Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > numclwwlkovg | Structured version Visualization version Unicode version |
Description: Value of operation , mapping a vertex v and an integer n greater than 1 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v" according to definition 6 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 29-May-2021.) |
Ref | Expression |
---|---|
numclwwlkovg.c | ClWWalksN |
Ref | Expression |
---|---|
numclwwlkovg | ClWWalksN |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 6657 | . . . 4 ClWWalksN ClWWalksN | |
2 | 1 | adantl 482 | . . 3 ClWWalksN ClWWalksN |
3 | eqeq2 2633 | . . . 4 | |
4 | oveq1 6657 | . . . . . 6 | |
5 | 4 | fveq2d 6195 | . . . . 5 |
6 | 5 | eqeq1d 2624 | . . . 4 |
7 | 3, 6 | bi2anan9 917 | . . 3 |
8 | 2, 7 | rabeqbidv 3195 | . 2 ClWWalksN ClWWalksN |
9 | numclwwlkovg.c | . 2 ClWWalksN | |
10 | ovex 6678 | . . 3 ClWWalksN | |
11 | 10 | rabex 4813 | . 2 ClWWalksN |
12 | 8, 9, 11 | ovmpt2a 6791 | 1 ClWWalksN |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 crab 2916 cfv 5888 (class class class)co 6650 cmpt2 6652 cc0 9936 cmin 10266 c2 11070 cuz 11687 ClWWalksN cclwwlksn 26876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 |
This theorem is referenced by: numclwwlkovgel 27221 extwwlkfab 27223 numclwwlk3lem 27241 |
Copyright terms: Public domain | W3C validator |