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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 25408 | . . . 4 β’ vol:dom volβΆ(0[,]+β) | |
2 | 1 | a1i 11 | . . 3 β’ (π β vol:dom volβΆ(0[,]+β)) |
3 | icof 44472 | . . . . . . 7 β’ [,):(β* Γ β*)βΆπ« β* | |
4 | 3 | a1i 11 | . . . . . 6 β’ (π β [,):(β* Γ β*)βΆπ« β*) |
5 | ressxr 11259 | . . . . . . . 8 β’ β β β* | |
6 | xpss1 5688 | . . . . . . . 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 44463 | . . . . 5 β’ (π β ([,) β πΉ):π΄βΆπ« β*) |
11 | 10 | ffnd 6711 | . . . 4 β’ (π β ([,) β πΉ) Fn π΄) |
12 | 9 | adantr 480 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β πΉ:π΄βΆ(β Γ β*)) |
13 | simpr 484 | . . . . . . 7 β’ ((π β§ π₯ β π΄) β π₯ β π΄) | |
14 | 12, 13 | fvovco 44446 | . . . . . 6 β’ ((π β§ π₯ β π΄) β (([,) β πΉ)βπ₯) = ((1st β(πΉβπ₯))[,)(2nd β(πΉβπ₯)))) |
15 | 9 | ffvelcdmda 7079 | . . . . . . . 8 β’ ((π β§ π₯ β π΄) β (πΉβπ₯) β (β Γ β*)) |
16 | xp1st 8003 | . . . . . . . 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 8004 | . . . . . . . 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 25443 | . . . . . . 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 583 | . . . . . 6 β’ ((π β§ π₯ β π΄) β ((1st β(πΉβπ₯))[,)(2nd β(πΉβπ₯))) β dom vol) |
22 | 14, 21 | eqeltrd 2827 | . . . . 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 7115 | . . . 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 583 | . . 3 β’ (π β ran ([,) β πΉ) β dom vol) |
26 | ffrn 6724 | . . . 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 44463 | . 2 β’ (π β (vol β ([,) β πΉ)):π΄βΆ(0[,]+β)) |
29 | coass 6257 | . . . 4 β’ ((vol β [,)) β πΉ) = (vol β ([,) β πΉ)) | |
30 | 29 | feq1i 6701 | . . 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 395 β wcel 2098 βwral 3055 β wss 3943 π« cpw 4597 Γ cxp 5667 dom cdm 5669 ran crn 5670 β ccom 5673 Fn wfn 6531 βΆwf 6532 βcfv 6536 (class class class)co 7404 1st c1st 7969 2nd c2nd 7970 βcr 11108 0cc0 11109 +βcpnf 11246 β*cxr 11248 [,)cico 13329 [,]cicc 13330 volcvol 25342 |
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-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-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-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 |
This theorem is referenced by: volicofmpt 45267 |
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