MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dlwwlknondlwlknonf1o Structured version   Visualization version   GIF version

Theorem dlwwlknondlwlknonf1o 29883
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 1087 . . . . 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 3436 . . . 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 2758 . . 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 2730 . . 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 2730 . . 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 12874 . . . 4 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 𝑁 ∈ β„•)
9 dlwwlknondlwlknonbij.v . . . . 5 𝑉 = (Vtxβ€˜πΊ)
109, 5, 7clwwlknonclwlknonf1o 29880 . . . 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 1163 . . 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 6891 . . . . . . 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 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†’ (π‘¦β€˜(𝑁 βˆ’ 2)) = (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)))
14 2fveq3 6897 . . . . . . . . . . . . 13 (𝑀 = 𝑐 β†’ (β™―β€˜(1st β€˜π‘€)) = (β™―β€˜(1st β€˜π‘)))
1514eqeq1d 2732 . . . . . . . . . . . 12 (𝑀 = 𝑐 β†’ ((β™―β€˜(1st β€˜π‘€)) = 𝑁 ↔ (β™―β€˜(1st β€˜π‘)) = 𝑁))
16 fveq2 6892 . . . . . . . . . . . . . 14 (𝑀 = 𝑐 β†’ (2nd β€˜π‘€) = (2nd β€˜π‘))
1716fveq1d 6894 . . . . . . . . . . . . 13 (𝑀 = 𝑐 β†’ ((2nd β€˜π‘€)β€˜0) = ((2nd β€˜π‘)β€˜0))
1817eqeq1d 2732 . . . . . . . . . . . 12 (𝑀 = 𝑐 β†’ (((2nd β€˜π‘€)β€˜0) = 𝑋 ↔ ((2nd β€˜π‘)β€˜0) = 𝑋))
1915, 18anbi12d 629 . . . . . . . . . . 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 3684 . . . . . . . . . 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 773 . . . . . . . . . . . 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 763 . . . . . . . . . . . 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 1194 . . . . . . . . . . . 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 1126 . . . . . . . . . . 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 411 . . . . . . . . . 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 406 . . . . . . . 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 29882 . . . . . . . 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 1130 . . . . . 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 2770 . . . . 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 2732 . . . 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 1915 . . . . 5 Ⅎ𝑀((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)) = 𝑋
3416fveq1d 6894 . . . . . 6 (𝑀 = 𝑐 β†’ ((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))
3534eqeq1d 2732 . . . . 5 (𝑀 = 𝑐 β†’ (((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)) = 𝑋))
3633, 35sbiev 2306 . . . 4 ([𝑐 / 𝑀]((2nd β€˜π‘€)β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)) = 𝑋)
3732, 36bitr4di 288 . . 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 7129 . 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 6891 . . . . . 6 (𝑀 = 𝑦 β†’ (π‘€β€˜(𝑁 βˆ’ 2)) = (π‘¦β€˜(𝑁 βˆ’ 2)))
4140eqeq1d 2732 . . . . 5 (𝑀 = 𝑦 β†’ ((π‘€β€˜(𝑁 βˆ’ 2)) = 𝑋 ↔ (π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋))
4241cbvrabv 3440 . . . 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 2758 . . 3 𝐷 = {𝑦 ∈ (𝑋(ClWWalksNOnβ€˜πΊ)𝑁) ∣ (π‘¦β€˜(𝑁 βˆ’ 2)) = 𝑋}
44 f1oeq3 6824 . . 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 394   ∧ w3a 1085   = wceq 1539  [wsb 2065   ∈ wcel 2104  {crab 3430   ↦ cmpt 5232  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7413  1st c1st 7977  2nd c2nd 7978  0cc0 11114   βˆ’ cmin 11450  β„•cn 12218  2c2 12273  β„€β‰₯cuz 12828  β™―chash 14296   prefix cpfx 14626  Vtxcvtx 28521  USPGraphcuspgr 28673  ClWalkscclwlks 29292  ClWWalksNOncclwwlknon 29605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-oadd 8474  df-er 8707  df-map 8826  df-pm 8827  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-dju 9900  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-nn 12219  df-2 12281  df-n0 12479  df-xnn0 12551  df-z 12565  df-uz 12829  df-rp 12981  df-fz 13491  df-fzo 13634  df-hash 14297  df-word 14471  df-lsw 14519  df-concat 14527  df-s1 14552  df-substr 14597  df-pfx 14627  df-edg 28573  df-uhgr 28583  df-upgr 28607  df-uspgr 28675  df-wlks 29121  df-clwlks 29293  df-clwwlk 29500  df-clwwlkn 29543  df-clwwlknon 29606
This theorem is referenced by:  dlwwlknondlwlknonen  29884
  Copyright terms: Public domain W3C validator