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Theorem clwwlknonclwlknonf1o 29348
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 2737 . . 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 2737 . . 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 2737 . . . . 5 (1st β€˜π‘) = (1st β€˜π‘)
6 eqid 2737 . . . . 5 (2nd β€˜π‘) = (2nd β€˜π‘)
75, 6, 2, 4clwlknf1oclwwlkn 29070 . . . 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 1132 . . 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 6846 . . . . . . 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 1136 . . . . . 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 6852 . . . . . . . . . . . 12 (𝑀 = 𝑐 β†’ (β™―β€˜(1st β€˜π‘€)) = (β™―β€˜(1st β€˜π‘)))
1211eqeq1d 2739 . . . . . . . . . . 11 (𝑀 = 𝑐 β†’ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
1312elrab 3650 . . . . . . . . . 10 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (β™―β€˜(1st β€˜π‘€)) = 𝑁} ↔ (𝑐 ∈ (ClWalksβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) = 𝑁))
14 clwlkwlk 28765 . . . . . . . . . . . 12 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ 𝑐 ∈ (Walksβ€˜πΊ))
15 wlkcpr 28619 . . . . . . . . . . . . 13 (𝑐 ∈ (Walksβ€˜πΊ) ↔ (1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘))
16 eqid 2737 . . . . . . . . . . . . . . . . 17 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
1716wlkpwrd 28607 . . . . . . . . . . . . . . . 16 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
18173ad2ant1 1134 . . . . . . . . . . . . . . 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 12814 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ β„• ↔ 𝑁 ∈ (β„€β‰₯β€˜1))
20 eluzfz2 13456 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ (β„€β‰₯β€˜1) β†’ 𝑁 ∈ (1...𝑁))
2119, 20sylbi 216 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...𝑁))
22 fzelp1 13500 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ (1...𝑁) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . 19 (𝑁 ∈ β„• β†’ 𝑁 ∈ (1...(𝑁 + 1)))
24233ad2ant3 1136 . . . . . . . . . . . . . . . . . 18 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝑁 ∈ (1...(𝑁 + 1)))
25243ad2ant3 1136 . . . . . . . . . . . . . . . . 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 7369 . . . . . . . . . . . . . . . . . . . 20 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((β™―β€˜(1st β€˜π‘)) + 1) = (𝑁 + 1))
2827oveq2d 7378 . . . . . . . . . . . . . . . . . . 19 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ (1...((β™―β€˜(1st β€˜π‘)) + 1)) = (1...(𝑁 + 1)))
2926, 28eleq12d 2832 . . . . . . . . . . . . . . . . . 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 1135 . . . . . . . . . . . . . . . . 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 28608 . . . . . . . . . . . . . . . . . . 19 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1))
3332oveq2d 7378 . . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . . . 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 513 . . . . . . . . . . . . . 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 1120 . . . . . . . . . . . . 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 408 . . . . . . . . . 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 409 . . . . . . . 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 14587 . . . . . . . 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 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†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜0) = ((2nd β€˜π‘)β€˜0))
4710, 46eqtrd 2777 . . . . 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 2739 . . . 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 1918 . . . . 5 Ⅎ𝑀((2nd β€˜π‘)β€˜0) = 𝑋
50 fveq2 6847 . . . . . . 7 (𝑀 = 𝑐 β†’ (2nd β€˜π‘€) = (2nd β€˜π‘))
5150fveq1d 6849 . . . . . 6 (𝑀 = 𝑐 β†’ ((2nd β€˜π‘€)β€˜0) = ((2nd β€˜π‘)β€˜0))
5251eqeq1d 2739 . . . . 5 (𝑀 = 𝑐 β†’ (((2nd β€˜π‘€)β€˜0) = 𝑋 ↔ ((2nd β€˜π‘)β€˜0) = 𝑋))
5349, 52sbiev 2309 . . . 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 7079 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ 𝐹:π‘Šβ€“1-1-ontoβ†’{𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘ β€˜0) = 𝑋})
56 clwwlknon 29076 . . 3 (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) = {𝑠 ∈ (𝑁 ClWWalksN 𝐺) ∣ (π‘ β€˜0) = 𝑋}
57 f1oeq3 6779 . . 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 397   ∧ w3a 1088   = wceq 1542  [wsb 2068   ∈ wcel 2107  {crab 3410   class class class wbr 5110   ↦ cmpt 5193  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  0cc0 11058  1c1 11059   + caddc 11061  β„•cn 12160  β„€β‰₯cuz 12770  ...cfz 13431  β™―chash 14237  Word cword 14409   prefix cpfx 14565  Vtxcvtx 27989  USPGraphcuspgr 28141  Walkscwlks 28586  ClWalkscclwlks 28760   ClWWalksN cclwwlkn 29010  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-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  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-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-2o 8418  df-oadd 8421  df-er 8655  df-map 8774  df-pm 8775  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-dju 9844  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-xnn0 12493  df-z 12507  df-uz 12771  df-rp 12923  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-lsw 14458  df-concat 14466  df-s1 14491  df-substr 14536  df-pfx 14566  df-edg 28041  df-uhgr 28051  df-upgr 28075  df-uspgr 28143  df-wlks 28589  df-clwlks 28761  df-clwwlk 28968  df-clwwlkn 29011  df-clwwlknon 29074
This theorem is referenced by:  clwwlknonclwlknonen  29349  dlwwlknondlwlknonf1o  29351
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