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Theorem dlwwlknondlwlknonf1o 29351
Description: 𝐹 is a bijection between the two representations of double loops of a fixed positive length on a fixed vertex. (Contributed by AV, 30-May-2022.) (Revised by AV, 1-Nov-2022.)
Hypotheses
Ref Expression
dlwwlknondlwlknonbij.v 𝑉 = (Vtxβ€˜πΊ)
dlwwlknondlwlknonbij.w π‘Š = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}
dlwwlknondlwlknonbij.d 𝐷 = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋}
dlwwlknondlwlknonf1o.f 𝐹 = (𝑐 ∈ π‘Š ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))))
Assertion
Ref Expression
dlwwlknondlwlknonf1o ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ 𝐹:π‘Šβ€“1-1-onto→𝐷)
Distinct variable groups:   𝐺,𝑐,𝑀   𝑁,𝑐,𝑀   𝑉,𝑐   π‘Š,𝑐   𝑋,𝑐,𝑀
Allowed substitution hints:   𝐷(𝑀,𝑐)   𝐹(𝑀,𝑐)   𝑉(𝑀)   π‘Š(𝑀)

Proof of Theorem dlwwlknondlwlknonf1o
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dlwwlknondlwlknonbij.w . . . 4 π‘Š = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}
2 df-3an 1090 . . . . 5 (((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋) ↔ (((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋))
32rabbii 3416 . . . 4 {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋 ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)} = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}
41, 3eqtri 2765 . . 3 π‘Š = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ (((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ∧ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋)}
5 eqid 2737 . . 3 {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}
6 dlwwlknondlwlknonf1o.f . . 3 𝐹 = (𝑐 ∈ π‘Š ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))))
7 eqid 2737 . . 3 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) = (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))))
8 eluz2nn 12816 . . . 4 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 𝑁 ∈ β„•)
9 dlwwlknondlwlknonbij.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
109, 5, 7clwwlknonclwlknonf1o 29348 . . . 4 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ β„•) β†’ (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))):{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
118, 10syl3an3 1166 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ↦ ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))):{𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}–1-1-ontoβ†’(𝑋(ClWWalksNOnβ€˜πΊ)𝑁))
12 fveq1 6846 . . . . . . 7 (𝑦 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘))) β†’ (π‘¦β€˜(𝑁 βˆ’ 2)) = (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)))
13123ad2ant3 1136 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ∧ 𝑦 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ (π‘¦β€˜(𝑁 βˆ’ 2)) = (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)))
14 2fveq3 6852 . . . . . . . . . . . . 13 (𝑀 = 𝑐 β†’ (β™―β€˜(1st β€˜π‘€)) = (β™―β€˜(1st β€˜π‘)))
1514eqeq1d 2739 . . . . . . . . . . . 12 (𝑀 = 𝑐 β†’ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
16 fveq2 6847 . . . . . . . . . . . . . 14 (𝑀 = 𝑐 β†’ (2nd β€˜π‘€) = (2nd β€˜π‘))
1716fveq1d 6849 . . . . . . . . . . . . 13 (𝑀 = 𝑐 β†’ ((2nd β€˜π‘€)β€˜0) = ((2nd β€˜π‘)β€˜0))
1817eqeq1d 2739 . . . . . . . . . . . 12 (𝑀 = 𝑐 β†’ (((2nd β€˜π‘€)β€˜0) = 𝑋 ↔ ((2nd β€˜π‘)β€˜0) = 𝑋))
1915, 18anbi12d 632 . . . . . . . . . . 11 (𝑀 = 𝑐 β†’ (((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋) ↔ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)))
2019elrab 3650 . . . . . . . . . 10 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ↔ (𝑐 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)))
21 simplrl 776 . . . . . . . . . . . 12 (((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2))) β†’ (β™―β€˜(1st β€˜π‘)) = 𝑁)
22 simpll 766 . . . . . . . . . . . 12 (((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2))) β†’ 𝑐 ∈ (ClWalksβ€˜πΊ))
23 simpr3 1197 . . . . . . . . . . . 12 (((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2))) β†’ 𝑁 ∈ (β„€β‰₯β€˜2))
2421, 22, 233jca 1129 . . . . . . . . . . 11 (((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)) ∧ (𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2))) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)))
2524ex 414 . . . . . . . . . 10 ((𝑐 ∈ (ClWalksβ€˜πΊ) ∧ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ ((2nd β€˜π‘)β€˜0) = 𝑋)) β†’ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2))))
2620, 25sylbi 216 . . . . . . . . 9 (𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} β†’ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2))))
2726impcom 409 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) β†’ ((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)))
28 dlwwlknondlwlknonf1olem1 29350 . . . . . . . 8 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))
2927, 28syl 17 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)}) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))
30293adant3 1133 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ∧ 𝑦 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))
3113, 30eqtrd 2777 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ∧ 𝑦 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ (π‘¦β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))
3231eqeq1d 2739 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ∧ 𝑦 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ ((π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)) = 𝑋))
33 nfv 1918 . . . . 5 Ⅎ𝑀((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)) = 𝑋
3416fveq1d 6849 . . . . . 6 (𝑀 = 𝑐 β†’ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))
3534eqeq1d 2739 . . . . 5 (𝑀 = 𝑐 β†’ (((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)) = 𝑋))
3633, 35sbiev 2309 . . . 4 ([𝑐 / 𝑀]((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)) = 𝑋)
3732, 36bitr4di 289 . . 3 (((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) ∧ 𝑐 ∈ {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ∧ ((2nd β€˜π‘€)β€˜0) = 𝑋)} ∧ 𝑦 = ((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))) β†’ ((π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ [𝑐 / 𝑀]((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋))
384, 5, 6, 7, 11, 37f1ossf1o 7079 . 2 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ 𝐹:π‘Šβ€“1-1-ontoβ†’{𝑦 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋})
39 dlwwlknondlwlknonbij.d . . . 4 𝐷 = {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋}
40 fveq1 6846 . . . . . 6 (𝑀 = 𝑦 β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = (π‘¦β€˜(𝑁 βˆ’ 2)))
4140eqeq1d 2739 . . . . 5 (𝑀 = 𝑦 β†’ ((π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ (π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋))
4241cbvrabv 3420 . . . 4 {𝑀 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋} = {𝑦 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋}
4339, 42eqtri 2765 . . 3 𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋}
44 f1oeq3 6779 . . 3 (𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋} β†’ (𝐹:π‘Šβ€“1-1-onto→𝐷 ↔ 𝐹:π‘Šβ€“1-1-ontoβ†’{𝑦 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋}))
4543, 44ax-mp 5 . 2 (𝐹:π‘Šβ€“1-1-onto→𝐷 ↔ 𝐹:π‘Šβ€“1-1-ontoβ†’{𝑦 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋})
4638, 45sylibr 233 1 ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ 𝐹:π‘Šβ€“1-1-onto→𝐷)
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   ↦ cmpt 5193  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  0cc0 11058   βˆ’ cmin 11392  β„•cn 12160  2c2 12215  β„€β‰₯cuz 12770  β™―chash 14237   prefix cpfx 14565  Vtxcvtx 27989  USPGraphcuspgr 28141  ClWalkscclwlks 28760  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:  dlwwlknondlwlknonen  29352
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