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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > volicoff | Structured version Visualization version GIF version |
Description: ((vol β [,)) β πΉ) expressed in maps-to notation. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
volicoff.1 | β’ (π β πΉ:π΄βΆ(β Γ β*)) |
Ref | Expression |
---|---|
volicoff | β’ (π β ((vol β [,)) β πΉ):π΄βΆ(0[,]+β)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | volf 25478 | . . . 4 β’ vol:dom volβΆ(0[,]+β) | |
2 | 1 | a1i 11 | . . 3 β’ (π β vol:dom volβΆ(0[,]+β)) |
3 | icof 44622 | . . . . . . 7 β’ [,):(β* Γ β*)βΆπ« β* | |
4 | 3 | a1i 11 | . . . . . 6 β’ (π β [,):(β* Γ β*)βΆπ« β*) |
5 | ressxr 11296 | . . . . . . . 8 β’ β β β* | |
6 | xpss1 5701 | . . . . . . . 8 β’ (β β β* β (β Γ β*) β (β* Γ β*)) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 β’ (β Γ β*) β (β* Γ β*) |
8 | 7 | a1i 11 | . . . . . 6 β’ (π β (β Γ β*) β (β* Γ β*)) |
9 | volicoff.1 | . . . . . 6 β’ (π β πΉ:π΄βΆ(β Γ β*)) | |
10 | 4, 8, 9 | fcoss 44613 | . . . . 5 β’ (π β ([,) β πΉ):π΄βΆπ« β*) |
11 | 10 | ffnd 6728 | . . . 4 β’ (π β ([,) β πΉ) Fn π΄) |
12 | 9 | adantr 479 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β πΉ:π΄βΆ(β Γ β*)) |
13 | simpr 483 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β π₯ β π΄) | |
14 | 12, 13 | fvovco 44596 | . . . . . 6 β’ ((π β§ π₯ β π΄) β (([,) β πΉ)βπ₯) = ((1st β(πΉβπ₯))[,)(2nd β(πΉβπ₯)))) |
15 | 9 | ffvelcdmda 7099 | . . . . . . . 8 β’ ((π β§ π₯ β π΄) β (πΉβπ₯) β (β Γ β*)) |
16 | xp1st 8031 | . . . . . . . 8 β’ ((πΉβπ₯) β (β Γ β*) β (1st β(πΉβπ₯)) β β) | |
17 | 15, 16 | syl 17 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β (1st β(πΉβπ₯)) β β) |
18 | xp2nd 8032 | . . . . . . . 8 β’ ((πΉβπ₯) β (β Γ β*) β (2nd β(πΉβπ₯)) β β*) | |
19 | 15, 18 | syl 17 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β (2nd β(πΉβπ₯)) β β*) |
20 | icombl 25513 | . . . . . . 7 β’ (((1st β(πΉβπ₯)) β β β§ (2nd β(πΉβπ₯)) β β*) β ((1st β(πΉβπ₯))[,)(2nd β(πΉβπ₯))) β dom vol) | |
21 | 17, 19, 20 | syl2anc 582 | . . . . . 6 β’ ((π β§ π₯ β π΄) β ((1st β(πΉβπ₯))[,)(2nd β(πΉβπ₯))) β dom vol) |
22 | 14, 21 | eqeltrd 2829 | . . . . 5 β’ ((π β§ π₯ β π΄) β (([,) β πΉ)βπ₯) β dom vol) |
23 | 22 | ralrimiva 3143 | . . . 4 β’ (π β βπ₯ β π΄ (([,) β πΉ)βπ₯) β dom vol) |
24 | fnfvrnss 7136 | . . . 4 β’ ((([,) β πΉ) Fn π΄ β§ βπ₯ β π΄ (([,) β πΉ)βπ₯) β dom vol) β ran ([,) β πΉ) β dom vol) | |
25 | 11, 23, 24 | syl2anc 582 | . . 3 β’ (π β ran ([,) β πΉ) β dom vol) |
26 | ffrn 6741 | . . . 4 β’ (([,) β πΉ):π΄βΆπ« β* β ([,) β πΉ):π΄βΆran ([,) β πΉ)) | |
27 | 10, 26 | syl 17 | . . 3 β’ (π β ([,) β πΉ):π΄βΆran ([,) β πΉ)) |
28 | 2, 25, 27 | fcoss 44613 | . 2 β’ (π β (vol β ([,) β πΉ)):π΄βΆ(0[,]+β)) |
29 | coass 6274 | . . . 4 β’ ((vol β [,)) β πΉ) = (vol β ([,) β πΉ)) | |
30 | 29 | feq1i 6718 | . . 3 β’ (((vol β [,)) β πΉ):π΄βΆ(0[,]+β) β (vol β ([,) β πΉ)):π΄βΆ(0[,]+β)) |
31 | 30 | a1i 11 | . 2 β’ (π β (((vol β [,)) β πΉ):π΄βΆ(0[,]+β) β (vol β ([,) β πΉ)):π΄βΆ(0[,]+β))) |
32 | 28, 31 | mpbird 256 | 1 β’ (π β ((vol β [,)) β πΉ):π΄βΆ(0[,]+β)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 394 β wcel 2098 βwral 3058 β wss 3949 π« cpw 4606 Γ cxp 5680 dom cdm 5682 ran crn 5683 β ccom 5686 Fn wfn 6548 βΆwf 6549 βcfv 6553 (class class class)co 7426 1st c1st 7997 2nd c2nd 7998 βcr 11145 0cc0 11146 +βcpnf 11283 β*cxr 11285 [,)cico 13366 [,]cicc 13367 volcvol 25412 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 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 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-q 12971 df-rp 13015 df-xadd 13133 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-rlim 15473 df-sum 15673 df-xmet 21279 df-met 21280 df-ovol 25413 df-vol 25414 |
This theorem is referenced by: volicofmpt 45414 |
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