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Mirrors > Home > ILE Home > Th. List > ecopovtrn | Unicode version |
Description: Assuming that operation is commutative (second hypothesis), closed (third hypothesis), associative (fourth hypothesis), and has the cancellation property (fifth hypothesis), show that the relation , specified by the first hypothesis, is transitive. (Contributed by NM, 11-Feb-1996.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
ecopopr.1 | |
ecopopr.com | |
ecopopr.cl | |
ecopopr.ass | |
ecopopr.can |
Ref | Expression |
---|---|
ecopovtrn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecopopr.1 | . . . . . . 7 | |
2 | opabssxp 4432 | . . . . . . 7 | |
3 | 1, 2 | eqsstri 3029 | . . . . . 6 |
4 | 3 | brel 4410 | . . . . 5 |
5 | 4 | simpld 110 | . . . 4 |
6 | 3 | brel 4410 | . . . 4 |
7 | 5, 6 | anim12i 331 | . . 3 |
8 | 3anass 923 | . . 3 | |
9 | 7, 8 | sylibr 132 | . 2 |
10 | eqid 2081 | . . 3 | |
11 | breq1 3788 | . . . . 5 | |
12 | 11 | anbi1d 452 | . . . 4 |
13 | breq1 3788 | . . . 4 | |
14 | 12, 13 | imbi12d 232 | . . 3 |
15 | breq2 3789 | . . . . 5 | |
16 | breq1 3788 | . . . . 5 | |
17 | 15, 16 | anbi12d 456 | . . . 4 |
18 | 17 | imbi1d 229 | . . 3 |
19 | breq2 3789 | . . . . 5 | |
20 | 19 | anbi2d 451 | . . . 4 |
21 | breq2 3789 | . . . 4 | |
22 | 20, 21 | imbi12d 232 | . . 3 |
23 | 1 | ecopoveq 6224 | . . . . . . . 8 |
24 | 23 | 3adant3 958 | . . . . . . 7 |
25 | 1 | ecopoveq 6224 | . . . . . . . 8 |
26 | 25 | 3adant1 956 | . . . . . . 7 |
27 | 24, 26 | anbi12d 456 | . . . . . 6 |
28 | oveq12 5541 | . . . . . . 7 | |
29 | simp2l 964 | . . . . . . . . 9 | |
30 | simp2r 965 | . . . . . . . . 9 | |
31 | simp1l 962 | . . . . . . . . 9 | |
32 | ecopopr.com | . . . . . . . . . 10 | |
33 | 32 | a1i 9 | . . . . . . . . 9 |
34 | ecopopr.ass | . . . . . . . . . 10 | |
35 | 34 | a1i 9 | . . . . . . . . 9 |
36 | simp3r 967 | . . . . . . . . 9 | |
37 | ecopopr.cl | . . . . . . . . . 10 | |
38 | 37 | adantl 271 | . . . . . . . . 9 |
39 | 29, 30, 31, 33, 35, 36, 38 | caov411d 5706 | . . . . . . . 8 |
40 | simp1r 963 | . . . . . . . . . 10 | |
41 | simp3l 966 | . . . . . . . . . 10 | |
42 | 40, 30, 29, 33, 35, 41, 38 | caov411d 5706 | . . . . . . . . 9 |
43 | 40, 30, 29, 33, 35, 41, 38 | caov4d 5705 | . . . . . . . . 9 |
44 | 42, 43 | eqtr3d 2115 | . . . . . . . 8 |
45 | 39, 44 | eqeq12d 2095 | . . . . . . 7 |
46 | 28, 45 | syl5ibr 154 | . . . . . 6 |
47 | 27, 46 | sylbid 148 | . . . . 5 |
48 | ecopopr.can | . . . . . . . . 9 | |
49 | 48 | 3adant3 958 | . . . . . . . 8 |
50 | oveq2 5540 | . . . . . . . 8 | |
51 | 49, 50 | impbid1 140 | . . . . . . 7 |
52 | 51 | adantl 271 | . . . . . 6 |
53 | 37 | caovcl 5675 | . . . . . . 7 |
54 | 29, 30, 53 | syl2anc 403 | . . . . . 6 |
55 | 37 | caovcl 5675 | . . . . . . 7 |
56 | 31, 36, 55 | syl2anc 403 | . . . . . 6 |
57 | 38, 40, 41 | caovcld 5674 | . . . . . 6 |
58 | 52, 54, 56, 57 | caovcand 5683 | . . . . 5 |
59 | 47, 58 | sylibd 147 | . . . 4 |
60 | 1 | ecopoveq 6224 | . . . . 5 |
61 | 60 | 3adant2 957 | . . . 4 |
62 | 59, 61 | sylibrd 167 | . . 3 |
63 | 10, 14, 18, 22, 62 | 3optocl 4436 | . 2 |
64 | 9, 63 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 102 wb 103 w3a 919 wceq 1284 wex 1421 wcel 1433 cop 3401 class class class wbr 3785 copab 3838 cxp 4361 (class class class)co 5532 |
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-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-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 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-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-iota 4887 df-fv 4930 df-ov 5535 |
This theorem is referenced by: ecopover 6227 |
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