MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwwlknonclwlknonf1o Structured version   Visualization version   GIF version

Theorem clwwlknonclwlknonf1o 30159
Description: 𝐹 is a bijection between the two representations of closed walks of a fixed positive length on a fixed vertex. (Contributed by AV, 26-May-2022.) (Proof shortened by AV, 7-Aug-2022.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
clwwlknonclwlknonf1o.v 𝑉 = (Vtxβ€˜πΊ)
clwwlknonclwlknonf1o.w π‘Š = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}
clwwlknonclwlknonf1o.f 𝐹 = (𝑐 ∈ π‘Š ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))))
Assertion
Ref Expression
clwwlknonclwlknonf1o ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝐹:π‘Šβ€“1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
Distinct variable groups:   𝐺,𝑐,𝑀   𝑁,𝑐,𝑀   𝑉,𝑐   π‘Š,𝑐   𝑋,𝑐,𝑀
Allowed substitution hints:   𝐹(𝑀,𝑐)   𝑉(𝑀)   π‘Š(𝑀)

Proof of Theorem clwwlknonclwlknonf1o
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 clwwlknonclwlknonf1o.w . . 3 π‘Š = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}
2 eqid 2727 . . 3 {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁}
3 clwwlknonclwlknonf1o.f . . 3 𝐹 = (𝑐 ∈ π‘Š ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))))
4 eqid 2727 . . 3 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) = (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))))
5 eqid 2727 . . . . 5 (1st β€˜π‘) = (1st β€˜π‘)
6 eqid 2727 . . . . 5 (2nd β€˜π‘) = (2nd β€˜π‘)
75, 6, 2, 4clwlknf1oclwwlkn 29881 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ β„•) β†’ (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))):{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁}–1-1-ontoβ†’(𝑁 ClWWalksN 𝐺))
873adant2 1129 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))):{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁}–1-1-ontoβ†’(𝑁 ClWWalksN 𝐺))
9 fveq1 6890 . . . . . . 7 (𝑠 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))) β†’ (π‘ β€˜0) = (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜0))
1093ad2ant3 1133 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ∧ 𝑠 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ (π‘ β€˜0) = (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜0))
11 2fveq3 6896 . . . . . . . . . . . 12 (𝑀 = 𝑐 β†’ (β™―β€˜(1st β€˜π‘€)) = (β™―β€˜(1st β€˜π‘)))
1211eqeq1d 2729 . . . . . . . . . . 11 (𝑀 = 𝑐 β†’ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
1312elrab 3680 . . . . . . . . . 10 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ↔ (𝑐 ∈ (ClWalksβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁))
14 clwlkwlk 29576 . . . . . . . . . . . 12 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ 𝑐 ∈ (Walksβ€˜πΊ))
15 wlkcpr 29430 . . . . . . . . . . . . 13 (𝑐 ∈ (Walksβ€˜πΊ) ↔ (1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘))
16 eqid 2727 . . . . . . . . . . . . . . . . 17 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
1716wlkpwrd 29418 . . . . . . . . . . . . . . . 16 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
18173ad2ant1 1131 . . . . . . . . . . . . . . 15 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
19 elnnuz 12888 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„• ↔ 𝑁 ∈ (β„€β‰₯β€˜1))
20 eluzfz2 13533 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (β„€β‰₯β€˜1) β†’ 𝑁 ∈ (1...𝑁))
2119, 20sylbi 216 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...𝑁))
22 fzelp1 13577 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (1...𝑁) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...(𝑁 + 1)))
24233ad2ant3 1133 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
25243ad2ant3 1133 . . . . . . . . . . . . . . . . 17 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
26 id 22 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ (β™―β€˜(1st β€˜π‘)) = 𝑁)
27 oveq1 7421 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((β™―β€˜(1st β€˜π‘)) + 1) = (𝑁 + 1))
2827oveq2d 7430 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ (1...((β™―β€˜(1st β€˜π‘)) + 1)) = (1...(𝑁 + 1)))
2926, 28eleq12d 2822 . . . . . . . . . . . . . . . . . 18 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((β™―β€˜(1st β€˜π‘)) ∈ (1...((β™―β€˜(1st β€˜π‘)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
30293ad2ant2 1132 . . . . . . . . . . . . . . . . 17 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ ((β™―β€˜(1st β€˜π‘)) ∈ (1...((β™―β€˜(1st β€˜π‘)) + 1)) ↔ 𝑁 ∈ (1...(𝑁 + 1))))
3125, 30mpbird 257 . . . . . . . . . . . . . . . 16 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜(1st β€˜π‘)) ∈ (1...((β™―β€˜(1st β€˜π‘)) + 1)))
32 wlklenvp1 29419 . . . . . . . . . . . . . . . . . . 19 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1))
3332oveq2d 7430 . . . . . . . . . . . . . . . . . 18 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (1...(β™―β€˜(2nd β€˜π‘))) = (1...((β™―β€˜(1st β€˜π‘)) + 1)))
3433eleq2d 2814 . . . . . . . . . . . . . . . . 17 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ ((β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘))) ↔ (β™―β€˜(1st β€˜π‘)) ∈ (1...((β™―β€˜(1st β€˜π‘)) + 1))))
35343ad2ant1 1131 . . . . . . . . . . . . . . . 16 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ ((β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘))) ↔ (β™―β€˜(1st β€˜π‘)) ∈ (1...((β™―β€˜(1st β€˜π‘)) + 1))))
3631, 35mpbird 257 . . . . . . . . . . . . . . 15 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ (β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘))))
3718, 36jca 511 . . . . . . . . . . . . . 14 (((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•)) β†’ ((2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘)))))
38373exp 1117 . . . . . . . . . . . . 13 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘)))))))
3915, 38sylbi 216 . . . . . . . . . . . 12 (𝑐 ∈ (Walksβ€˜πΊ) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘)))))))
4014, 39syl 17 . . . . . . . . . . 11 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘)))))))
4140imp 406 . . . . . . . . . 10 ((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁) β†’ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘))))))
4213, 41sylbi 216 . . . . . . . . 9 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} β†’ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ ((2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘))))))
4342impcom 407 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁}) β†’ ((2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘)))))
44 pfxfv0 14666 . . . . . . . 8 (((2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) ∈ (1...(β™―β€˜(2nd β€˜π‘)))) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜0) = ((2nd β€˜π‘)β€˜0))
4543, 44syl 17 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁}) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜0) = ((2nd β€˜π‘)β€˜0))
46453adant3 1130 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ∧ 𝑠 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜0) = ((2nd β€˜π‘)β€˜0))
4710, 46eqtrd 2767 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ∧ 𝑠 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ (π‘ β€˜0) = ((2nd β€˜π‘)β€˜0))
4847eqeq1d 2729 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ∧ 𝑠 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ ((π‘ β€˜0) = 𝑋 ↔ ((2nd β€˜π‘)β€˜0) = 𝑋))
49 nfv 1910 . . . . 5 Ⅎ𝑀((2nd β€˜π‘)β€˜0) = 𝑋
50 fveq2 6891 . . . . . . 7 (𝑀 = 𝑐 β†’ (2nd β€˜π‘€) = (2nd β€˜π‘))
5150fveq1d 6893 . . . . . 6 (𝑀 = 𝑐 β†’ ((2nd β€˜π‘€)β€˜0) = ((2nd β€˜π‘)β€˜0))
5251eqeq1d 2729 . . . . 5 (𝑀 = 𝑐 β†’ (((2nd β€˜π‘€)β€˜0) = 𝑋 ↔ ((2nd β€˜π‘)β€˜0) = 𝑋))
5349, 52sbiev 2303 . . . 4 ([𝑐 / 𝑀]((2nd β€˜π‘€)β€˜0) = 𝑋 ↔ ((2nd β€˜π‘)β€˜0) = 𝑋)
5448, 53bitr4di 289 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ∧ 𝑠 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ ((π‘ β€˜0) = 𝑋 ↔ [𝑐 / 𝑀]((2nd β€˜π‘€)β€˜0) = 𝑋))
551, 2, 3, 4, 8, 54f1ossf1o 7131 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝐹:π‘Šβ€“1-1-ontoβ†’{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘ β€˜0) = 𝑋})
56 clwwlknon 29887 . . 3 (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘ β€˜0) = 𝑋}
57 f1oeq3 6823 . . 3 ((𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘ β€˜0) = 𝑋} β†’ (𝐹:π‘Šβ€“1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ 𝐹:π‘Šβ€“1-1-ontoβ†’{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘ β€˜0) = 𝑋}))
5856, 57ax-mp 5 . 2 (𝐹:π‘Šβ€“1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ↔ 𝐹:π‘Šβ€“1-1-ontoβ†’{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘ β€˜0) = 𝑋})
5955, 58sylibr 233 1 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝐹:π‘Šβ€“1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534  [wsb 2060   ∈ wcel 2099  {crab 3427   class class class wbr 5142   ↦ cmpt 5225  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7414  1st c1st 7985  2nd c2nd 7986  0cc0 11130  1c1 11131   + caddc 11133  β„•cn 12234  β„€β‰₯cuz 12844  ...cfz 13508  β™―chash 14313  Word cword 14488   prefix cpfx 14644  Vtxcvtx 28796  USPGraphcuspgr 28948  Walkscwlks 29397  ClWalkscclwlks 29571   ClWWalksN cclwwlkn 29821  ClWWalksNOncclwwlknon 29884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-cnex 11186  ax-resscn 11187  ax-1cn 11188  ax-icn 11189  ax-addcl 11190  ax-addrcl 11191  ax-mulcl 11192  ax-mulrcl 11193  ax-mulcom 11194  ax-addass 11195  ax-mulass 11196  ax-distr 11197  ax-i2m1 11198  ax-1ne0 11199  ax-1rid 11200  ax-rnegex 11201  ax-rrecex 11202  ax-cnre 11203  ax-pre-lttri 11204  ax-pre-lttrn 11205  ax-pre-ltadd 11206  ax-pre-mulgt0 11207
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-ifp 1062  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-oadd 8484  df-er 8718  df-map 8838  df-pm 8839  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9916  df-card 9954  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11468  df-neg 11469  df-nn 12235  df-2 12297  df-n0 12495  df-xnn0 12567  df-z 12581  df-uz 12845  df-rp 12999  df-fz 13509  df-fzo 13652  df-hash 14314  df-word 14489  df-lsw 14537  df-concat 14545  df-s1 14570  df-substr 14615  df-pfx 14645  df-edg 28848  df-uhgr 28858  df-upgr 28882  df-uspgr 28950  df-wlks 29400  df-clwlks 29572  df-clwwlk 29779  df-clwwlkn 29822  df-clwwlknon 29885
This theorem is referenced by:  clwwlknonclwlknonen  30160  dlwwlknondlwlknonf1o  30162
  Copyright terms: Public domain W3C validator