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Mirrors > Home > ILE Home > Th. List > lt2add | Unicode version |
Description: Adding both sides of two 'less than' relations. Theorem I.25 of [Apostol] p. 20. (Contributed by NM, 15-Aug-1999.) (Proof shortened by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
lt2add |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpll 527 |
. . . 4
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2 | simprl 529 |
. . . 4
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3 | simplr 528 |
. . . 4
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4 | ltadd1 8416 |
. . . 4
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5 | 1, 2, 3, 4 | syl3anc 1249 |
. . 3
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6 | simprr 531 |
. . . 4
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7 | 3, 6, 2 | ltadd2d 8408 |
. . 3
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8 | 5, 7 | anbi12d 473 |
. 2
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9 | 1, 3 | readdcld 8017 |
. . 3
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10 | 2, 3 | readdcld 8017 |
. . 3
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11 | 2, 6 | readdcld 8017 |
. . 3
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12 | lttr 8061 |
. . 3
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13 | 9, 10, 11, 12 | syl3anc 1249 |
. 2
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14 | 8, 13 | sylbid 150 |
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 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-1cn 7934 ax-icn 7936 ax-addcl 7937 ax-addrcl 7938 ax-mulcl 7939 ax-addcom 7941 ax-addass 7943 ax-i2m1 7946 ax-0id 7949 ax-rnegex 7950 ax-pre-lttrn 7955 ax-pre-ltadd 7957 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-xp 4650 df-iota 5196 df-fv 5243 df-ov 5899 df-pnf 8024 df-mnf 8025 df-ltxr 8027 |
This theorem is referenced by: addgt0 8435 lt2addi 8497 lt2halves 9184 |
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