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Mirrors > Home > MPE Home > Th. List > volf | Structured version Visualization version Unicode version |
Description: The domain and range of the Lebesgue measure function. (Contributed by Mario Carneiro, 19-Mar-2014.) |
Ref | Expression |
---|---|
volf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovolf 22435 |
. . . . . 6
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2 | ffun 5731 |
. . . . . 6
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3 | funres 5621 |
. . . . . 6
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4 | 1, 2, 3 | mp2b 10 |
. . . . 5
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5 | volres 22482 |
. . . . . 6
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6 | 5 | funeqi 5602 |
. . . . 5
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7 | 4, 6 | mpbir 213 |
. . . 4
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8 | resss 5128 |
. . . . . 6
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9 | 5, 8 | eqsstri 3462 |
. . . . 5
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10 | fssxp 5741 |
. . . . . 6
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11 | 1, 10 | ax-mp 5 |
. . . . 5
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12 | 9, 11 | sstri 3441 |
. . . 4
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13 | 7, 12 | pm3.2i 457 |
. . 3
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14 | funssxp 5742 |
. . 3
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15 | 13, 14 | mpbi 212 |
. 2
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16 | 15 | simpli 460 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-pre-sup 9617 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-iun 4280 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-er 7363 df-map 7474 df-en 7570 df-dom 7571 df-sdom 7572 df-sup 7956 df-inf 7957 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-div 10270 df-nn 10610 df-2 10668 df-3 10669 df-n0 10870 df-z 10938 df-uz 11160 df-rp 11303 df-ico 11641 df-icc 11642 df-fz 11785 df-seq 12214 df-exp 12273 df-cj 13162 df-re 13163 df-im 13164 df-sqrt 13298 df-abs 13299 df-ovol 22416 df-vol 22418 |
This theorem is referenced by: volsup 22509 volsup2 22563 volivth 22565 itg1climres 22672 itg2const2 22699 itg2gt0 22718 areambl 23884 voliune 29052 volfiniune 29053 volmeas 29054 volsupnfl 31985 areacirc 32037 arearect 36100 areaquad 36101 fourierdlem87 38057 hoidmv1lelem1 38413 hoidmv1lelem2 38414 hoidmv1lelem3 38415 |
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