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Mirrors > Home > MPE Home > Th. List > ftc1lem2 | Structured version Visualization version GIF version |
Description: Lemma for ftc1 25409. (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 6858 | . . 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 482 | . . . . . . 7 β’ ((π β§ π₯ β (π΄[,]π΅)) β π΅ β β) |
4 | 3 | rexrd 11206 | . . . . . 6 β’ ((π β§ π₯ β (π΄[,]π΅)) β π΅ β β*) |
5 | ftc1.a | . . . . . . . . 9 β’ (π β π΄ β β) | |
6 | elicc2 13330 | . . . . . . . . 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 585 | . . . . . . . 8 β’ (π β (π₯ β (π΄[,]π΅) β (π₯ β β β§ π΄ β€ π₯ β§ π₯ β€ π΅))) |
8 | 7 | biimpa 478 | . . . . . . 7 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π₯ β β β§ π΄ β€ π₯ β§ π₯ β€ π΅)) |
9 | 8 | simp3d 1145 | . . . . . 6 β’ ((π β§ π₯ β (π΄[,]π΅)) β π₯ β€ π΅) |
10 | iooss2 13301 | . . . . . 6 β’ ((π΅ β β* β§ π₯ β€ π΅) β (π΄(,)π₯) β (π΄(,)π΅)) | |
11 | 4, 9, 10 | syl2anc 585 | . . . . 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 482 | . . . . 5 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π΄(,)π΅) β π·) |
14 | 11, 13 | sstrd 3955 | . . . 4 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π΄(,)π₯) β π·) |
15 | ioombl 24932 | . . . . 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 6858 | . . . 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 6911 | . . . . . 6 β’ (π β πΉ = (π‘ β π· β¦ (πΉβπ‘))) |
20 | ftc1.i | . . . . . 6 β’ (π β πΉ β πΏ1) | |
21 | 19, 20 | eqeltrrd 2839 | . . . . 5 β’ (π β (π‘ β π· β¦ (πΉβπ‘)) β πΏ1) |
22 | 21 | adantr 482 | . . . 4 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π‘ β π· β¦ (πΉβπ‘)) β πΏ1) |
23 | 14, 16, 17, 22 | iblss 25172 | . . 3 β’ ((π β§ π₯ β (π΄[,]π΅)) β (π‘ β (π΄(,)π₯) β¦ (πΉβπ‘)) β πΏ1) |
24 | 1, 23 | itgcl 25151 | . 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 7063 | 1 β’ (π β πΊ:(π΄[,]π΅)βΆβ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3446 β wss 3911 class class class wbr 5106 β¦ cmpt 5189 dom cdm 5634 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcc 11050 βcr 11051 β*cxr 11189 β€ cle 11191 (,)cioo 13265 [,]cicc 13268 volcvol 24830 πΏ1cibl 24984 β«citg 24985 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-inf2 9578 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 ax-pre-sup 11130 |
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 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-se 5590 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7618 df-ofr 7619 df-om 7804 df-1st 7922 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-2o 8414 df-er 8649 df-map 8768 df-pm 8769 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-sup 9379 df-inf 9380 df-oi 9447 df-dju 9838 df-card 9876 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-div 11814 df-nn 12155 df-2 12217 df-3 12218 df-4 12219 df-n0 12415 df-z 12501 df-uz 12765 df-q 12875 df-rp 12917 df-xadd 13035 df-ioo 13269 df-ico 13271 df-icc 13272 df-fz 13426 df-fzo 13569 df-fl 13698 df-mod 13776 df-seq 13908 df-exp 13969 df-hash 14232 df-cj 14985 df-re 14986 df-im 14987 df-sqrt 15121 df-abs 15122 df-clim 15371 df-rlim 15372 df-sum 15572 df-xmet 20792 df-met 20793 df-ovol 24831 df-vol 24832 df-mbf 24986 df-itg1 24987 df-itg2 24988 df-ibl 24989 df-itg 24990 |
This theorem is referenced by: ftc1a 25404 ftc1lem5 25407 ftc1lem6 25408 ftc1 25409 ftc1cn 25410 ftc1cnnc 36153 ftc1anc 36162 |
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