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Mirrors > Home > MPE Home > Th. List > cnicciblnc | Structured version Visualization version GIF version |
Description: Choice-free proof of cniccibl 25720. (Contributed by Brendan Leahy, 2-Nov-2017.) |
Ref | Expression |
---|---|
cnicciblnc | β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β πΏ1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccmbl 25445 | . . 3 β’ ((π΄ β β β§ π΅ β β) β (π΄[,]π΅) β dom vol) | |
2 | cnmbf 25538 | . . 3 β’ (((π΄[,]π΅) β dom vol β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β MblFn) | |
3 | 1, 2 | stoic3 1770 | . 2 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β MblFn) |
4 | simp3 1135 | . . . . 5 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β ((π΄[,]π΅)βcnββ)) | |
5 | cncff 24763 | . . . . 5 β’ (πΉ β ((π΄[,]π΅)βcnββ) β πΉ:(π΄[,]π΅)βΆβ) | |
6 | fdm 6719 | . . . . 5 β’ (πΉ:(π΄[,]π΅)βΆβ β dom πΉ = (π΄[,]π΅)) | |
7 | 4, 5, 6 | 3syl 18 | . . . 4 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β dom πΉ = (π΄[,]π΅)) |
8 | 7 | fveq2d 6888 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (volβdom πΉ) = (volβ(π΄[,]π΅))) |
9 | iccvolcl 25446 | . . . 4 β’ ((π΄ β β β§ π΅ β β) β (volβ(π΄[,]π΅)) β β) | |
10 | 9 | 3adant3 1129 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (volβ(π΄[,]π΅)) β β) |
11 | 8, 10 | eqeltrd 2827 | . 2 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (volβdom πΉ) β β) |
12 | cniccbdd 25340 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β βπ₯ β β βπ¦ β (π΄[,]π΅)(absβ(πΉβπ¦)) β€ π₯) | |
13 | 7 | raleqdv 3319 | . . . 4 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯ β βπ¦ β (π΄[,]π΅)(absβ(πΉβπ¦)) β€ π₯)) |
14 | 13 | rexbidv 3172 | . . 3 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β (βπ₯ β β βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯ β βπ₯ β β βπ¦ β (π΄[,]π΅)(absβ(πΉβπ¦)) β€ π₯)) |
15 | 12, 14 | mpbird 257 | . 2 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β βπ₯ β β βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯) |
16 | bddiblnc 25721 | . 2 β’ ((πΉ β MblFn β§ (volβdom πΉ) β β β§ βπ₯ β β βπ¦ β dom πΉ(absβ(πΉβπ¦)) β€ π₯) β πΉ β πΏ1) | |
17 | 3, 11, 15, 16 | syl3anc 1368 | 1 β’ ((π΄ β β β§ π΅ β β β§ πΉ β ((π΄[,]π΅)βcnββ)) β πΉ β πΏ1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ w3a 1084 = wceq 1533 β wcel 2098 βwral 3055 βwrex 3064 class class class wbr 5141 dom cdm 5669 βΆwf 6532 βcfv 6536 (class class class)co 7404 βcc 11107 βcr 11108 β€ cle 11250 [,]cicc 13330 abscabs 15184 βcnβccncf 24746 volcvol 25342 MblFncmbf 25493 πΏ1cibl 25496 |
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 ax-addf 11188 |
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-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 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-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-dju 9895 df-card 9933 df-acn 9936 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-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 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-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-hom 17227 df-cco 17228 df-rest 17374 df-topn 17375 df-0g 17393 df-gsum 17394 df-topgen 17395 df-pt 17396 df-prds 17399 df-xrs 17454 df-qtop 17459 df-imas 17460 df-xps 17462 df-mre 17536 df-mrc 17537 df-acs 17539 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-submnd 18711 df-mulg 18993 df-cntz 19230 df-cmn 19699 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-cnfld 21236 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-cn 23081 df-cnp 23082 df-cmp 23241 df-tx 23416 df-hmeo 23609 df-xms 24176 df-ms 24177 df-tms 24178 df-cncf 24748 df-ovol 25343 df-vol 25344 df-mbf 25498 df-itg1 25499 df-itg2 25500 df-ibl 25501 df-0p 25549 |
This theorem is referenced by: itgpowd 25935 areacirclem3 37090 3factsumint1 41401 3factsumint3 41403 lcmineqlem10 41418 lcmineqlem12 41420 |
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