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Mirrors > Home > ILE Home > Th. List > ltaprg | Unicode version |
Description: Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by Jim Kingdon, 26-Dec-2019.) |
Ref | Expression |
---|---|
ltaprg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltaprlem 6947 |
. . 3
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2 | 1 | 3ad2ant3 962 |
. 2
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3 | ltexpri 6942 |
. . . . 5
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4 | 3 | adantl 271 |
. . . 4
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5 | simpl1 942 |
. . . . . . 7
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6 | simprl 498 |
. . . . . . 7
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7 | ltaddpr 6926 |
. . . . . . 7
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8 | 5, 6, 7 | syl2anc 403 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | addassprg 6908 |
. . . . . . . . . . . 12
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10 | 9 | 3com12 1143 |
. . . . . . . . . . 11
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11 | 10 | 3expa 1139 |
. . . . . . . . . 10
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12 | 11 | adantrr 463 |
. . . . . . . . 9
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13 | simprr 499 |
. . . . . . . . 9
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14 | 12, 13 | eqtr3d 2117 |
. . . . . . . 8
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15 | 14 | 3adantl2 1096 |
. . . . . . 7
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16 | simpl3 944 |
. . . . . . . 8
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17 | addclpr 6866 |
. . . . . . . . 9
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18 | 5, 6, 17 | syl2anc 403 |
. . . . . . . 8
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19 | simpl2 943 |
. . . . . . . 8
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20 | addcanprg 6945 |
. . . . . . . 8
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21 | 16, 18, 19, 20 | syl3anc 1170 |
. . . . . . 7
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22 | 15, 21 | mpd 13 |
. . . . . 6
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23 | 8, 22 | breqtrd 3830 |
. . . . 5
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24 | 23 | adantlr 461 |
. . . 4
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25 | 4, 24 | rexlimddv 2487 |
. . 3
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26 | 25 | ex 113 |
. 2
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27 | 2, 26 | impbid 127 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3914 ax-sep 3917 ax-nul 3925 ax-pow 3969 ax-pr 3993 ax-un 4217 ax-setind 4309 ax-iinf 4358 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2613 df-sbc 2826 df-csb 2919 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-nul 3269 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-int 3658 df-iun 3701 df-br 3807 df-opab 3861 df-mpt 3862 df-tr 3897 df-eprel 4073 df-id 4077 df-po 4080 df-iso 4081 df-iord 4150 df-on 4152 df-suc 4155 df-iom 4361 df-xp 4398 df-rel 4399 df-cnv 4400 df-co 4401 df-dm 4402 df-rn 4403 df-res 4404 df-ima 4405 df-iota 4918 df-fun 4955 df-fn 4956 df-f 4957 df-f1 4958 df-fo 4959 df-f1o 4960 df-fv 4961 df-ov 5568 df-oprab 5569 df-mpt2 5570 df-1st 5820 df-2nd 5821 df-recs 5976 df-irdg 6041 df-1o 6087 df-2o 6088 df-oadd 6091 df-omul 6092 df-er 6195 df-ec 6197 df-qs 6201 df-ni 6633 df-pli 6634 df-mi 6635 df-lti 6636 df-plpq 6673 df-mpq 6674 df-enq 6676 df-nqqs 6677 df-plqqs 6678 df-mqqs 6679 df-1nqqs 6680 df-rq 6681 df-ltnqqs 6682 df-enq0 6753 df-nq0 6754 df-0nq0 6755 df-plq0 6756 df-mq0 6757 df-inp 6795 df-iplp 6797 df-iltp 6799 |
This theorem is referenced by: prplnqu 6949 addextpr 6950 caucvgprlemcanl 6973 caucvgprprlemnkltj 7018 caucvgprprlemnbj 7022 caucvgprprlemmu 7024 caucvgprprlemloc 7032 caucvgprprlemexbt 7035 caucvgprprlemexb 7036 caucvgprprlemaddq 7037 caucvgprprlem1 7038 caucvgprprlem2 7039 ltsrprg 7063 gt0srpr 7064 lttrsr 7078 ltsosr 7080 ltasrg 7086 prsrlt 7102 |
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