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Mirrors > Home > ILE Home > Th. List > ltsosr | Unicode version |
Description: Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) |
Ref | Expression |
---|---|
ltsosr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltposr 7695 | . 2 | |
2 | df-nr 7659 | . . . 4 | |
3 | breq1 3979 | . . . . 5 | |
4 | breq1 3979 | . . . . . 6 | |
5 | 4 | orbi1d 781 | . . . . 5 |
6 | 3, 5 | imbi12d 233 | . . . 4 |
7 | breq2 3980 | . . . . 5 | |
8 | breq2 3980 | . . . . . 6 | |
9 | 8 | orbi2d 780 | . . . . 5 |
10 | 7, 9 | imbi12d 233 | . . . 4 |
11 | breq2 3980 | . . . . . 6 | |
12 | breq1 3979 | . . . . . 6 | |
13 | 11, 12 | orbi12d 783 | . . . . 5 |
14 | 13 | imbi2d 229 | . . . 4 |
15 | simp1l 1010 | . . . . . . . . 9 | |
16 | simp3r 1015 | . . . . . . . . 9 | |
17 | addclpr 7469 | . . . . . . . . 9 | |
18 | 15, 16, 17 | syl2anc 409 | . . . . . . . 8 |
19 | simp2r 1013 | . . . . . . . 8 | |
20 | addclpr 7469 | . . . . . . . 8 | |
21 | 18, 19, 20 | syl2anc 409 | . . . . . . 7 |
22 | simp2l 1012 | . . . . . . . . 9 | |
23 | addclpr 7469 | . . . . . . . . 9 | |
24 | 16, 22, 23 | syl2anc 409 | . . . . . . . 8 |
25 | simp1r 1011 | . . . . . . . 8 | |
26 | addclpr 7469 | . . . . . . . 8 | |
27 | 24, 25, 26 | syl2anc 409 | . . . . . . 7 |
28 | simp3l 1014 | . . . . . . . . 9 | |
29 | addclpr 7469 | . . . . . . . . 9 | |
30 | 25, 28, 29 | syl2anc 409 | . . . . . . . 8 |
31 | addclpr 7469 | . . . . . . . 8 | |
32 | 30, 19, 31 | syl2anc 409 | . . . . . . 7 |
33 | ltsopr 7528 | . . . . . . . 8 | |
34 | sowlin 4292 | . . . . . . . 8 | |
35 | 33, 34 | mpan 421 | . . . . . . 7 |
36 | 21, 27, 32, 35 | syl3anc 1227 | . . . . . 6 |
37 | addclpr 7469 | . . . . . . . . 9 | |
38 | 15, 19, 37 | syl2anc 409 | . . . . . . . 8 |
39 | addclpr 7469 | . . . . . . . . 9 | |
40 | 25, 22, 39 | syl2anc 409 | . . . . . . . 8 |
41 | ltaprg 7551 | . . . . . . . 8 | |
42 | 38, 40, 16, 41 | syl3anc 1227 | . . . . . . 7 |
43 | addcomprg 7510 | . . . . . . . . . . 11 | |
44 | 43 | adantl 275 | . . . . . . . . . 10 |
45 | addassprg 7511 | . . . . . . . . . . 11 | |
46 | 45 | adantl 275 | . . . . . . . . . 10 |
47 | 16, 15, 19, 44, 46 | caov12d 6014 | . . . . . . . . 9 |
48 | 46, 15, 16, 19 | caovassd 5992 | . . . . . . . . 9 |
49 | 47, 48 | eqtr4d 2200 | . . . . . . . 8 |
50 | 46, 16, 25, 22 | caovassd 5992 | . . . . . . . . 9 |
51 | 16, 25, 22, 44, 46 | caov32d 6013 | . . . . . . . . 9 |
52 | 50, 51 | eqtr3d 2199 | . . . . . . . 8 |
53 | 49, 52 | breq12d 3989 | . . . . . . 7 |
54 | 42, 53 | bitrd 187 | . . . . . 6 |
55 | ltaprg 7551 | . . . . . . . . 9 | |
56 | 55 | adantl 275 | . . . . . . . 8 |
57 | 56, 18, 30, 19, 44 | caovord2d 6002 | . . . . . . 7 |
58 | addclpr 7469 | . . . . . . . . . 10 | |
59 | 28, 19, 58 | syl2anc 409 | . . . . . . . . 9 |
60 | 56, 59, 24, 25, 44 | caovord2d 6002 | . . . . . . . 8 |
61 | 46, 25, 28, 19 | caovassd 5992 | . . . . . . . . . 10 |
62 | 44, 25, 59 | caovcomd 5989 | . . . . . . . . . 10 |
63 | 61, 62 | eqtrd 2197 | . . . . . . . . 9 |
64 | 63 | breq1d 3986 | . . . . . . . 8 |
65 | 60, 64 | bitr4d 190 | . . . . . . 7 |
66 | 57, 65 | orbi12d 783 | . . . . . 6 |
67 | 36, 54, 66 | 3imtr4d 202 | . . . . 5 |
68 | ltsrprg 7679 | . . . . . 6 | |
69 | 68 | 3adant3 1006 | . . . . 5 |
70 | ltsrprg 7679 | . . . . . . 7 | |
71 | 70 | 3adant2 1005 | . . . . . 6 |
72 | ltsrprg 7679 | . . . . . . . 8 | |
73 | 72 | ancoms 266 | . . . . . . 7 |
74 | 73 | 3adant1 1004 | . . . . . 6 |
75 | 71, 74 | orbi12d 783 | . . . . 5 |
76 | 67, 69, 75 | 3imtr4d 202 | . . . 4 |
77 | 2, 6, 10, 14, 76 | 3ecoptocl 6581 | . . 3 |
78 | 77 | rgen3 2551 | . 2 |
79 | df-iso 4269 | . 2 | |
80 | 1, 78, 79 | mpbir2an 931 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wo 698 w3a 967 wceq 1342 wcel 2135 wral 2442 cop 3573 class class class wbr 3976 wpo 4266 wor 4267 (class class class)co 5836 cec 6490 cnp 7223 cpp 7225 cltp 7227 cer 7228 cnr 7229 cltr 7235 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-2o 6376 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 df-enq0 7356 df-nq0 7357 df-0nq0 7358 df-plq0 7359 df-mq0 7360 df-inp 7398 df-iplp 7400 df-iltp 7402 df-enr 7658 df-nr 7659 df-ltr 7662 |
This theorem is referenced by: 1ne0sr 7698 addgt0sr 7707 caucvgsrlemcl 7721 caucvgsrlemfv 7723 suplocsrlemb 7738 suplocsrlempr 7739 suplocsrlem 7740 axpre-ltirr 7814 axpre-ltwlin 7815 axpre-lttrn 7816 |
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