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Mirrors > Home > ILE Home > Th. List > div2subap | Unicode version |
Description: Swap the order of subtraction in a division. (Contributed by Scott Fenton, 24-Jun-2013.) |
Ref | Expression |
---|---|
div2subap | # |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subcl 7929 | . . 3 | |
2 | subcl 7929 | . . . . 5 | |
3 | 2 | 3adant3 986 | . . . 4 # |
4 | apneg 8341 | . . . . . . . 8 # # | |
5 | 4 | biimp3a 1308 | . . . . . . 7 # # |
6 | simp1 966 | . . . . . . . . 9 # | |
7 | 6 | negcld 8028 | . . . . . . . 8 # |
8 | simp2 967 | . . . . . . . . 9 # | |
9 | 8 | negcld 8028 | . . . . . . . 8 # |
10 | apadd2 8339 | . . . . . . . 8 # # | |
11 | 7, 9, 6, 10 | syl3anc 1201 | . . . . . . 7 # # # |
12 | 5, 11 | mpbid 146 | . . . . . 6 # # |
13 | 6 | negidd 8031 | . . . . . 6 # |
14 | 6, 8 | negsubd 8047 | . . . . . 6 # |
15 | 12, 13, 14 | 3brtr3d 3929 | . . . . 5 # # |
16 | 0cnd 7727 | . . . . . 6 # | |
17 | apsym 8336 | . . . . . 6 # # | |
18 | 16, 3, 17 | syl2anc 408 | . . . . 5 # # # |
19 | 15, 18 | mpbid 146 | . . . 4 # # |
20 | 3, 19 | jca 304 | . . 3 # # |
21 | div2negap 8463 | . . . 4 # | |
22 | 21 | 3expb 1167 | . . 3 # |
23 | 1, 20, 22 | syl2an 287 | . 2 # |
24 | negsubdi2 7989 | . . 3 | |
25 | negsubdi2 7989 | . . . 4 | |
26 | 25 | 3adant3 986 | . . 3 # |
27 | 24, 26 | oveqan12d 5761 | . 2 # |
28 | 23, 27 | eqtr3d 2152 | 1 # |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 class class class wbr 3899 (class class class)co 5742 cc 7586 cc0 7588 caddc 7591 cmin 7901 cneg 7902 # cap 8311 cdiv 8400 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8305 df-ap 8312 df-div 8401 |
This theorem is referenced by: div2subapd 8565 |
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