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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eucrct2eupth Structured version   Visualization version   GIF version

Theorem eucrct2eupth 41413
Description: Removing one edge (𝐼‘(𝐹𝐽)) from a graph 𝐺 with an Eulerian circuit 𝐹, 𝑃 results in a graph 𝑆 with an Eulerian path 𝐻, 𝑄. (Contributed by AV, 17-Mar-2021.)
Hypotheses
Ref Expression
eucrct2eupth1.v 𝑉 = (Vtx‘𝐺)
eucrct2eupth1.i 𝐼 = (iEdg‘𝐺)
eucrct2eupth1.d (𝜑𝐹(EulerPaths‘𝐺)𝑃)
eucrct2eupth1.c (𝜑𝐹(CircuitS‘𝐺)𝑃)
eucrct2eupth1.s (Vtx‘𝑆) = 𝑉
eucrct2eupth.n (𝜑𝑁 = (#‘𝐹))
eucrct2eupth.j (𝜑𝐽 ∈ (0..^𝑁))
eucrct2eupth.e (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ ((0..^𝑁) ∖ {𝐽}))))
eucrct2eupth.k 𝐾 = (𝐽 + 1)
eucrct2eupth.h 𝐻 = ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1)))
eucrct2eupth.q 𝑄 = (𝑥 ∈ (0..^𝑁) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁))))
Assertion
Ref Expression
eucrct2eupth (𝜑𝐻(EulerPaths‘𝑆)𝑄)
Distinct variable groups:   𝑥,𝐹   𝑥,𝐼   𝑥,𝐽   𝑥,𝐾   𝑥,𝑁   𝑥,𝑃   𝑥,𝑉   𝜑,𝑥
Allowed substitution hints:   𝑄(𝑥)   𝑆(𝑥)   𝐺(𝑥)   𝐻(𝑥)

Proof of Theorem eucrct2eupth
StepHypRef Expression
1 eucrct2eupth1.v . . . 4 𝑉 = (Vtx‘𝐺)
2 eucrct2eupth1.i . . . 4 𝐼 = (iEdg‘𝐺)
3 eucrct2eupth1.d . . . . . 6 (𝜑𝐹(EulerPaths‘𝐺)𝑃)
43adantl 481 . . . . 5 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝐹(EulerPaths‘𝐺)𝑃)
5 eucrct2eupth.k . . . . . . . 8 𝐾 = (𝐽 + 1)
65eqcomi 2619 . . . . . . 7 (𝐽 + 1) = 𝐾
76oveq2i 6560 . . . . . 6 (𝐹 cyclShift (𝐽 + 1)) = (𝐹 cyclShift 𝐾)
8 oveq1 6556 . . . . . . . . 9 (𝐽 = (𝑁 − 1) → (𝐽 + 1) = ((𝑁 − 1) + 1))
9 eucrct2eupth.j . . . . . . . . . 10 (𝜑𝐽 ∈ (0..^𝑁))
10 elfzo0 12376 . . . . . . . . . . 11 (𝐽 ∈ (0..^𝑁) ↔ (𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁))
11 nncn 10905 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℂ)
12113ad2ant2 1076 . . . . . . . . . . 11 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → 𝑁 ∈ ℂ)
1310, 12sylbi 206 . . . . . . . . . 10 (𝐽 ∈ (0..^𝑁) → 𝑁 ∈ ℂ)
14 npcan1 10334 . . . . . . . . . 10 (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁)
159, 13, 143syl 18 . . . . . . . . 9 (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
168, 15sylan9eq 2664 . . . . . . . 8 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐽 + 1) = 𝑁)
1716oveq2d 6565 . . . . . . 7 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 cyclShift (𝐽 + 1)) = (𝐹 cyclShift 𝑁))
18 eucrct2eupth.n . . . . . . . . . 10 (𝜑𝑁 = (#‘𝐹))
1918oveq2d 6565 . . . . . . . . 9 (𝜑 → (𝐹 cyclShift 𝑁) = (𝐹 cyclShift (#‘𝐹)))
20 eucrct2eupth1.c . . . . . . . . . . 11 (𝜑𝐹(CircuitS‘𝐺)𝑃)
21 crctis1wlk 41002 . . . . . . . . . . . 12 (𝐹(CircuitS‘𝐺)𝑃𝐹(1Walks‘𝐺)𝑃)
2221wlkf 40819 . . . . . . . . . . . 12 (𝐹(1Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
2321, 22syl 17 . . . . . . . . . . 11 (𝐹(CircuitS‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
2420, 23syl 17 . . . . . . . . . 10 (𝜑𝐹 ∈ Word dom 𝐼)
25 cshwn 13394 . . . . . . . . . 10 (𝐹 ∈ Word dom 𝐼 → (𝐹 cyclShift (#‘𝐹)) = 𝐹)
2624, 25syl 17 . . . . . . . . 9 (𝜑 → (𝐹 cyclShift (#‘𝐹)) = 𝐹)
2719, 26eqtrd 2644 . . . . . . . 8 (𝜑 → (𝐹 cyclShift 𝑁) = 𝐹)
2827adantl 481 . . . . . . 7 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 cyclShift 𝑁) = 𝐹)
2917, 28eqtrd 2644 . . . . . 6 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 cyclShift (𝐽 + 1)) = 𝐹)
307, 29syl5eqr 2658 . . . . 5 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 cyclShift 𝐾) = 𝐹)
31 eqid 2610 . . . . . . . . . . . . . 14 (#‘𝐹) = (#‘𝐹)
321, 2, 20, 31crctcshlem1 41020 . . . . . . . . . . . . 13 (𝜑 → (#‘𝐹) ∈ ℕ0)
33 fz0sn0fz1 12325 . . . . . . . . . . . . 13 ((#‘𝐹) ∈ ℕ0 → (0...(#‘𝐹)) = ({0} ∪ (1...(#‘𝐹))))
3432, 33syl 17 . . . . . . . . . . . 12 (𝜑 → (0...(#‘𝐹)) = ({0} ∪ (1...(#‘𝐹))))
3534eleq2d 2673 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (0...(#‘𝐹)) ↔ 𝑥 ∈ ({0} ∪ (1...(#‘𝐹)))))
36 elun 3715 . . . . . . . . . . 11 (𝑥 ∈ ({0} ∪ (1...(#‘𝐹))) ↔ (𝑥 ∈ {0} ∨ 𝑥 ∈ (1...(#‘𝐹))))
3735, 36syl6bb 275 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (0...(#‘𝐹)) ↔ (𝑥 ∈ {0} ∨ 𝑥 ∈ (1...(#‘𝐹)))))
38 elsni 4142 . . . . . . . . . . . . . . . 16 (𝑥 ∈ {0} → 𝑥 = 0)
39 0le0 10987 . . . . . . . . . . . . . . . 16 0 ≤ 0
4038, 39syl6eqbr 4622 . . . . . . . . . . . . . . 15 (𝑥 ∈ {0} → 𝑥 ≤ 0)
4140adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ {0}) → 𝑥 ≤ 0)
4241iftrued 4044 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ {0}) → if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))) = (𝑃‘(𝑥 + 𝑁)))
4318fveq2d 6107 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃𝑁) = (𝑃‘(#‘𝐹)))
44 crctprop 40998 . . . . . . . . . . . . . . . . . 18 (𝐹(CircuitS‘𝐺)𝑃 → (𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))
45 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑃‘0) = (𝑃‘(#‘𝐹)))
4645eqcomd 2616 . . . . . . . . . . . . . . . . . 18 ((𝐹(TrailS‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))) → (𝑃‘(#‘𝐹)) = (𝑃‘0))
4720, 44, 463syl 18 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃‘(#‘𝐹)) = (𝑃‘0))
4843, 47eqtrd 2644 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃𝑁) = (𝑃‘0))
4948adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑥 = 0) → (𝑃𝑁) = (𝑃‘0))
50 oveq1 6556 . . . . . . . . . . . . . . . . 17 (𝑥 = 0 → (𝑥 + 𝑁) = (0 + 𝑁))
519, 13syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℂ)
5251addid2d 10116 . . . . . . . . . . . . . . . . 17 (𝜑 → (0 + 𝑁) = 𝑁)
5350, 52sylan9eqr 2666 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 = 0) → (𝑥 + 𝑁) = 𝑁)
5453fveq2d 6107 . . . . . . . . . . . . . . 15 ((𝜑𝑥 = 0) → (𝑃‘(𝑥 + 𝑁)) = (𝑃𝑁))
55 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑥 = 0 → (𝑃𝑥) = (𝑃‘0))
5655adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑥 = 0) → (𝑃𝑥) = (𝑃‘0))
5749, 54, 563eqtr4d 2654 . . . . . . . . . . . . . 14 ((𝜑𝑥 = 0) → (𝑃‘(𝑥 + 𝑁)) = (𝑃𝑥))
5838, 57sylan2 490 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ {0}) → (𝑃‘(𝑥 + 𝑁)) = (𝑃𝑥))
5942, 58eqtrd 2644 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ {0}) → if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))) = (𝑃𝑥))
6059ex 449 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ {0} → if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))) = (𝑃𝑥)))
61 elfznn 12241 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1...(#‘𝐹)) → 𝑥 ∈ ℕ)
62 nnnle0 10928 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ℕ → ¬ 𝑥 ≤ 0)
6361, 62syl 17 . . . . . . . . . . . . . . 15 (𝑥 ∈ (1...(#‘𝐹)) → ¬ 𝑥 ≤ 0)
6463adantl 481 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1...(#‘𝐹))) → ¬ 𝑥 ≤ 0)
6564iffalsed 4047 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1...(#‘𝐹))) → if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))) = (𝑃‘((𝑥 + 𝑁) − 𝑁)))
6661nncnd 10913 . . . . . . . . . . . . . . . 16 (𝑥 ∈ (1...(#‘𝐹)) → 𝑥 ∈ ℂ)
6766adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1...(#‘𝐹))) → 𝑥 ∈ ℂ)
6851adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (1...(#‘𝐹))) → 𝑁 ∈ ℂ)
6967, 68pncand 10272 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (1...(#‘𝐹))) → ((𝑥 + 𝑁) − 𝑁) = 𝑥)
7069fveq2d 6107 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (1...(#‘𝐹))) → (𝑃‘((𝑥 + 𝑁) − 𝑁)) = (𝑃𝑥))
7165, 70eqtrd 2644 . . . . . . . . . . . 12 ((𝜑𝑥 ∈ (1...(#‘𝐹))) → if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))) = (𝑃𝑥))
7271ex 449 . . . . . . . . . . 11 (𝜑 → (𝑥 ∈ (1...(#‘𝐹)) → if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))) = (𝑃𝑥)))
7360, 72jaod 394 . . . . . . . . . 10 (𝜑 → ((𝑥 ∈ {0} ∨ 𝑥 ∈ (1...(#‘𝐹))) → if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))) = (𝑃𝑥)))
7437, 73sylbid 229 . . . . . . . . 9 (𝜑 → (𝑥 ∈ (0...(#‘𝐹)) → if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))) = (𝑃𝑥)))
7574imp 444 . . . . . . . 8 ((𝜑𝑥 ∈ (0...(#‘𝐹))) → if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))) = (𝑃𝑥))
7675mpteq2dva 4672 . . . . . . 7 (𝜑 → (𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁)))) = (𝑥 ∈ (0...(#‘𝐹)) ↦ (𝑃𝑥)))
7776adantl 481 . . . . . 6 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁)))) = (𝑥 ∈ (0...(#‘𝐹)) ↦ (𝑃𝑥)))
785oveq2i 6560 . . . . . . . . . 10 (𝑁𝐾) = (𝑁 − (𝐽 + 1))
798oveq2d 6565 . . . . . . . . . . 11 (𝐽 = (𝑁 − 1) → (𝑁 − (𝐽 + 1)) = (𝑁 − ((𝑁 − 1) + 1)))
8015oveq2d 6565 . . . . . . . . . . . 12 (𝜑 → (𝑁 − ((𝑁 − 1) + 1)) = (𝑁𝑁))
8151subidd 10259 . . . . . . . . . . . 12 (𝜑 → (𝑁𝑁) = 0)
8280, 81eqtrd 2644 . . . . . . . . . . 11 (𝜑 → (𝑁 − ((𝑁 − 1) + 1)) = 0)
8379, 82sylan9eq 2664 . . . . . . . . . 10 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑁 − (𝐽 + 1)) = 0)
8478, 83syl5eq 2656 . . . . . . . . 9 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑁𝐾) = 0)
8584breq2d 4595 . . . . . . . 8 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑥 ≤ (𝑁𝐾) ↔ 𝑥 ≤ 0))
865oveq2i 6560 . . . . . . . . . 10 (𝑥 + 𝐾) = (𝑥 + (𝐽 + 1))
8786fveq2i 6106 . . . . . . . . 9 (𝑃‘(𝑥 + 𝐾)) = (𝑃‘(𝑥 + (𝐽 + 1)))
888oveq2d 6565 . . . . . . . . . . 11 (𝐽 = (𝑁 − 1) → (𝑥 + (𝐽 + 1)) = (𝑥 + ((𝑁 − 1) + 1)))
8915oveq2d 6565 . . . . . . . . . . 11 (𝜑 → (𝑥 + ((𝑁 − 1) + 1)) = (𝑥 + 𝑁))
9088, 89sylan9eq 2664 . . . . . . . . . 10 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑥 + (𝐽 + 1)) = (𝑥 + 𝑁))
9190fveq2d 6107 . . . . . . . . 9 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑃‘(𝑥 + (𝐽 + 1))) = (𝑃‘(𝑥 + 𝑁)))
9287, 91syl5eq 2656 . . . . . . . 8 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑃‘(𝑥 + 𝐾)) = (𝑃‘(𝑥 + 𝑁)))
9386oveq1i 6559 . . . . . . . . . 10 ((𝑥 + 𝐾) − 𝑁) = ((𝑥 + (𝐽 + 1)) − 𝑁)
9493fveq2i 6106 . . . . . . . . 9 (𝑃‘((𝑥 + 𝐾) − 𝑁)) = (𝑃‘((𝑥 + (𝐽 + 1)) − 𝑁))
9588oveq1d 6564 . . . . . . . . . . 11 (𝐽 = (𝑁 − 1) → ((𝑥 + (𝐽 + 1)) − 𝑁) = ((𝑥 + ((𝑁 − 1) + 1)) − 𝑁))
9689oveq1d 6564 . . . . . . . . . . 11 (𝜑 → ((𝑥 + ((𝑁 − 1) + 1)) − 𝑁) = ((𝑥 + 𝑁) − 𝑁))
9795, 96sylan9eq 2664 . . . . . . . . . 10 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → ((𝑥 + (𝐽 + 1)) − 𝑁) = ((𝑥 + 𝑁) − 𝑁))
9897fveq2d 6107 . . . . . . . . 9 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑃‘((𝑥 + (𝐽 + 1)) − 𝑁)) = (𝑃‘((𝑥 + 𝑁) − 𝑁)))
9994, 98syl5eq 2656 . . . . . . . 8 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑃‘((𝑥 + 𝐾) − 𝑁)) = (𝑃‘((𝑥 + 𝑁) − 𝑁)))
10085, 92, 99ifbieq12d 4063 . . . . . . 7 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁))) = if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁))))
101100mpteq2dv 4673 . . . . . 6 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) = (𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ 0, (𝑃‘(𝑥 + 𝑁)), (𝑃‘((𝑥 + 𝑁) − 𝑁)))))
10220, 21syl 17 . . . . . . . . 9 (𝜑𝐹(1Walks‘𝐺)𝑃)
10311wlkp 40821 . . . . . . . . 9 (𝐹(1Walks‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
104 ffn 5958 . . . . . . . . 9 (𝑃:(0...(#‘𝐹))⟶𝑉𝑃 Fn (0...(#‘𝐹)))
105102, 103, 1043syl 18 . . . . . . . 8 (𝜑𝑃 Fn (0...(#‘𝐹)))
106105adantl 481 . . . . . . 7 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝑃 Fn (0...(#‘𝐹)))
107 dffn5 6151 . . . . . . 7 (𝑃 Fn (0...(#‘𝐹)) ↔ 𝑃 = (𝑥 ∈ (0...(#‘𝐹)) ↦ (𝑃𝑥)))
108106, 107sylib 207 . . . . . 6 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝑃 = (𝑥 ∈ (0...(#‘𝐹)) ↦ (𝑃𝑥)))
10977, 101, 1083eqtr4d 2654 . . . . 5 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) = 𝑃)
1104, 30, 1093brtr4d 4615 . . . 4 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))))
11120adantl 481 . . . . 5 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝐹(CircuitS‘𝐺)𝑃)
112111, 30, 1093brtr4d 4615 . . . 4 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))))
113 eucrct2eupth1.s . . . 4 (Vtx‘𝑆) = 𝑉
114 elfzolt3 12349 . . . . . . 7 (𝐽 ∈ (0..^𝑁) → 0 < 𝑁)
1159, 114syl 17 . . . . . 6 (𝜑 → 0 < 𝑁)
116 elfzoelz 12339 . . . . . . . . . . 11 (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℤ)
1179, 116syl 17 . . . . . . . . . 10 (𝜑𝐽 ∈ ℤ)
118117peano2zd 11361 . . . . . . . . 9 (𝜑 → (𝐽 + 1) ∈ ℤ)
1195, 118syl5eqel 2692 . . . . . . . 8 (𝜑𝐾 ∈ ℤ)
120 cshwlen 13396 . . . . . . . . 9 ((𝐹 ∈ Word dom 𝐼𝐾 ∈ ℤ) → (#‘(𝐹 cyclShift 𝐾)) = (#‘𝐹))
121120eqcomd 2616 . . . . . . . 8 ((𝐹 ∈ Word dom 𝐼𝐾 ∈ ℤ) → (#‘𝐹) = (#‘(𝐹 cyclShift 𝐾)))
12224, 119, 121syl2anc 691 . . . . . . 7 (𝜑 → (#‘𝐹) = (#‘(𝐹 cyclShift 𝐾)))
12318, 122eqtrd 2644 . . . . . 6 (𝜑𝑁 = (#‘(𝐹 cyclShift 𝐾)))
124115, 123breqtrd 4609 . . . . 5 (𝜑 → 0 < (#‘(𝐹 cyclShift 𝐾)))
125124adantl 481 . . . 4 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → 0 < (#‘(𝐹 cyclShift 𝐾)))
126123adantl 481 . . . . 5 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝑁 = (#‘(𝐹 cyclShift 𝐾)))
127126oveq1d 6564 . . . 4 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑁 − 1) = ((#‘(𝐹 cyclShift 𝐾)) − 1))
128 eucrct2eupth.e . . . . . 6 (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ ((0..^𝑁) ∖ {𝐽}))))
129128adantl 481 . . . . 5 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ ((0..^𝑁) ∖ {𝐽}))))
13024, 18, 93jca 1235 . . . . . . . . 9 (𝜑 → (𝐹 ∈ Word dom 𝐼𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)))
131130adantl 481 . . . . . . . 8 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 ∈ Word dom 𝐼𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)))
132 cshimadifsn0 13427 . . . . . . . 8 ((𝐹 ∈ Word dom 𝐼𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))
133131, 132syl 17 . . . . . . 7 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))
1347imaeq1i 5382 . . . . . . 7 ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))) = ((𝐹 cyclShift 𝐾) “ (0..^(𝑁 − 1)))
135133, 134syl6eq 2660 . . . . . 6 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐾) “ (0..^(𝑁 − 1))))
136135reseq2d 5317 . . . . 5 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐼 ↾ (𝐹 “ ((0..^𝑁) ∖ {𝐽}))) = (𝐼 ↾ ((𝐹 cyclShift 𝐾) “ (0..^(𝑁 − 1)))))
137129, 136eqtrd 2644 . . . 4 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → (iEdg‘𝑆) = (𝐼 ↾ ((𝐹 cyclShift 𝐾) “ (0..^(𝑁 − 1)))))
138 eqid 2610 . . . 4 ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1))) = ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1)))
139 eqid 2610 . . . 4 ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0...(𝑁 − 1))) = ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0...(𝑁 − 1)))
1401, 2, 110, 112, 113, 125, 127, 137, 138, 139eucrct2eupth1 41412 . . 3 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1)))(EulerPaths‘𝑆)((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0...(𝑁 − 1))))
141 eucrct2eupth.h . . . 4 𝐻 = ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1)))
142141a1i 11 . . 3 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝐻 = ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1))))
143 eucrct2eupth.q . . . . 5 𝑄 = (𝑥 ∈ (0..^𝑁) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁))))
144 fzossfz 12357 . . . . . . . 8 (0..^𝑁) ⊆ (0...𝑁)
14518oveq2d 6565 . . . . . . . 8 (𝜑 → (0...𝑁) = (0...(#‘𝐹)))
146144, 145syl5sseq 3616 . . . . . . 7 (𝜑 → (0..^𝑁) ⊆ (0...(#‘𝐹)))
147146resmptd 5371 . . . . . 6 (𝜑 → ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0..^𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))))
148 elfzoel2 12338 . . . . . . . 8 (𝐽 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
149 fzoval 12340 . . . . . . . 8 (𝑁 ∈ ℤ → (0..^𝑁) = (0...(𝑁 − 1)))
1509, 148, 1493syl 18 . . . . . . 7 (𝜑 → (0..^𝑁) = (0...(𝑁 − 1)))
151150reseq2d 5317 . . . . . 6 (𝜑 → ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0..^𝑁)) = ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0...(𝑁 − 1))))
152147, 151eqtr3d 2646 . . . . 5 (𝜑 → (𝑥 ∈ (0..^𝑁) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) = ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0...(𝑁 − 1))))
153143, 152syl5eq 2656 . . . 4 (𝜑𝑄 = ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0...(𝑁 − 1))))
154153adantl 481 . . 3 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝑄 = ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0...(𝑁 − 1))))
155140, 142, 1543brtr4d 4615 . 2 ((𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝐻(EulerPaths‘𝑆)𝑄)
15620adantl 481 . . . 4 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝐹(CircuitS‘𝐺)𝑃)
157 peano2nn0 11210 . . . . . . . . . . . . 13 (𝐽 ∈ ℕ0 → (𝐽 + 1) ∈ ℕ0)
1581573ad2ant1 1075 . . . . . . . . . . . 12 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → (𝐽 + 1) ∈ ℕ0)
159158adantr 480 . . . . . . . . . . 11 (((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) ∧ ¬ 𝐽 = (𝑁 − 1)) → (𝐽 + 1) ∈ ℕ0)
160 simpl2 1058 . . . . . . . . . . 11 (((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) ∧ ¬ 𝐽 = (𝑁 − 1)) → 𝑁 ∈ ℕ)
161 1cnd 9935 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → 1 ∈ ℂ)
162 nn0cn 11179 . . . . . . . . . . . . . . . . 17 (𝐽 ∈ ℕ0𝐽 ∈ ℂ)
1631623ad2ant1 1075 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → 𝐽 ∈ ℂ)
16412, 161, 163subadd2d 10290 . . . . . . . . . . . . . . 15 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → ((𝑁 − 1) = 𝐽 ↔ (𝐽 + 1) = 𝑁))
165 eqcom 2617 . . . . . . . . . . . . . . 15 (𝐽 = (𝑁 − 1) ↔ (𝑁 − 1) = 𝐽)
166 eqcom 2617 . . . . . . . . . . . . . . 15 (𝑁 = (𝐽 + 1) ↔ (𝐽 + 1) = 𝑁)
167164, 165, 1663bitr4g 302 . . . . . . . . . . . . . 14 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → (𝐽 = (𝑁 − 1) ↔ 𝑁 = (𝐽 + 1)))
168167necon3bbid 2819 . . . . . . . . . . . . 13 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → (¬ 𝐽 = (𝑁 − 1) ↔ 𝑁 ≠ (𝐽 + 1)))
169157nn0red 11229 . . . . . . . . . . . . . . . 16 (𝐽 ∈ ℕ0 → (𝐽 + 1) ∈ ℝ)
1701693ad2ant1 1075 . . . . . . . . . . . . . . 15 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → (𝐽 + 1) ∈ ℝ)
171 nnre 10904 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
1721713ad2ant2 1076 . . . . . . . . . . . . . . 15 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → 𝑁 ∈ ℝ)
173 nn0z 11277 . . . . . . . . . . . . . . . . 17 (𝐽 ∈ ℕ0𝐽 ∈ ℤ)
174 nnz 11276 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
175 zltp1le 11304 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐽 < 𝑁 ↔ (𝐽 + 1) ≤ 𝑁))
176173, 174, 175syl2an 493 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ) → (𝐽 < 𝑁 ↔ (𝐽 + 1) ≤ 𝑁))
177176biimp3a 1424 . . . . . . . . . . . . . . 15 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → (𝐽 + 1) ≤ 𝑁)
178170, 172, 177leltned 10069 . . . . . . . . . . . . . 14 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → ((𝐽 + 1) < 𝑁𝑁 ≠ (𝐽 + 1)))
179178biimprd 237 . . . . . . . . . . . . 13 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → (𝑁 ≠ (𝐽 + 1) → (𝐽 + 1) < 𝑁))
180168, 179sylbid 229 . . . . . . . . . . . 12 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → (¬ 𝐽 = (𝑁 − 1) → (𝐽 + 1) < 𝑁))
181180imp 444 . . . . . . . . . . 11 (((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) ∧ ¬ 𝐽 = (𝑁 − 1)) → (𝐽 + 1) < 𝑁)
182159, 160, 1813jca 1235 . . . . . . . . . 10 (((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) ∧ ¬ 𝐽 = (𝑁 − 1)) → ((𝐽 + 1) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝐽 + 1) < 𝑁))
183182ex 449 . . . . . . . . 9 ((𝐽 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝐽 < 𝑁) → (¬ 𝐽 = (𝑁 − 1) → ((𝐽 + 1) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝐽 + 1) < 𝑁)))
18410, 183sylbi 206 . . . . . . . 8 (𝐽 ∈ (0..^𝑁) → (¬ 𝐽 = (𝑁 − 1) → ((𝐽 + 1) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝐽 + 1) < 𝑁)))
185 elfzo0 12376 . . . . . . . 8 ((𝐽 + 1) ∈ (0..^𝑁) ↔ ((𝐽 + 1) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝐽 + 1) < 𝑁))
186184, 185syl6ibr 241 . . . . . . 7 (𝐽 ∈ (0..^𝑁) → (¬ 𝐽 = (𝑁 − 1) → (𝐽 + 1) ∈ (0..^𝑁)))
1879, 186syl 17 . . . . . 6 (𝜑 → (¬ 𝐽 = (𝑁 − 1) → (𝐽 + 1) ∈ (0..^𝑁)))
188187impcom 445 . . . . 5 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐽 + 1) ∈ (0..^𝑁))
1895a1i 11 . . . . 5 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝐾 = (𝐽 + 1))
19018eqcomd 2616 . . . . . . 7 (𝜑 → (#‘𝐹) = 𝑁)
191190oveq2d 6565 . . . . . 6 (𝜑 → (0..^(#‘𝐹)) = (0..^𝑁))
192191adantl 481 . . . . 5 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (0..^(#‘𝐹)) = (0..^𝑁))
193188, 189, 1923eltr4d 2703 . . . 4 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝐾 ∈ (0..^(#‘𝐹)))
194 eqid 2610 . . . 4 (𝐹 cyclShift 𝐾) = (𝐹 cyclShift 𝐾)
195 eqid 2610 . . . 4 (𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) = (𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹)))))
1963adantl 481 . . . 4 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝐹(EulerPaths‘𝐺)𝑃)
1971, 2, 156, 31, 193, 194, 195, 196eucrctshift 41411 . . 3 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹)))))))
198 simprl 790 . . . . 5 (((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) ∧ ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))) → (𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))
199 simprr 792 . . . . 5 (((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) ∧ ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))) → (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))
200124ad2antlr 759 . . . . 5 (((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) ∧ ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))) → 0 < (#‘(𝐹 cyclShift 𝐾)))
201123oveq1d 6564 . . . . . 6 (𝜑 → (𝑁 − 1) = ((#‘(𝐹 cyclShift 𝐾)) − 1))
202201ad2antlr 759 . . . . 5 (((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) ∧ ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))) → (𝑁 − 1) = ((#‘(𝐹 cyclShift 𝐾)) − 1))
203128adantl 481 . . . . . . 7 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ ((0..^𝑁) ∖ {𝐽}))))
204130adantl 481 . . . . . . . . . 10 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 ∈ Word dom 𝐼𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)))
205204, 132syl 17 . . . . . . . . 9 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))
206205, 134syl6eq 2660 . . . . . . . 8 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐾) “ (0..^(𝑁 − 1))))
207206reseq2d 5317 . . . . . . 7 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝐼 ↾ (𝐹 “ ((0..^𝑁) ∖ {𝐽}))) = (𝐼 ↾ ((𝐹 cyclShift 𝐾) “ (0..^(𝑁 − 1)))))
208203, 207eqtrd 2644 . . . . . 6 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (iEdg‘𝑆) = (𝐼 ↾ ((𝐹 cyclShift 𝐾) “ (0..^(𝑁 − 1)))))
209208adantr 480 . . . . 5 (((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) ∧ ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))) → (iEdg‘𝑆) = (𝐼 ↾ ((𝐹 cyclShift 𝐾) “ (0..^(𝑁 − 1)))))
210 eqid 2610 . . . . 5 ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ↾ (0...(𝑁 − 1))) = ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ↾ (0...(𝑁 − 1)))
2111, 2, 198, 199, 113, 200, 202, 209, 138, 210eucrct2eupth1 41412 . . . 4 (((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) ∧ ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))) → ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1)))(EulerPaths‘𝑆)((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ↾ (0...(𝑁 − 1))))
212141a1i 11 . . . 4 (((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) ∧ ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))) → 𝐻 = ((𝐹 cyclShift 𝐾) ↾ (0..^(𝑁 − 1))))
213190oveq1d 6564 . . . . . . . . . . . 12 (𝜑 → ((#‘𝐹) − 𝐾) = (𝑁𝐾))
214213breq2d 4595 . . . . . . . . . . 11 (𝜑 → (𝑥 ≤ ((#‘𝐹) − 𝐾) ↔ 𝑥 ≤ (𝑁𝐾)))
215214adantl 481 . . . . . . . . . 10 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑥 ≤ ((#‘𝐹) − 𝐾) ↔ 𝑥 ≤ (𝑁𝐾)))
216190oveq2d 6565 . . . . . . . . . . . 12 (𝜑 → ((𝑥 + 𝐾) − (#‘𝐹)) = ((𝑥 + 𝐾) − 𝑁))
217216fveq2d 6107 . . . . . . . . . . 11 (𝜑 → (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))) = (𝑃‘((𝑥 + 𝐾) − 𝑁)))
218217adantl 481 . . . . . . . . . 10 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))) = (𝑃‘((𝑥 + 𝐾) − 𝑁)))
219215, 218ifbieq2d 4061 . . . . . . . . 9 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹)))) = if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁))))
220219mpteq2dv 4673 . . . . . . . 8 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) = (𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))))
221150eqcomd 2616 . . . . . . . . 9 (𝜑 → (0...(𝑁 − 1)) = (0..^𝑁))
222221adantl 481 . . . . . . . 8 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (0...(𝑁 − 1)) = (0..^𝑁))
223220, 222reseq12d 5318 . . . . . . 7 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ↾ (0...(𝑁 − 1))) = ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0..^𝑁)))
22418adantl 481 . . . . . . . . . 10 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝑁 = (#‘𝐹))
225224oveq2d 6565 . . . . . . . . 9 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (0...𝑁) = (0...(#‘𝐹)))
226144, 225syl5sseq 3616 . . . . . . . 8 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → (0..^𝑁) ⊆ (0...(#‘𝐹)))
227226resmptd 5371 . . . . . . 7 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))) ↾ (0..^𝑁)) = (𝑥 ∈ (0..^𝑁) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))))
228223, 227eqtrd 2644 . . . . . 6 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ↾ (0...(𝑁 − 1))) = (𝑥 ∈ (0..^𝑁) ↦ if(𝑥 ≤ (𝑁𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − 𝑁)))))
229228, 143syl6reqr 2663 . . . . 5 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝑄 = ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ↾ (0...(𝑁 − 1))))
230229adantr 480 . . . 4 (((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) ∧ ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))) → 𝑄 = ((𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ↾ (0...(𝑁 − 1))))
231211, 212, 2303brtr4d 4615 . . 3 (((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) ∧ ((𝐹 cyclShift 𝐾)(EulerPaths‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))) ∧ (𝐹 cyclShift 𝐾)(CircuitS‘𝐺)(𝑥 ∈ (0...(#‘𝐹)) ↦ if(𝑥 ≤ ((#‘𝐹) − 𝐾), (𝑃‘(𝑥 + 𝐾)), (𝑃‘((𝑥 + 𝐾) − (#‘𝐹))))))) → 𝐻(EulerPaths‘𝑆)𝑄)
232197, 231mpdan 699 . 2 ((¬ 𝐽 = (𝑁 − 1) ∧ 𝜑) → 𝐻(EulerPaths‘𝑆)𝑄)
233155, 232pm2.61ian 827 1 (𝜑𝐻(EulerPaths‘𝑆)𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 195  wo 382  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  cdif 3537  cun 3538  ifcif 4036  {csn 4125   class class class wbr 4583  cmpt 4643  dom cdm 5038  cres 5040  cima 5041   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  cc 9813  cr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953  cle 9954  cmin 10145  cn 10897  0cn0 11169  cz 11254  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   cyclShift ccsh 13385  Vtxcvtx 25673  iEdgciedg 25674  1Walksc1wlks 40796  TrailSctrls 40899  CircuitSccrcts 40990  EulerPathsceupth 41364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-ico 12052  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386  df-1wlks 40800  df-trls 40901  df-crcts 40992  df-eupth 41365
This theorem is referenced by: (None)
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