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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 24916 | . . . 4 β’ vol:dom volβΆ(0[,]+β) | |
2 | 1 | a1i 11 | . . 3 β’ (π β vol:dom volβΆ(0[,]+β)) |
3 | icof 43531 | . . . . . . 7 β’ [,):(β* Γ β*)βΆπ« β* | |
4 | 3 | a1i 11 | . . . . . 6 β’ (π β [,):(β* Γ β*)βΆπ« β*) |
5 | ressxr 11207 | . . . . . . . 8 β’ β β β* | |
6 | xpss1 5656 | . . . . . . . 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 43522 | . . . . 5 β’ (π β ([,) β πΉ):π΄βΆπ« β*) |
11 | 10 | ffnd 6673 | . . . 4 β’ (π β ([,) β πΉ) Fn π΄) |
12 | 9 | adantr 482 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β πΉ:π΄βΆ(β Γ β*)) |
13 | simpr 486 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β π₯ β π΄) | |
14 | 12, 13 | fvovco 43505 | . . . . . 6 β’ ((π β§ π₯ β π΄) β (([,) β πΉ)βπ₯) = ((1st β(πΉβπ₯))[,)(2nd β(πΉβπ₯)))) |
15 | 9 | ffvelcdmda 7039 | . . . . . . . 8 β’ ((π β§ π₯ β π΄) β (πΉβπ₯) β (β Γ β*)) |
16 | xp1st 7957 | . . . . . . . 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 7958 | . . . . . . . 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 24951 | . . . . . . 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 585 | . . . . . 6 β’ ((π β§ π₯ β π΄) β ((1st β(πΉβπ₯))[,)(2nd β(πΉβπ₯))) β dom vol) |
22 | 14, 21 | eqeltrd 2834 | . . . . 5 β’ ((π β§ π₯ β π΄) β (([,) β πΉ)βπ₯) β dom vol) |
23 | 22 | ralrimiva 3140 | . . . 4 β’ (π β βπ₯ β π΄ (([,) β πΉ)βπ₯) β dom vol) |
24 | fnfvrnss 7072 | . . . 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 585 | . . 3 β’ (π β ran ([,) β πΉ) β dom vol) |
26 | ffrn 6686 | . . . 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 43522 | . 2 β’ (π β (vol β ([,) β πΉ)):π΄βΆ(0[,]+β)) |
29 | coass 6221 | . . . 4 β’ ((vol β [,)) β πΉ) = (vol β ([,) β πΉ)) | |
30 | 29 | feq1i 6663 | . . 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 257 | 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 397 β wcel 2107 βwral 3061 β wss 3914 π« cpw 4564 Γ cxp 5635 dom cdm 5637 ran crn 5638 β ccom 5641 Fn wfn 6495 βΆwf 6496 βcfv 6500 (class class class)co 7361 1st c1st 7923 2nd c2nd 7924 βcr 11058 0cc0 11059 +βcpnf 11194 β*cxr 11196 [,)cico 13275 [,]cicc 13276 volcvol 24850 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-inf2 9585 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-pm 8774 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-sup 9386 df-inf 9387 df-oi 9454 df-dju 9845 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-q 12882 df-rp 12924 df-xadd 13042 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-fl 13706 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-clim 15379 df-rlim 15380 df-sum 15580 df-xmet 20812 df-met 20813 df-ovol 24851 df-vol 24852 |
This theorem is referenced by: volicofmpt 44328 |
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