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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 4672 | . . . . . . 7 | |
3 | 1, 2 | eqsstri 3169 | . . . . . 6 |
4 | 3 | brel 4650 | . . . . 5 |
5 | 4 | simpld 111 | . . . 4 |
6 | 3 | brel 4650 | . . . 4 |
7 | 5, 6 | anim12i 336 | . . 3 |
8 | 3anass 971 | . . 3 | |
9 | 7, 8 | sylibr 133 | . 2 |
10 | eqid 2164 | . . 3 | |
11 | breq1 3979 | . . . . 5 | |
12 | 11 | anbi1d 461 | . . . 4 |
13 | breq1 3979 | . . . 4 | |
14 | 12, 13 | imbi12d 233 | . . 3 |
15 | breq2 3980 | . . . . 5 | |
16 | breq1 3979 | . . . . 5 | |
17 | 15, 16 | anbi12d 465 | . . . 4 |
18 | 17 | imbi1d 230 | . . 3 |
19 | breq2 3980 | . . . . 5 | |
20 | 19 | anbi2d 460 | . . . 4 |
21 | breq2 3980 | . . . 4 | |
22 | 20, 21 | imbi12d 233 | . . 3 |
23 | 1 | ecopoveq 6587 | . . . . . . . 8 |
24 | 23 | 3adant3 1006 | . . . . . . 7 |
25 | 1 | ecopoveq 6587 | . . . . . . . 8 |
26 | 25 | 3adant1 1004 | . . . . . . 7 |
27 | 24, 26 | anbi12d 465 | . . . . . 6 |
28 | oveq12 5845 | . . . . . . 7 | |
29 | simp2l 1012 | . . . . . . . . 9 | |
30 | simp2r 1013 | . . . . . . . . 9 | |
31 | simp1l 1010 | . . . . . . . . 9 | |
32 | ecopoprg.com | . . . . . . . . . 10 | |
33 | 32 | adantl 275 | . . . . . . . . 9 |
34 | ecopoprg.ass | . . . . . . . . . 10 | |
35 | 34 | adantl 275 | . . . . . . . . 9 |
36 | simp3r 1015 | . . . . . . . . 9 | |
37 | ecopoprg.cl | . . . . . . . . . 10 | |
38 | 37 | adantl 275 | . . . . . . . . 9 |
39 | 29, 30, 31, 33, 35, 36, 38 | caov411d 6018 | . . . . . . . 8 |
40 | simp1r 1011 | . . . . . . . . . 10 | |
41 | simp3l 1014 | . . . . . . . . . 10 | |
42 | 40, 30, 29, 33, 35, 41, 38 | caov411d 6018 | . . . . . . . . 9 |
43 | 40, 30, 29, 33, 35, 41, 38 | caov4d 6017 | . . . . . . . . 9 |
44 | 42, 43 | eqtr3d 2199 | . . . . . . . 8 |
45 | 39, 44 | eqeq12d 2179 | . . . . . . 7 |
46 | 28, 45 | syl5ibr 155 | . . . . . 6 |
47 | 27, 46 | sylbid 149 | . . . . 5 |
48 | ecopoprg.can | . . . . . . . 8 | |
49 | oveq2 5844 | . . . . . . . 8 | |
50 | 48, 49 | impbid1 141 | . . . . . . 7 |
51 | 50 | adantl 275 | . . . . . 6 |
52 | 37 | caovcl 5987 | . . . . . . 7 |
53 | 29, 30, 52 | syl2anc 409 | . . . . . 6 |
54 | 37 | caovcl 5987 | . . . . . . 7 |
55 | 31, 36, 54 | syl2anc 409 | . . . . . 6 |
56 | 38, 40, 41 | caovcld 5986 | . . . . . 6 |
57 | 51, 53, 55, 56 | caovcand 5995 | . . . . 5 |
58 | 47, 57 | sylibd 148 | . . . 4 |
59 | 1 | ecopoveq 6587 | . . . . 5 |
60 | 59 | 3adant2 1005 | . . . 4 |
61 | 58, 60 | sylibrd 168 | . . 3 |
62 | 10, 14, 18, 22, 61 | 3optocl 4676 | . 2 |
63 | 9, 62 | mpcom 36 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wex 1479 wcel 2135 cop 3573 class class class wbr 3976 copab 4036 cxp 4596 (class class class)co 5836 |
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 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-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 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-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-iota 5147 df-fv 5190 df-ov 5839 |
This theorem is referenced by: ecopoverg 6593 |
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