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Mirrors > Home > MPE Home > Th. List > clwwlkprop | Structured version Visualization version Unicode version |
Description: Properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) |
Ref | Expression |
---|---|
clwwlkprop |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clwwlk 25558 |
. . 3
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2 | 1 | elmpt2cl 6530 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
3 | pm3.22 456 |
. . . . . 6
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4 | df-3an 1009 |
. . . . . 6
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5 | 3, 4 | sylibr 217 |
. . . . 5
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6 | 5 | a1d 25 |
. . . 4
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7 | 6 | ex 441 |
. . 3
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8 | clwwlk 25573 |
. . . . . 6
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9 | 8 | eleq2d 2534 |
. . . . 5
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10 | fveq2 5879 |
. . . . . . . . . . 11
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11 | 10 | oveq1d 6323 |
. . . . . . . . . 10
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12 | 11 | oveq2d 6324 |
. . . . . . . . 9
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13 | fveq1 5878 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
14 | fveq1 5878 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 13, 14 | preq12d 4050 |
. . . . . . . . . 10
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16 | 15 | eleq1d 2533 |
. . . . . . . . 9
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17 | 12, 16 | raleqbidv 2987 |
. . . . . . . 8
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18 | fveq2 5879 |
. . . . . . . . . 10
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19 | fveq1 5878 |
. . . . . . . . . 10
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20 | 18, 19 | preq12d 4050 |
. . . . . . . . 9
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21 | 20 | eleq1d 2533 |
. . . . . . . 8
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22 | 17, 21 | anbi12d 725 |
. . . . . . 7
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23 | 22 | elrab 3184 |
. . . . . 6
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24 | pm2.24 112 |
. . . . . . 7
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25 | 24 | adantr 472 |
. . . . . 6
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26 | 23, 25 | sylbi 200 |
. . . . 5
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27 | 9, 26 | syl6bi 236 |
. . . 4
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28 | 27 | com3r 81 |
. . 3
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29 | 7, 28 | pm2.61i 169 |
. 2
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30 | 2, 29 | mpcom 36 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-rep 4508 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 ax-un 6602 ax-cnex 9613 ax-resscn 9614 ax-1cn 9615 ax-icn 9616 ax-addcl 9617 ax-addrcl 9618 ax-mulcl 9619 ax-mulrcl 9620 ax-mulcom 9621 ax-addass 9622 ax-mulass 9623 ax-distr 9624 ax-i2m1 9625 ax-1ne0 9626 ax-1rid 9627 ax-rnegex 9628 ax-rrecex 9629 ax-cnre 9630 ax-pre-lttri 9631 ax-pre-lttrn 9632 ax-pre-ltadd 9633 ax-pre-mulgt0 9634 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3or 1008 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-nel 2644 df-ral 2761 df-rex 2762 df-reu 2763 df-rmo 2764 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-pss 3406 df-nul 3723 df-if 3873 df-pw 3944 df-sn 3960 df-pr 3962 df-tp 3964 df-op 3966 df-uni 4191 df-int 4227 df-iun 4271 df-br 4396 df-opab 4455 df-mpt 4456 df-tr 4491 df-eprel 4750 df-id 4754 df-po 4760 df-so 4761 df-fr 4798 df-we 4800 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-pred 5387 df-ord 5433 df-on 5434 df-lim 5435 df-suc 5436 df-iota 5553 df-fun 5591 df-fn 5592 df-f 5593 df-f1 5594 df-fo 5595 df-f1o 5596 df-fv 5597 df-riota 6270 df-ov 6311 df-oprab 6312 df-mpt2 6313 df-om 6712 df-1st 6812 df-2nd 6813 df-wrecs 7046 df-recs 7108 df-rdg 7146 df-1o 7200 df-oadd 7204 df-er 7381 df-map 7492 df-pm 7493 df-en 7588 df-dom 7589 df-sdom 7590 df-fin 7591 df-card 8391 df-cda 8616 df-pnf 9695 df-mnf 9696 df-xr 9697 df-ltxr 9698 df-le 9699 df-sub 9882 df-neg 9883 df-nn 10632 df-2 10690 df-n0 10894 df-z 10962 df-uz 11183 df-fz 11811 df-fzo 11943 df-hash 12554 df-word 12711 df-clwwlk 25558 |
This theorem is referenced by: clwwlkgt0 25578 clwwlknprop 25579 clwwlkn0 25581 clwwisshclww 25614 clwwisshclwwn 25615 erclwwlkeqlen 25619 erclwwlkref 25620 erclwwlksym 25621 erclwwlktr 25622 |
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