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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 30109 | . . 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 2813 | . 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 6884 | . . . 4 β’ (π€ = π β (π€β(π β 2)) = (πβ(π β 2))) | |
5 | 4 | eqeq1d 2728 | . . 3 β’ (π€ = π β ((π€β(π β 2)) = π β (πβ(π β 2)) = π)) |
6 | 5 | elrab 3678 | . 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 395 = wceq 1533 β wcel 2098 {crab 3426 βcfv 6537 (class class class)co 7405 β cmpo 7407 β cmin 11448 2c2 12271 β€β₯cuz 12826 ClWWalksNOncclwwlknon 29849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6489 df-fun 6539 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 |
This theorem is referenced by: 2clwwlk2clwwlk 30112 numclwwlk1lem2f1 30119 |
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