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Mirrors > Home > ILE Home > Th. List > ltleadd | Unicode version |
Description: Adding both sides of two orderings. (Contributed by NM, 23-Dec-2007.) |
Ref | Expression |
---|---|
ltleadd |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltadd1 8184 | . . . . . 6 | |
2 | 1 | 3com23 1187 | . . . . 5 |
3 | 2 | 3expa 1181 | . . . 4 |
4 | 3 | adantrr 470 | . . 3 |
5 | leadd2 8186 | . . . . . 6 | |
6 | 5 | 3com23 1187 | . . . . 5 |
7 | 6 | 3expb 1182 | . . . 4 |
8 | 7 | adantll 467 | . . 3 |
9 | 4, 8 | anbi12d 464 | . 2 |
10 | readdcl 7739 | . . . 4 | |
11 | 10 | adantr 274 | . . 3 |
12 | readdcl 7739 | . . . . 5 | |
13 | 12 | ancoms 266 | . . . 4 |
14 | 13 | ad2ant2lr 501 | . . 3 |
15 | readdcl 7739 | . . . 4 | |
16 | 15 | adantl 275 | . . 3 |
17 | ltletr 7846 | . . 3 | |
18 | 11, 14, 16, 17 | syl3anc 1216 | . 2 |
19 | 9, 18 | sylbid 149 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 1480 class class class wbr 3924 (class class class)co 5767 cr 7612 caddc 7616 clt 7793 cle 7794 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-pre-ltwlin 7726 ax-pre-ltadd 7729 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-iota 5083 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 |
This theorem is referenced by: leltadd 8202 addgtge0 8205 ltleaddd 8320 |
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