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Mirrors  >  Home  >  MPE Home  >  Th. List  >  2pthnloop Structured version   Visualization version   GIF version
Theorem 2pthnloop 29252
Description: A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon 29668. (Contributed by AV, 6-Feb-2021.)
Hypothesis
Ref Expression
2pthnloop.i 𝐼 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
2pthnloop ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ 1 < (β™―β€˜πΉ)) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–))))
Distinct variable groups:   𝑖,𝐹   𝑖,𝐺   𝑖,𝐼   𝑃,𝑖
Proof of Theorem 2pthnloop
StepHypRef Expression
1 pthiswlk 29248 . . . . 5 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
2 wlkv 29133 . . . . 5 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
31, 2syl 17 . . . 4 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
4 ispth 29244 . . . . . 6 (𝐹(Pathsβ€˜πΊ)𝑃 ↔ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…))
5 istrl 29217 . . . . . . . . . . . 12 (𝐹(Trailsβ€˜πΊ)𝑃 ↔ (𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹))
6 eqid 2731 . . . . . . . . . . . . . 14 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
7 2pthnloop.i . . . . . . . . . . . . . 14 𝐼 = (iEdgβ€˜πΊ)
86, 7iswlkg 29134 . . . . . . . . . . . . 13 (𝐺 ∈ V β†’ (𝐹(Walksβ€˜πΊ)𝑃 ↔ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))))))
98anbi1d 629 . . . . . . . . . . . 12 (𝐺 ∈ V β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ Fun ◑𝐹) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–)))) ∧ Fun ◑𝐹)))
105, 9bitrid 282 . . . . . . . . . . 11 (𝐺 ∈ V β†’ (𝐹(Trailsβ€˜πΊ)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–)))) ∧ Fun ◑𝐹)))
11 pthdadjvtx 29251 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ 1 < (β™―β€˜πΉ) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜π‘–) β‰  (π‘ƒβ€˜(𝑖 + 1)))
1211ad5ant245 1360 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐹(Pathsβ€˜πΊ)𝑃) ∧ ((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))) ∧ 1 < (β™―β€˜πΉ)) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜π‘–) β‰  (π‘ƒβ€˜(𝑖 + 1)))
1312neneqd 2944 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐹(Pathsβ€˜πΊ)𝑃) ∧ ((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))) ∧ 1 < (β™―β€˜πΉ)) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)))
14 ifpfal 1074 . . . . . . . . . . . . . . . . . . . . . 22 (Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)) β†’ (if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))) ↔ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))))
1514adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐹(Pathsβ€˜πΊ)𝑃) ∧ ((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))) ∧ 1 < (β™―β€˜πΉ)) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) ∧ Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1))) β†’ (if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))) ↔ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))))
16 fvexd 6907 . . . . . . . . . . . . . . . . . . . . . . 23 (Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)) β†’ (π‘ƒβ€˜π‘–) ∈ V)
17 fvexd 6907 . . . . . . . . . . . . . . . . . . . . . . 23 (Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)) β†’ (π‘ƒβ€˜(𝑖 + 1)) ∈ V)
18 neqne 2947 . . . . . . . . . . . . . . . . . . . . . . 23 (Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)) β†’ (π‘ƒβ€˜π‘–) β‰  (π‘ƒβ€˜(𝑖 + 1)))
19 fvexd 6907 . . . . . . . . . . . . . . . . . . . . . . 23 (Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)) β†’ (πΌβ€˜(πΉβ€˜π‘–)) ∈ V)
20 prsshashgt1 14375 . . . . . . . . . . . . . . . . . . . . . . 23 ((((π‘ƒβ€˜π‘–) ∈ V ∧ (π‘ƒβ€˜(𝑖 + 1)) ∈ V ∧ (π‘ƒβ€˜π‘–) β‰  (π‘ƒβ€˜(𝑖 + 1))) ∧ (πΌβ€˜(πΉβ€˜π‘–)) ∈ V) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–)) β†’ 2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))
2116, 17, 18, 19, 20syl31anc 1372 . . . . . . . . . . . . . . . . . . . . . 22 (Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–)) β†’ 2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))
2221adantl 481 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐹(Pathsβ€˜πΊ)𝑃) ∧ ((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))) ∧ 1 < (β™―β€˜πΉ)) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) ∧ Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1))) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–)) β†’ 2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))
2315, 22sylbid 239 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐹(Pathsβ€˜πΊ)𝑃) ∧ ((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))) ∧ 1 < (β™―β€˜πΉ)) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) ∧ Β¬ (π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1))) β†’ (if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))) β†’ 2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))
2413, 23mpdan 684 . . . . . . . . . . . . . . . . . . 19 ((((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐹(Pathsβ€˜πΊ)𝑃) ∧ ((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))) ∧ 1 < (β™―β€˜πΉ)) ∧ 𝑖 ∈ (0..^(β™―β€˜πΉ))) β†’ (if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))) β†’ 2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))
2524ralimdva 3166 . . . . . . . . . . . . . . . . . 18 (((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐹(Pathsβ€˜πΊ)𝑃) ∧ ((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))) ∧ 1 < (β™―β€˜πΉ)) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))
2625ex 412 . . . . . . . . . . . . . . . . 17 ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐹(Pathsβ€˜πΊ)𝑃) ∧ ((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))) β†’ (1 < (β™―β€˜πΉ) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–))))))
2726com23 86 . . . . . . . . . . . . . . . 16 ((((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) ∧ 𝐹(Pathsβ€˜πΊ)𝑃) ∧ ((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))))) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))) β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–))))))
2827exp31 419 . . . . . . . . . . . . . . 15 ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ)))) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))) β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–))))))))
2928com24 95 . . . . . . . . . . . . . 14 ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ)) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–))) β†’ (((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ)))) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–))))))))
30293impia 1116 . . . . . . . . . . . . 13 ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–)))) β†’ (((Fun ◑𝐹 ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ)))) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))))
3130exp4c 432 . . . . . . . . . . . 12 ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–)))) β†’ (Fun ◑𝐹 β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))))))
3231imp 406 . . . . . . . . . . 11 (((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))⟢(Vtxβ€˜πΊ) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘–) = (π‘ƒβ€˜(𝑖 + 1)), (πΌβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–)}, {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘–)))) ∧ Fun ◑𝐹) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–))))))))
3310, 32syl6bi 252 . . . . . . . . . 10 (𝐺 ∈ V β†’ (𝐹(Trailsβ€˜πΊ)𝑃 β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))))))
3433com24 95 . . . . . . . . 9 (𝐺 ∈ V β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ (𝐹(Trailsβ€˜πΊ)𝑃 β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))))))
3534com14 96 . . . . . . . 8 (𝐹(Trailsβ€˜πΊ)𝑃 β†’ (Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) β†’ (((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ… β†’ (𝐺 ∈ V β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))))))
36353imp 1110 . . . . . . 7 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (𝐺 ∈ V β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))))
3736com12 32 . . . . . 6 (𝐺 ∈ V β†’ ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ Fun β—‘(𝑃 β†Ύ (1..^(β™―β€˜πΉ))) ∧ ((𝑃 β€œ {0, (β™―β€˜πΉ)}) ∩ (𝑃 β€œ (1..^(β™―β€˜πΉ)))) = βˆ…) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))))
384, 37biimtrid 241 . . . . 5 (𝐺 ∈ V β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))))
39383ad2ant1 1132 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))))
403, 39mpcom 38 . . 3 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–))))))
4140pm2.43i 52 . 2 (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (1 < (β™―β€˜πΉ) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–)))))
4241imp 406 1 ((𝐹(Pathsβ€˜πΊ)𝑃 ∧ 1 < (β™―β€˜πΉ)) β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))2 ≀ (β™―β€˜(πΌβ€˜(πΉβ€˜π‘–))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395  if-wif 1060   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  Vcvv 3473   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  {csn 4629  {cpr 4631   class class class wbr 5149  β—‘ccnv 5676  dom cdm 5677   β†Ύ cres 5679   β€œ cima 5680  Fun wfun 6538  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412  0cc0 11113  1c1 11114   + caddc 11116   < clt 11253   ≀ cle 11254  2c2 12272  ...cfz 13489  ..^cfzo 13632  β™―chash 14295  Word cword 14469  Vtxcvtx 28520  iEdgciedg 28521  Walkscwlks 29117  Trailsctrls 29211  Pathscpths 29233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-oadd 8473  df-er 8706  df-map 8825  df-pm 8826  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-dju 9899  df-card 9937  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-n0 12478  df-xnn0 12550  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-wlks 29120  df-trls 29213  df-pths 29237
This theorem is referenced by:  upgr2pthnlp  29253
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