Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6940 | . 2 | |
2 | df-nr 6904 | . . . 4 | |
3 | breq1 3788 | . . . . 5 | |
4 | breq1 3788 | . . . . . 6 | |
5 | 4 | orbi1d 737 | . . . . 5 |
6 | 3, 5 | imbi12d 232 | . . . 4 |
7 | breq2 3789 | . . . . 5 | |
8 | breq2 3789 | . . . . . 6 | |
9 | 8 | orbi2d 736 | . . . . 5 |
10 | 7, 9 | imbi12d 232 | . . . 4 |
11 | breq2 3789 | . . . . . 6 | |
12 | breq1 3788 | . . . . . 6 | |
13 | 11, 12 | orbi12d 739 | . . . . 5 |
14 | 13 | imbi2d 228 | . . . 4 |
15 | simp1l 962 | . . . . . . . . 9 | |
16 | simp3r 967 | . . . . . . . . 9 | |
17 | addclpr 6727 | . . . . . . . . 9 | |
18 | 15, 16, 17 | syl2anc 403 | . . . . . . . 8 |
19 | simp2r 965 | . . . . . . . 8 | |
20 | addclpr 6727 | . . . . . . . 8 | |
21 | 18, 19, 20 | syl2anc 403 | . . . . . . 7 |
22 | simp2l 964 | . . . . . . . . 9 | |
23 | addclpr 6727 | . . . . . . . . 9 | |
24 | 16, 22, 23 | syl2anc 403 | . . . . . . . 8 |
25 | simp1r 963 | . . . . . . . 8 | |
26 | addclpr 6727 | . . . . . . . 8 | |
27 | 24, 25, 26 | syl2anc 403 | . . . . . . 7 |
28 | simp3l 966 | . . . . . . . . 9 | |
29 | addclpr 6727 | . . . . . . . . 9 | |
30 | 25, 28, 29 | syl2anc 403 | . . . . . . . 8 |
31 | addclpr 6727 | . . . . . . . 8 | |
32 | 30, 19, 31 | syl2anc 403 | . . . . . . 7 |
33 | ltsopr 6786 | . . . . . . . 8 | |
34 | sowlin 4075 | . . . . . . . 8 | |
35 | 33, 34 | mpan 414 | . . . . . . 7 |
36 | 21, 27, 32, 35 | syl3anc 1169 | . . . . . 6 |
37 | addclpr 6727 | . . . . . . . . 9 | |
38 | 15, 19, 37 | syl2anc 403 | . . . . . . . 8 |
39 | addclpr 6727 | . . . . . . . . 9 | |
40 | 25, 22, 39 | syl2anc 403 | . . . . . . . 8 |
41 | ltaprg 6809 | . . . . . . . 8 | |
42 | 38, 40, 16, 41 | syl3anc 1169 | . . . . . . 7 |
43 | addcomprg 6768 | . . . . . . . . . . 11 | |
44 | 43 | adantl 271 | . . . . . . . . . 10 |
45 | addassprg 6769 | . . . . . . . . . . 11 | |
46 | 45 | adantl 271 | . . . . . . . . . 10 |
47 | 16, 15, 19, 44, 46 | caov12d 5702 | . . . . . . . . 9 |
48 | 46, 15, 16, 19 | caovassd 5680 | . . . . . . . . 9 |
49 | 47, 48 | eqtr4d 2116 | . . . . . . . 8 |
50 | 46, 16, 25, 22 | caovassd 5680 | . . . . . . . . 9 |
51 | 16, 25, 22, 44, 46 | caov32d 5701 | . . . . . . . . 9 |
52 | 50, 51 | eqtr3d 2115 | . . . . . . . 8 |
53 | 49, 52 | breq12d 3798 | . . . . . . 7 |
54 | 42, 53 | bitrd 186 | . . . . . 6 |
55 | ltaprg 6809 | . . . . . . . . 9 | |
56 | 55 | adantl 271 | . . . . . . . 8 |
57 | 56, 18, 30, 19, 44 | caovord2d 5690 | . . . . . . 7 |
58 | addclpr 6727 | . . . . . . . . . 10 | |
59 | 28, 19, 58 | syl2anc 403 | . . . . . . . . 9 |
60 | 56, 59, 24, 25, 44 | caovord2d 5690 | . . . . . . . 8 |
61 | 46, 25, 28, 19 | caovassd 5680 | . . . . . . . . . 10 |
62 | 44, 25, 59 | caovcomd 5677 | . . . . . . . . . 10 |
63 | 61, 62 | eqtrd 2113 | . . . . . . . . 9 |
64 | 63 | breq1d 3795 | . . . . . . . 8 |
65 | 60, 64 | bitr4d 189 | . . . . . . 7 |
66 | 57, 65 | orbi12d 739 | . . . . . 6 |
67 | 36, 54, 66 | 3imtr4d 201 | . . . . 5 |
68 | ltsrprg 6924 | . . . . . 6 | |
69 | 68 | 3adant3 958 | . . . . 5 |
70 | ltsrprg 6924 | . . . . . . 7 | |
71 | 70 | 3adant2 957 | . . . . . 6 |
72 | ltsrprg 6924 | . . . . . . . 8 | |
73 | 72 | ancoms 264 | . . . . . . 7 |
74 | 73 | 3adant1 956 | . . . . . 6 |
75 | 71, 74 | orbi12d 739 | . . . . 5 |
76 | 67, 69, 75 | 3imtr4d 201 | . . . 4 |
77 | 2, 6, 10, 14, 76 | 3ecoptocl 6218 | . . 3 |
78 | 77 | rgen3 2448 | . 2 |
79 | df-iso 4052 | . 2 | |
80 | 1, 78, 79 | mpbir2an 883 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 wo 661 w3a 919 wceq 1284 wcel 1433 wral 2348 cop 3401 class class class wbr 3785 wpo 4049 wor 4050 (class class class)co 5532 cec 6127 cnp 6481 cpp 6483 cltp 6485 cer 6486 cnr 6487 cltr 6493 |
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 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-eprel 4044 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-irdg 5980 df-1o 6024 df-2o 6025 df-oadd 6028 df-omul 6029 df-er 6129 df-ec 6131 df-qs 6135 df-ni 6494 df-pli 6495 df-mi 6496 df-lti 6497 df-plpq 6534 df-mpq 6535 df-enq 6537 df-nqqs 6538 df-plqqs 6539 df-mqqs 6540 df-1nqqs 6541 df-rq 6542 df-ltnqqs 6543 df-enq0 6614 df-nq0 6615 df-0nq0 6616 df-plq0 6617 df-mq0 6618 df-inp 6656 df-iplp 6658 df-iltp 6660 df-enr 6903 df-nr 6904 df-ltr 6907 |
This theorem is referenced by: 1ne0sr 6943 addgt0sr 6952 caucvgsrlemcl 6965 caucvgsrlemfv 6967 axpre-ltirr 7048 axpre-ltwlin 7049 axpre-lttrn 7050 |
Copyright terms: Public domain | W3C validator |