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Mirrors > Home > ILE Home > Th. List > ecopovtrng | 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 Jim Kingdon, 1-Sep-2019.) |
Ref | Expression |
---|---|
ecopopr.1 | |
ecopoprg.com | |
ecopoprg.cl | |
ecopoprg.ass | |
ecopoprg.can |
Ref | Expression |
---|---|
ecopovtrng |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ecopopr.1 | . . . . . . 7 | |
2 | opabssxp 4613 | . . . . . . 7 | |
3 | 1, 2 | eqsstri 3129 | . . . . . 6 |
4 | 3 | brel 4591 | . . . . 5 |
5 | 4 | simpld 111 | . . . 4 |
6 | 3 | brel 4591 | . . . 4 |
7 | 5, 6 | anim12i 336 | . . 3 |
8 | 3anass 966 | . . 3 | |
9 | 7, 8 | sylibr 133 | . 2 |
10 | eqid 2139 | . . 3 | |
11 | breq1 3932 | . . . . 5 | |
12 | 11 | anbi1d 460 | . . . 4 |
13 | breq1 3932 | . . . 4 | |
14 | 12, 13 | imbi12d 233 | . . 3 |
15 | breq2 3933 | . . . . 5 | |
16 | breq1 3932 | . . . . 5 | |
17 | 15, 16 | anbi12d 464 | . . . 4 |
18 | 17 | imbi1d 230 | . . 3 |
19 | breq2 3933 | . . . . 5 | |
20 | 19 | anbi2d 459 | . . . 4 |
21 | breq2 3933 | . . . 4 | |
22 | 20, 21 | imbi12d 233 | . . 3 |
23 | 1 | ecopoveq 6524 | . . . . . . . 8 |
24 | 23 | 3adant3 1001 | . . . . . . 7 |
25 | 1 | ecopoveq 6524 | . . . . . . . 8 |
26 | 25 | 3adant1 999 | . . . . . . 7 |
27 | 24, 26 | anbi12d 464 | . . . . . 6 |
28 | oveq12 5783 | . . . . . . 7 | |
29 | simp2l 1007 | . . . . . . . . 9 | |
30 | simp2r 1008 | . . . . . . . . 9 | |
31 | simp1l 1005 | . . . . . . . . 9 | |
32 | ecopoprg.com | . . . . . . . . . 10 | |
33 | 32 | adantl 275 | . . . . . . . . 9 |
34 | ecopoprg.ass | . . . . . . . . . 10 | |
35 | 34 | adantl 275 | . . . . . . . . 9 |
36 | simp3r 1010 | . . . . . . . . 9 | |
37 | ecopoprg.cl | . . . . . . . . . 10 | |
38 | 37 | adantl 275 | . . . . . . . . 9 |
39 | 29, 30, 31, 33, 35, 36, 38 | caov411d 5956 | . . . . . . . 8 |
40 | simp1r 1006 | . . . . . . . . . 10 | |
41 | simp3l 1009 | . . . . . . . . . 10 | |
42 | 40, 30, 29, 33, 35, 41, 38 | caov411d 5956 | . . . . . . . . 9 |
43 | 40, 30, 29, 33, 35, 41, 38 | caov4d 5955 | . . . . . . . . 9 |
44 | 42, 43 | eqtr3d 2174 | . . . . . . . 8 |
45 | 39, 44 | eqeq12d 2154 | . . . . . . 7 |
46 | 28, 45 | syl5ibr 155 | . . . . . 6 |
47 | 27, 46 | sylbid 149 | . . . . 5 |
48 | ecopoprg.can | . . . . . . . 8 | |
49 | oveq2 5782 | . . . . . . . 8 | |
50 | 48, 49 | impbid1 141 | . . . . . . 7 |
51 | 50 | adantl 275 | . . . . . 6 |
52 | 37 | caovcl 5925 | . . . . . . 7 |
53 | 29, 30, 52 | syl2anc 408 | . . . . . 6 |
54 | 37 | caovcl 5925 | . . . . . . 7 |
55 | 31, 36, 54 | syl2anc 408 | . . . . . 6 |
56 | 38, 40, 41 | caovcld 5924 | . . . . . 6 |
57 | 51, 53, 55, 56 | caovcand 5933 | . . . . 5 |
58 | 47, 57 | sylibd 148 | . . . 4 |
59 | 1 | ecopoveq 6524 | . . . . 5 |
60 | 59 | 3adant2 1000 | . . . 4 |
61 | 58, 60 | sylibrd 168 | . . 3 |
62 | 10, 14, 18, 22, 61 | 3optocl 4617 | . 2 |
63 | 9, 62 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wex 1468 wcel 1480 cop 3530 class class class wbr 3929 copab 3988 cxp 4537 (class class class)co 5774 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-iota 5088 df-fv 5131 df-ov 5777 |
This theorem is referenced by: ecopoverg 6530 |
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