25928 ftc1cnnc 37072 ftc1anc 37081 |
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