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Mirrors > Home > ILE Home > Th. List > addlelt | Unicode version |
Description: If the sum of a real number and a positive real number is less than or equal to a third real number, the first real number is less than the third real number. (Contributed by AV, 1-Jul-2021.) |
Ref | Expression |
---|---|
addlelt |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpgt0 9622 | . . . 4 | |
2 | 1 | 3ad2ant3 1015 | . . 3 |
3 | rpre 9617 | . . . . 5 | |
4 | 3 | 3ad2ant3 1015 | . . . 4 |
5 | simp1 992 | . . . 4 | |
6 | 4, 5 | ltaddposd 8448 | . . 3 |
7 | 2, 6 | mpbid 146 | . 2 |
8 | simpl 108 | . . . . 5 | |
9 | 3 | adantl 275 | . . . . 5 |
10 | 8, 9 | readdcld 7949 | . . . 4 |
11 | 10 | 3adant2 1011 | . . 3 |
12 | simp2 993 | . . 3 | |
13 | ltletr 8009 | . . 3 | |
14 | 5, 11, 12, 13 | syl3anc 1233 | . 2 |
15 | 7, 14 | mpand 427 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wcel 2141 class class class wbr 3989 (class class class)co 5853 cr 7773 cc0 7774 caddc 7777 clt 7954 cle 7955 crp 9610 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-i2m1 7879 ax-0id 7882 ax-rnegex 7883 ax-pre-ltwlin 7887 ax-pre-ltadd 7890 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-cnv 4619 df-iota 5160 df-fv 5206 df-ov 5856 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-rp 9611 |
This theorem is referenced by: zltaddlt1le 9964 |
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