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Mirrors > Home > ILE Home > Th. List > max0addsup | Unicode version |
Description: The sum of the positive and negative part functions is the absolute value function over the reals. (Contributed by Jim Kingdon, 30-Jan-2022.) |
Ref | Expression |
---|---|
max0addsup |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 7790 |
. . . . . 6
![]() ![]() ![]() ![]() | |
2 | maxabs 11013 |
. . . . . 6
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3 | 1, 2 | mpan2 422 |
. . . . 5
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4 | recn 7777 |
. . . . . . . 8
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5 | 4 | addid1d 7935 |
. . . . . . 7
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6 | 4 | subid1d 8086 |
. . . . . . . 8
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7 | 6 | fveq2d 5433 |
. . . . . . 7
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8 | 5, 7 | oveq12d 5800 |
. . . . . 6
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9 | 8 | oveq1d 5797 |
. . . . 5
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10 | 3, 9 | eqtrd 2173 |
. . . 4
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11 | renegcl 8047 |
. . . . . 6
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12 | maxabs 11013 |
. . . . . 6
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13 | 11, 1, 12 | sylancl 410 |
. . . . 5
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14 | 11 | recnd 7818 |
. . . . . . . 8
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15 | 14 | addid1d 7935 |
. . . . . . 7
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16 | 14 | subid1d 8086 |
. . . . . . . . 9
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17 | 16 | fveq2d 5433 |
. . . . . . . 8
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18 | 4 | absnegd 10993 |
. . . . . . . 8
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19 | 17, 18 | eqtrd 2173 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 15, 19 | oveq12d 5800 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
21 | 20 | oveq1d 5797 |
. . . . 5
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22 | 13, 21 | eqtrd 2173 |
. . . 4
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23 | 10, 22 | oveq12d 5800 |
. . 3
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24 | 4 | abscld 10985 |
. . . . . 6
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25 | 24 | recnd 7818 |
. . . . 5
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26 | 4, 25 | addcld 7809 |
. . . 4
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27 | 14, 25 | addcld 7809 |
. . . 4
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28 | 2cnd 8817 |
. . . 4
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29 | 2ap0 8837 |
. . . . 5
![]() ![]() ![]() | |
30 | 29 | a1i 9 |
. . . 4
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31 | 26, 27, 28, 30 | divdirapd 8613 |
. . 3
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32 | 4, 25, 14, 25 | add4d 7955 |
. . . . 5
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33 | 4 | negidd 8087 |
. . . . . 6
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34 | 33 | oveq1d 5797 |
. . . . 5
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35 | 25, 25 | addcld 7809 |
. . . . . 6
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36 | 35 | addid2d 7936 |
. . . . 5
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37 | 32, 34, 36 | 3eqtrd 2177 |
. . . 4
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38 | 37 | oveq1d 5797 |
. . 3
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39 | 23, 31, 38 | 3eqtr2d 2179 |
. 2
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40 | 25 | 2timesd 8986 |
. . 3
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41 | 40 | oveq1d 5797 |
. 2
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42 | 25, 28, 30 | divcanap3d 8579 |
. 2
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43 | 39, 41, 42 | 3eqtr2d 2179 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-sup 6879 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-rp 9471 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 |
This theorem is referenced by: (None) |
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