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Mirrors > Home > MPE Home > Th. List > itg1cl | Structured version Visualization version GIF version |
Description: Closure of the integral on simple functions. (Contributed by Mario Carneiro, 26-Jun-2014.) |
Ref | Expression |
---|---|
itg1cl | β’ (πΉ β dom β«1 β (β«1βπΉ) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | itg1val 25050 | . 2 β’ (πΉ β dom β«1 β (β«1βπΉ) = Ξ£π₯ β (ran πΉ β {0})(π₯ Β· (volβ(β‘πΉ β {π₯})))) | |
2 | i1frn 25044 | . . . 4 β’ (πΉ β dom β«1 β ran πΉ β Fin) | |
3 | difss 4092 | . . . 4 β’ (ran πΉ β {0}) β ran πΉ | |
4 | ssfi 9118 | . . . 4 β’ ((ran πΉ β Fin β§ (ran πΉ β {0}) β ran πΉ) β (ran πΉ β {0}) β Fin) | |
5 | 2, 3, 4 | sylancl 587 | . . 3 β’ (πΉ β dom β«1 β (ran πΉ β {0}) β Fin) |
6 | i1ff 25043 | . . . . . . 7 β’ (πΉ β dom β«1 β πΉ:ββΆβ) | |
7 | 6 | frnd 6677 | . . . . . 6 β’ (πΉ β dom β«1 β ran πΉ β β) |
8 | 7 | ssdifssd 4103 | . . . . 5 β’ (πΉ β dom β«1 β (ran πΉ β {0}) β β) |
9 | 8 | sselda 3945 | . . . 4 β’ ((πΉ β dom β«1 β§ π₯ β (ran πΉ β {0})) β π₯ β β) |
10 | i1fima2sn 25047 | . . . 4 β’ ((πΉ β dom β«1 β§ π₯ β (ran πΉ β {0})) β (volβ(β‘πΉ β {π₯})) β β) | |
11 | 9, 10 | remulcld 11186 | . . 3 β’ ((πΉ β dom β«1 β§ π₯ β (ran πΉ β {0})) β (π₯ Β· (volβ(β‘πΉ β {π₯}))) β β) |
12 | 5, 11 | fsumrecl 15620 | . 2 β’ (πΉ β dom β«1 β Ξ£π₯ β (ran πΉ β {0})(π₯ Β· (volβ(β‘πΉ β {π₯}))) β β) |
13 | 1, 12 | eqeltrd 2838 | 1 β’ (πΉ β dom β«1 β (β«1βπΉ) β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β wcel 2107 β cdif 3908 β wss 3911 {csn 4587 β‘ccnv 5633 dom cdm 5634 ran crn 5635 β cima 5637 βcfv 6497 (class class class)co 7358 Fincfn 8884 βcr 11051 0cc0 11052 Β· cmul 11057 Ξ£csu 15571 volcvol 24830 β«1citg1 24982 |
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-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-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-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-sum 15572 df-xmet 20792 df-met 20793 df-ovol 24831 df-vol 24832 df-mbf 24986 df-itg1 24987 |
This theorem is referenced by: itg1mulc 25072 itg1sub 25077 itg1lea 25080 itg2lcl 25095 itg2itg1 25104 itg2seq 25110 itg2uba 25111 itg2mulclem 25114 itg2splitlem 25116 itg2split 25117 itg2monolem1 25118 itg2monolem2 25119 itg2monolem3 25120 itg2i1fseq2 25124 itg2addlem 25126 i1fibl 25175 itg2addnclem 36132 itg2addnc 36135 ftc1anclem5 36158 ftc1anclem7 36160 ftc1anclem8 36161 |
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