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Theorem wlkres 26623
Description: The restriction 𝐻, 𝑄 of a walk 𝐹, 𝑃 to an initial segment of the walk (of length 𝑁) forms a walk on the subgraph 𝑆 consisting of the edges in the initial segment. Formerly proven directly for Eulerian paths, see eupthres 27193. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.)
Hypotheses
Ref Expression
wlkres.v 𝑉 = (Vtx‘𝐺)
wlkres.i 𝐼 = (iEdg‘𝐺)
wlkres.d (𝜑𝐹(Walks‘𝐺)𝑃)
wlkres.n (𝜑𝑁 ∈ (0..^(#‘𝐹)))
wlkres.s (𝜑 → (Vtx‘𝑆) = 𝑉)
wlkres.e (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
wlkres.h 𝐻 = (𝐹 ↾ (0..^𝑁))
wlkres.q 𝑄 = (𝑃 ↾ (0...𝑁))
Assertion
Ref Expression
wlkres (𝜑𝐻(Walks‘𝑆)𝑄)

Proof of Theorem wlkres
Dummy variables 𝑘 𝑖 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wlkres.h . . 3 𝐻 = (𝐹 ↾ (0..^𝑁))
2 wlkres.d . . . . . . . 8 (𝜑𝐹(Walks‘𝐺)𝑃)
3 wlkres.i . . . . . . . . 9 𝐼 = (iEdg‘𝐺)
43wlkf 26566 . . . . . . . 8 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
5 wrdfn 13351 . . . . . . . 8 (𝐹 ∈ Word dom 𝐼𝐹 Fn (0..^(#‘𝐹)))
62, 4, 53syl 18 . . . . . . 7 (𝜑𝐹 Fn (0..^(#‘𝐹)))
7 wlkres.n . . . . . . . 8 (𝜑𝑁 ∈ (0..^(#‘𝐹)))
8 elfzouz2 12523 . . . . . . . 8 (𝑁 ∈ (0..^(#‘𝐹)) → (#‘𝐹) ∈ (ℤ𝑁))
9 fzoss2 12535 . . . . . . . 8 ((#‘𝐹) ∈ (ℤ𝑁) → (0..^𝑁) ⊆ (0..^(#‘𝐹)))
107, 8, 93syl 18 . . . . . . 7 (𝜑 → (0..^𝑁) ⊆ (0..^(#‘𝐹)))
11 fnssres 6042 . . . . . . 7 ((𝐹 Fn (0..^(#‘𝐹)) ∧ (0..^𝑁) ⊆ (0..^(#‘𝐹))) → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
126, 10, 11syl2anc 694 . . . . . 6 (𝜑 → (𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁))
13 simpr 476 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
14 fveq2 6229 . . . . . . . . . . . . . 14 (𝑖 = 𝑥 → (𝐹𝑖) = (𝐹𝑥))
1514eqeq1d 2653 . . . . . . . . . . . . 13 (𝑖 = 𝑥 → ((𝐹𝑖) = ((𝐹 ↾ (0..^𝑁))‘𝑥) ↔ (𝐹𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥)))
1615adantl 481 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ (0..^𝑁)) ∧ 𝑖 = 𝑥) → ((𝐹𝑖) = ((𝐹 ↾ (0..^𝑁))‘𝑥) ↔ (𝐹𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥)))
17 fvres 6245 . . . . . . . . . . . . . 14 (𝑥 ∈ (0..^𝑁) → ((𝐹 ↾ (0..^𝑁))‘𝑥) = (𝐹𝑥))
1817adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝐹 ↾ (0..^𝑁))‘𝑥) = (𝐹𝑥))
1918eqcomd 2657 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (0..^𝑁)) → (𝐹𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
2013, 16, 19rspcedvd 3348 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (0..^𝑁)) → ∃𝑖 ∈ (0..^𝑁)(𝐹𝑖) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
216adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (0..^𝑁)) → 𝐹 Fn (0..^(#‘𝐹)))
2210adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (0..^𝑁)) → (0..^𝑁) ⊆ (0..^(#‘𝐹)))
2321, 22fvelimabd 6293 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (0..^𝑁)) → (((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ (𝐹 “ (0..^𝑁)) ↔ ∃𝑖 ∈ (0..^𝑁)(𝐹𝑖) = ((𝐹 ↾ (0..^𝑁))‘𝑥)))
2420, 23mpbird 247 . . . . . . . . . 10 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ (𝐹 “ (0..^𝑁)))
252, 4syl 17 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ Word dom 𝐼)
26 wrdf 13342 . . . . . . . . . . . . . 14 (𝐹 ∈ Word dom 𝐼𝐹:(0..^(#‘𝐹))⟶dom 𝐼)
27 ffvelrn 6397 . . . . . . . . . . . . . . . . 17 ((𝐹:(0..^(#‘𝐹))⟶dom 𝐼𝑥 ∈ (0..^(#‘𝐹))) → (𝐹𝑥) ∈ dom 𝐼)
2827expcom 450 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (0..^(#‘𝐹)) → (𝐹:(0..^(#‘𝐹))⟶dom 𝐼 → (𝐹𝑥) ∈ dom 𝐼))
2910sselda 3636 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^(#‘𝐹)))
3028, 29syl11 33 . . . . . . . . . . . . . . 15 (𝐹:(0..^(#‘𝐹))⟶dom 𝐼 → ((𝜑𝑥 ∈ (0..^𝑁)) → (𝐹𝑥) ∈ dom 𝐼))
3130expd 451 . . . . . . . . . . . . . 14 (𝐹:(0..^(#‘𝐹))⟶dom 𝐼 → (𝜑 → (𝑥 ∈ (0..^𝑁) → (𝐹𝑥) ∈ dom 𝐼)))
3226, 31syl 17 . . . . . . . . . . . . 13 (𝐹 ∈ Word dom 𝐼 → (𝜑 → (𝑥 ∈ (0..^𝑁) → (𝐹𝑥) ∈ dom 𝐼)))
3325, 32mpcom 38 . . . . . . . . . . . 12 (𝜑 → (𝑥 ∈ (0..^𝑁) → (𝐹𝑥) ∈ dom 𝐼))
3433imp 444 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (0..^𝑁)) → (𝐹𝑥) ∈ dom 𝐼)
3518, 34eqeltrd 2730 . . . . . . . . . 10 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ dom 𝐼)
3624, 35elind 3831 . . . . . . . . 9 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼))
37 dmres 5454 . . . . . . . . 9 dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = ((𝐹 “ (0..^𝑁)) ∩ dom 𝐼)
3836, 37syl6eleqr 2741 . . . . . . . 8 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))
39 wlkres.e . . . . . . . . . . 11 (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
4039dmeqd 5358 . . . . . . . . . 10 (𝜑 → dom (iEdg‘𝑆) = dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))
4140eleq2d 2716 . . . . . . . . 9 (𝜑 → (((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ dom (iEdg‘𝑆) ↔ ((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))))
4241adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ (0..^𝑁)) → (((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ dom (iEdg‘𝑆) ↔ ((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))))
4338, 42mpbird 247 . . . . . . 7 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ dom (iEdg‘𝑆))
4443ralrimiva 2995 . . . . . 6 (𝜑 → ∀𝑥 ∈ (0..^𝑁)((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ dom (iEdg‘𝑆))
45 ffnfv 6428 . . . . . 6 ((𝐹 ↾ (0..^𝑁)):(0..^𝑁)⟶dom (iEdg‘𝑆) ↔ ((𝐹 ↾ (0..^𝑁)) Fn (0..^𝑁) ∧ ∀𝑥 ∈ (0..^𝑁)((𝐹 ↾ (0..^𝑁))‘𝑥) ∈ dom (iEdg‘𝑆)))
4612, 44, 45sylanbrc 699 . . . . 5 (𝜑 → (𝐹 ↾ (0..^𝑁)):(0..^𝑁)⟶dom (iEdg‘𝑆))
47 fzossfz 12527 . . . . . . . . 9 (0..^(#‘𝐹)) ⊆ (0...(#‘𝐹))
4847, 7sseldi 3634 . . . . . . . 8 (𝜑𝑁 ∈ (0...(#‘𝐹)))
49 wlkreslem0 26621 . . . . . . . 8 ((𝐹 ∈ Word dom 𝐼𝑁 ∈ (0...(#‘𝐹))) → (#‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
5025, 48, 49syl2anc 694 . . . . . . 7 (𝜑 → (#‘(𝐹 ↾ (0..^𝑁))) = 𝑁)
5150oveq2d 6706 . . . . . 6 (𝜑 → (0..^(#‘(𝐹 ↾ (0..^𝑁)))) = (0..^𝑁))
5251feq2d 6069 . . . . 5 (𝜑 → ((𝐹 ↾ (0..^𝑁)):(0..^(#‘(𝐹 ↾ (0..^𝑁))))⟶dom (iEdg‘𝑆) ↔ (𝐹 ↾ (0..^𝑁)):(0..^𝑁)⟶dom (iEdg‘𝑆)))
5346, 52mpbird 247 . . . 4 (𝜑 → (𝐹 ↾ (0..^𝑁)):(0..^(#‘(𝐹 ↾ (0..^𝑁))))⟶dom (iEdg‘𝑆))
54 iswrdb 13343 . . . 4 ((𝐹 ↾ (0..^𝑁)) ∈ Word dom (iEdg‘𝑆) ↔ (𝐹 ↾ (0..^𝑁)):(0..^(#‘(𝐹 ↾ (0..^𝑁))))⟶dom (iEdg‘𝑆))
5553, 54sylibr 224 . . 3 (𝜑 → (𝐹 ↾ (0..^𝑁)) ∈ Word dom (iEdg‘𝑆))
561, 55syl5eqel 2734 . 2 (𝜑𝐻 ∈ Word dom (iEdg‘𝑆))
57 wlkres.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
5857wlkp 26568 . . . . . . 7 (𝐹(Walks‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
592, 58syl 17 . . . . . 6 (𝜑𝑃:(0...(#‘𝐹))⟶𝑉)
60 wlkres.s . . . . . . 7 (𝜑 → (Vtx‘𝑆) = 𝑉)
6160feq3d 6070 . . . . . 6 (𝜑 → (𝑃:(0...(#‘𝐹))⟶(Vtx‘𝑆) ↔ 𝑃:(0...(#‘𝐹))⟶𝑉))
6259, 61mpbird 247 . . . . 5 (𝜑𝑃:(0...(#‘𝐹))⟶(Vtx‘𝑆))
63 elfzuz3 12377 . . . . . 6 (𝑁 ∈ (0...(#‘𝐹)) → (#‘𝐹) ∈ (ℤ𝑁))
64 fzss2 12419 . . . . . 6 ((#‘𝐹) ∈ (ℤ𝑁) → (0...𝑁) ⊆ (0...(#‘𝐹)))
6548, 63, 643syl 18 . . . . 5 (𝜑 → (0...𝑁) ⊆ (0...(#‘𝐹)))
6662, 65fssresd 6109 . . . 4 (𝜑 → (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶(Vtx‘𝑆))
671fveq2i 6232 . . . . . . 7 (#‘𝐻) = (#‘(𝐹 ↾ (0..^𝑁)))
6867, 50syl5eq 2697 . . . . . 6 (𝜑 → (#‘𝐻) = 𝑁)
6968oveq2d 6706 . . . . 5 (𝜑 → (0...(#‘𝐻)) = (0...𝑁))
7069feq2d 6069 . . . 4 (𝜑 → ((𝑃 ↾ (0...𝑁)):(0...(#‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ↾ (0...𝑁)):(0...𝑁)⟶(Vtx‘𝑆)))
7166, 70mpbird 247 . . 3 (𝜑 → (𝑃 ↾ (0...𝑁)):(0...(#‘𝐻))⟶(Vtx‘𝑆))
72 wlkres.q . . . 4 𝑄 = (𝑃 ↾ (0...𝑁))
7372feq1i 6074 . . 3 (𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆) ↔ (𝑃 ↾ (0...𝑁)):(0...(#‘𝐻))⟶(Vtx‘𝑆))
7471, 73sylibr 224 . 2 (𝜑𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆))
75 wlkv 26564 . . . . . . 7 (𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))
7657, 3iswlk 26562 . . . . . . . 8 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
7776biimpd 219 . . . . . . 7 ((𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V) → (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
7875, 77mpcom 38 . . . . . 6 (𝐹(Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
792, 78syl 17 . . . . 5 (𝜑 → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
8079adantr 480 . . . 4 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
8168oveq2d 6706 . . . . . . . . . . 11 (𝜑 → (0..^(#‘𝐻)) = (0..^𝑁))
8281eleq2d 2716 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (0..^(#‘𝐻)) ↔ 𝑥 ∈ (0..^𝑁)))
8372fveq1i 6230 . . . . . . . . . . . . 13 (𝑄𝑥) = ((𝑃 ↾ (0...𝑁))‘𝑥)
84 fzossfz 12527 . . . . . . . . . . . . . . . 16 (0..^𝑁) ⊆ (0...𝑁)
8584a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (0..^𝑁) ⊆ (0...𝑁))
8685sselda 3636 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0...𝑁))
8786fvresd 6246 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘𝑥) = (𝑃𝑥))
8883, 87syl5req 2698 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (0..^𝑁)) → (𝑃𝑥) = (𝑄𝑥))
8972fveq1i 6230 . . . . . . . . . . . . 13 (𝑄‘(𝑥 + 1)) = ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1))
90 fzofzp1 12605 . . . . . . . . . . . . . . 15 (𝑥 ∈ (0..^𝑁) → (𝑥 + 1) ∈ (0...𝑁))
9190adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (0..^𝑁)) → (𝑥 + 1) ∈ (0...𝑁))
9291fvresd 6246 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝑃 ↾ (0...𝑁))‘(𝑥 + 1)) = (𝑃‘(𝑥 + 1)))
9389, 92syl5req 2698 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (0..^𝑁)) → (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))
9488, 93jca 553 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
9594ex 449 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (0..^𝑁) → ((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))))
9682, 95sylbid 230 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (0..^(#‘𝐻)) → ((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1)))))
9796imp 444 . . . . . . . 8 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → ((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))))
9825ancli 573 . . . . . . . . . . . . . 14 (𝜑 → (𝜑𝐹 ∈ Word dom 𝐼))
9926ffund 6087 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ Word dom 𝐼 → Fun 𝐹)
10099adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ Word dom 𝐼) → Fun 𝐹)
101100adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → Fun 𝐹)
102 fdm 6089 . . . . . . . . . . . . . . . . . 18 (𝐹:(0..^(#‘𝐹))⟶dom 𝐼 → dom 𝐹 = (0..^(#‘𝐹)))
103 sseq2 3660 . . . . . . . . . . . . . . . . . . 19 (dom 𝐹 = (0..^(#‘𝐹)) → ((0..^𝑁) ⊆ dom 𝐹 ↔ (0..^𝑁) ⊆ (0..^(#‘𝐹))))
10410, 103syl5ibr 236 . . . . . . . . . . . . . . . . . 18 (dom 𝐹 = (0..^(#‘𝐹)) → (𝜑 → (0..^𝑁) ⊆ dom 𝐹))
10526, 102, 1043syl 18 . . . . . . . . . . . . . . . . 17 (𝐹 ∈ Word dom 𝐼 → (𝜑 → (0..^𝑁) ⊆ dom 𝐹))
106105impcom 445 . . . . . . . . . . . . . . . 16 ((𝜑𝐹 ∈ Word dom 𝐼) → (0..^𝑁) ⊆ dom 𝐹)
107106adantr 480 . . . . . . . . . . . . . . 15 (((𝜑𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → (0..^𝑁) ⊆ dom 𝐹)
108 simpr 476 . . . . . . . . . . . . . . 15 (((𝜑𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → 𝑥 ∈ (0..^𝑁))
109101, 107, 108resfvresima 6534 . . . . . . . . . . . . . 14 (((𝜑𝐹 ∈ Word dom 𝐼) ∧ 𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹𝑥)))
11098, 109sylan 487 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (0..^𝑁)) → ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)) = (𝐼‘(𝐹𝑥)))
111110eqcomd 2657 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (0..^𝑁)) → (𝐼‘(𝐹𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
112111ex 449 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0..^𝑁) → (𝐼‘(𝐹𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥))))
11382, 112sylbid 230 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (0..^(#‘𝐻)) → (𝐼‘(𝐹𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥))))
114113imp 444 . . . . . . . . 9 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → (𝐼‘(𝐹𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
11539adantr 480 . . . . . . . . . 10 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))
1161fveq1i 6230 . . . . . . . . . . 11 (𝐻𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥)
117116a1i 11 . . . . . . . . . 10 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → (𝐻𝑥) = ((𝐹 ↾ (0..^𝑁))‘𝑥))
118115, 117fveq12d 6235 . . . . . . . . 9 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → ((iEdg‘𝑆)‘(𝐻𝑥)) = ((𝐼 ↾ (𝐹 “ (0..^𝑁)))‘((𝐹 ↾ (0..^𝑁))‘𝑥)))
119114, 118eqtr4d 2688 . . . . . . . 8 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥)))
12097, 119jca 553 . . . . . . 7 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → (((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥))))
1217, 8syl 17 . . . . . . . . . . 11 (𝜑 → (#‘𝐹) ∈ (ℤ𝑁))
1221a1i 11 . . . . . . . . . . . . . 14 (𝜑𝐻 = (𝐹 ↾ (0..^𝑁)))
123122fveq2d 6233 . . . . . . . . . . . . 13 (𝜑 → (#‘𝐻) = (#‘(𝐹 ↾ (0..^𝑁))))
124123, 50eqtrd 2685 . . . . . . . . . . . 12 (𝜑 → (#‘𝐻) = 𝑁)
125124fveq2d 6233 . . . . . . . . . . 11 (𝜑 → (ℤ‘(#‘𝐻)) = (ℤ𝑁))
126121, 125eleqtrrd 2733 . . . . . . . . . 10 (𝜑 → (#‘𝐹) ∈ (ℤ‘(#‘𝐻)))
127 fzoss2 12535 . . . . . . . . . 10 ((#‘𝐹) ∈ (ℤ‘(#‘𝐻)) → (0..^(#‘𝐻)) ⊆ (0..^(#‘𝐹)))
128126, 127syl 17 . . . . . . . . 9 (𝜑 → (0..^(#‘𝐻)) ⊆ (0..^(#‘𝐹)))
129128sselda 3636 . . . . . . . 8 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → 𝑥 ∈ (0..^(#‘𝐹)))
130 wkslem1 26559 . . . . . . . . 9 (𝑘 = 𝑥 → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ↔ if-((𝑃𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹𝑥)) = {(𝑃𝑥)}, {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹𝑥)))))
131130rspcv 3336 . . . . . . . 8 (𝑥 ∈ (0..^(#‘𝐹)) → (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → if-((𝑃𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹𝑥)) = {(𝑃𝑥)}, {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹𝑥)))))
132129, 131syl 17 . . . . . . 7 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → if-((𝑃𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹𝑥)) = {(𝑃𝑥)}, {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹𝑥)))))
133 eqeq12 2664 . . . . . . . . . 10 (((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → ((𝑃𝑥) = (𝑃‘(𝑥 + 1)) ↔ (𝑄𝑥) = (𝑄‘(𝑥 + 1))))
134133adantr 480 . . . . . . . . 9 ((((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥))) → ((𝑃𝑥) = (𝑃‘(𝑥 + 1)) ↔ (𝑄𝑥) = (𝑄‘(𝑥 + 1))))
135 simpr 476 . . . . . . . . . 10 ((((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥))) → (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥)))
136 sneq 4220 . . . . . . . . . . . 12 ((𝑃𝑥) = (𝑄𝑥) → {(𝑃𝑥)} = {(𝑄𝑥)})
137136adantr 480 . . . . . . . . . . 11 (((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃𝑥)} = {(𝑄𝑥)})
138137adantr 480 . . . . . . . . . 10 ((((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥))) → {(𝑃𝑥)} = {(𝑄𝑥)})
139135, 138eqeq12d 2666 . . . . . . . . 9 ((((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥))) → ((𝐼‘(𝐹𝑥)) = {(𝑃𝑥)} ↔ ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}))
140 preq12 4302 . . . . . . . . . . 11 (((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) → {(𝑃𝑥), (𝑃‘(𝑥 + 1))} = {(𝑄𝑥), (𝑄‘(𝑥 + 1))})
141140adantr 480 . . . . . . . . . 10 ((((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥))) → {(𝑃𝑥), (𝑃‘(𝑥 + 1))} = {(𝑄𝑥), (𝑄‘(𝑥 + 1))})
142141, 135sseq12d 3667 . . . . . . . . 9 ((((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥))) → ({(𝑃𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹𝑥)) ↔ {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥))))
143134, 139, 142ifpbi123d 1047 . . . . . . . 8 ((((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥))) → (if-((𝑃𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹𝑥)) = {(𝑃𝑥)}, {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹𝑥))) ↔ if-((𝑄𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}, {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥)))))
144143biimpd 219 . . . . . . 7 ((((𝑃𝑥) = (𝑄𝑥) ∧ (𝑃‘(𝑥 + 1)) = (𝑄‘(𝑥 + 1))) ∧ (𝐼‘(𝐹𝑥)) = ((iEdg‘𝑆)‘(𝐻𝑥))) → (if-((𝑃𝑥) = (𝑃‘(𝑥 + 1)), (𝐼‘(𝐹𝑥)) = {(𝑃𝑥)}, {(𝑃𝑥), (𝑃‘(𝑥 + 1))} ⊆ (𝐼‘(𝐹𝑥))) → if-((𝑄𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}, {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥)))))
145120, 132, 144sylsyld 61 . . . . . 6 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → if-((𝑄𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}, {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥)))))
146145com12 32 . . . . 5 (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) → ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → if-((𝑄𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}, {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥)))))
1471463ad2ant3 1104 . . . 4 ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))) → ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → if-((𝑄𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}, {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥)))))
14880, 147mpcom 38 . . 3 ((𝜑𝑥 ∈ (0..^(#‘𝐻))) → if-((𝑄𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}, {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥))))
149148ralrimiva 2995 . 2 (𝜑 → ∀𝑥 ∈ (0..^(#‘𝐻))if-((𝑄𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}, {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥))))
15057, 3, 2, 7, 60, 39, 1, 72wlkreslem 26622 . . 3 (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
151 eqid 2651 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
152 eqid 2651 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
153151, 152iswlk 26562 . . 3 ((𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V) → (𝐻(Walks‘𝑆)𝑄 ↔ (𝐻 ∈ Word dom (iEdg‘𝑆) ∧ 𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆) ∧ ∀𝑥 ∈ (0..^(#‘𝐻))if-((𝑄𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}, {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥))))))
154150, 153syl 17 . 2 (𝜑 → (𝐻(Walks‘𝑆)𝑄 ↔ (𝐻 ∈ Word dom (iEdg‘𝑆) ∧ 𝑄:(0...(#‘𝐻))⟶(Vtx‘𝑆) ∧ ∀𝑥 ∈ (0..^(#‘𝐻))if-((𝑄𝑥) = (𝑄‘(𝑥 + 1)), ((iEdg‘𝑆)‘(𝐻𝑥)) = {(𝑄𝑥)}, {(𝑄𝑥), (𝑄‘(𝑥 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑥))))))
15556, 74, 149, 154mpbir3and 1264 1 (𝜑𝐻(Walks‘𝑆)𝑄)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  if-wif 1032  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  {csn 4210  {cpr 4212   class class class wbr 4685  dom cdm 5143  cres 5145  cima 5146  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  cuz 11725  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323  Vtxcvtx 25919  iEdgciedg 25920  Walkscwlks 26548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-substr 13335  df-wlks 26551
This theorem is referenced by:  trlres  26653  eupthres  27193
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