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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 4683 | . . . . . . 7 | |
3 | 1, 2 | eqsstri 3179 | . . . . . 6 |
4 | 3 | brel 4661 | . . . . 5 |
5 | 4 | simpld 111 | . . . 4 |
6 | 3 | brel 4661 | . . . 4 |
7 | 5, 6 | anim12i 336 | . . 3 |
8 | 3anass 977 | . . 3 | |
9 | 7, 8 | sylibr 133 | . 2 |
10 | eqid 2170 | . . 3 | |
11 | breq1 3990 | . . . . 5 | |
12 | 11 | anbi1d 462 | . . . 4 |
13 | breq1 3990 | . . . 4 | |
14 | 12, 13 | imbi12d 233 | . . 3 |
15 | breq2 3991 | . . . . 5 | |
16 | breq1 3990 | . . . . 5 | |
17 | 15, 16 | anbi12d 470 | . . . 4 |
18 | 17 | imbi1d 230 | . . 3 |
19 | breq2 3991 | . . . . 5 | |
20 | 19 | anbi2d 461 | . . . 4 |
21 | breq2 3991 | . . . 4 | |
22 | 20, 21 | imbi12d 233 | . . 3 |
23 | 1 | ecopoveq 6604 | . . . . . . . 8 |
24 | 23 | 3adant3 1012 | . . . . . . 7 |
25 | 1 | ecopoveq 6604 | . . . . . . . 8 |
26 | 25 | 3adant1 1010 | . . . . . . 7 |
27 | 24, 26 | anbi12d 470 | . . . . . 6 |
28 | oveq12 5859 | . . . . . . 7 | |
29 | simp2l 1018 | . . . . . . . . 9 | |
30 | simp2r 1019 | . . . . . . . . 9 | |
31 | simp1l 1016 | . . . . . . . . 9 | |
32 | ecopopr.com | . . . . . . . . . 10 | |
33 | 32 | a1i 9 | . . . . . . . . 9 |
34 | ecopopr.ass | . . . . . . . . . 10 | |
35 | 34 | a1i 9 | . . . . . . . . 9 |
36 | simp3r 1021 | . . . . . . . . 9 | |
37 | ecopopr.cl | . . . . . . . . . 10 | |
38 | 37 | adantl 275 | . . . . . . . . 9 |
39 | 29, 30, 31, 33, 35, 36, 38 | caov411d 6035 | . . . . . . . 8 |
40 | simp1r 1017 | . . . . . . . . . 10 | |
41 | simp3l 1020 | . . . . . . . . . 10 | |
42 | 40, 30, 29, 33, 35, 41, 38 | caov411d 6035 | . . . . . . . . 9 |
43 | 40, 30, 29, 33, 35, 41, 38 | caov4d 6034 | . . . . . . . . 9 |
44 | 42, 43 | eqtr3d 2205 | . . . . . . . 8 |
45 | 39, 44 | eqeq12d 2185 | . . . . . . 7 |
46 | 28, 45 | syl5ibr 155 | . . . . . 6 |
47 | 27, 46 | sylbid 149 | . . . . 5 |
48 | ecopopr.can | . . . . . . . . 9 | |
49 | 48 | 3adant3 1012 | . . . . . . . 8 |
50 | oveq2 5858 | . . . . . . . 8 | |
51 | 49, 50 | impbid1 141 | . . . . . . 7 |
52 | 51 | adantl 275 | . . . . . 6 |
53 | 37 | caovcl 6004 | . . . . . . 7 |
54 | 29, 30, 53 | syl2anc 409 | . . . . . 6 |
55 | 37 | caovcl 6004 | . . . . . . 7 |
56 | 31, 36, 55 | syl2anc 409 | . . . . . 6 |
57 | 38, 40, 41 | caovcld 6003 | . . . . . 6 |
58 | 52, 54, 56, 57 | caovcand 6012 | . . . . 5 |
59 | 47, 58 | sylibd 148 | . . . 4 |
60 | 1 | ecopoveq 6604 | . . . . 5 |
61 | 60 | 3adant2 1011 | . . . 4 |
62 | 59, 61 | sylibrd 168 | . . 3 |
63 | 10, 14, 18, 22, 62 | 3optocl 4687 | . 2 |
64 | 9, 63 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wex 1485 wcel 2141 cop 3584 class class class wbr 3987 copab 4047 cxp 4607 (class class class)co 5850 |
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-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-xp 4615 df-iota 5158 df-fv 5204 df-ov 5853 |
This theorem is referenced by: ecopover 6607 |
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