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Mirrors > Home > ILE Home > Th. List > ltadd2 | Unicode version |
Description: Addition to both sides of 'less than'. (Contributed by NM, 12-Nov-1999.) (Revised by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltadd2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | axltadd 7834 | . 2 | |
2 | ax-rnegex 7729 | . . . 4 | |
3 | 2 | 3ad2ant3 1004 | . . 3 |
4 | simpl3 986 | . . . . . . 7 | |
5 | simpl1 984 | . . . . . . 7 | |
6 | 4, 5 | readdcld 7795 | . . . . . 6 |
7 | simpl2 985 | . . . . . . 7 | |
8 | 4, 7 | readdcld 7795 | . . . . . 6 |
9 | simprl 520 | . . . . . 6 | |
10 | axltadd 7834 | . . . . . 6 | |
11 | 6, 8, 9, 10 | syl3anc 1216 | . . . . 5 |
12 | 9 | recnd 7794 | . . . . . . 7 |
13 | 4 | recnd 7794 | . . . . . . 7 |
14 | 5 | recnd 7794 | . . . . . . 7 |
15 | 12, 13, 14 | addassd 7788 | . . . . . 6 |
16 | 7 | recnd 7794 | . . . . . . 7 |
17 | 12, 13, 16 | addassd 7788 | . . . . . 6 |
18 | 15, 17 | breq12d 3942 | . . . . 5 |
19 | 11, 18 | sylibrd 168 | . . . 4 |
20 | simprr 521 | . . . . . . . 8 | |
21 | addcom 7899 | . . . . . . . . . 10 | |
22 | 21 | eqeq1d 2148 | . . . . . . . . 9 |
23 | 13, 12, 22 | syl2anc 408 | . . . . . . . 8 |
24 | 20, 23 | mpbid 146 | . . . . . . 7 |
25 | 24 | oveq1d 5789 | . . . . . 6 |
26 | 14 | addid2d 7912 | . . . . . 6 |
27 | 25, 26 | eqtrd 2172 | . . . . 5 |
28 | 24 | oveq1d 5789 | . . . . . 6 |
29 | 16 | addid2d 7912 | . . . . . 6 |
30 | 28, 29 | eqtrd 2172 | . . . . 5 |
31 | 27, 30 | breq12d 3942 | . . . 4 |
32 | 19, 31 | sylibd 148 | . . 3 |
33 | 3, 32 | rexlimddv 2554 | . 2 |
34 | 1, 33 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wrex 2417 class class class wbr 3929 (class class class)co 5774 cc 7618 cr 7619 cc0 7620 caddc 7623 clt 7800 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-iota 5088 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-ltxr 7805 |
This theorem is referenced by: ltadd2i 8182 ltadd2d 8183 ltaddneg 8186 ltadd1 8191 ltaddpos 8214 ltsub2 8221 ltaddsublt 8333 avglt1 8958 flqbi2 10064 |
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