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Mirrors > Home > MPE Home > Th. List > 2clwwlkel | Structured version Visualization version GIF version |
Description: Characterization of an element of the value of operation πΆ, i.e., of a word being a double loop of length π on vertex π. (Contributed by Alexander van der Vekens, 24-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.) |
Ref | Expression |
---|---|
2clwwlk.c | β’ πΆ = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£}) |
Ref | Expression |
---|---|
2clwwlkel | β’ ((π β π β§ π β (β€β₯β2)) β (π β (ππΆπ) β (π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2clwwlk.c | . . . 4 β’ πΆ = (π£ β π, π β (β€β₯β2) β¦ {π€ β (π£(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π£}) | |
2 | 1 | 2clwwlk 29333 | . . 3 β’ ((π β π β§ π β (β€β₯β2)) β (ππΆπ) = {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π}) |
3 | 2 | eleq2d 2824 | . 2 β’ ((π β π β§ π β (β€β₯β2)) β (π β (ππΆπ) β π β {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π})) |
4 | fveq1 6846 | . . . 4 β’ (π€ = π β (π€β(π β 2)) = (πβ(π β 2))) | |
5 | 4 | eqeq1d 2739 | . . 3 β’ (π€ = π β ((π€β(π β 2)) = π β (πβ(π β 2)) = π)) |
6 | 5 | elrab 3650 | . 2 β’ (π β {π€ β (π(ClWWalksNOnβπΊ)π) β£ (π€β(π β 2)) = π} β (π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π)) |
7 | 3, 6 | bitrdi 287 | 1 β’ ((π β π β§ π β (β€β₯β2)) β (π β (ππΆπ) β (π β (π(ClWWalksNOnβπΊ)π) β§ (πβ(π β 2)) = π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3410 βcfv 6501 (class class class)co 7362 β cmpo 7364 β cmin 11392 2c2 12215 β€β₯cuz 12770 ClWWalksNOncclwwlknon 29073 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-ov 7365 df-oprab 7366 df-mpo 7367 |
This theorem is referenced by: 2clwwlk2clwwlk 29336 numclwwlk1lem2f1 29343 |
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