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Theorem eucrct2eupth 29192
Description: Removing one edge (πΌβ€˜(πΉβ€˜π½)) from a graph 𝐺 with an Eulerian circuit ⟨𝐹, π‘ƒβŸ© results in a graph 𝑆 with an Eulerian path ⟨𝐻, π‘„βŸ©. (Contributed by AV, 17-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)
Hypotheses
Ref Expression
eucrct2eupth1.v 𝑉 = (Vtxβ€˜πΊ)
eucrct2eupth1.i 𝐼 = (iEdgβ€˜πΊ)
eucrct2eupth1.d (πœ‘ β†’ 𝐹(EulerPathsβ€˜πΊ)𝑃)
eucrct2eupth1.c (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)
eucrct2eupth1.s (Vtxβ€˜π‘†) = 𝑉
eucrct2eupth.n (πœ‘ β†’ 𝑁 = (β™―β€˜πΉ))
eucrct2eupth.j (πœ‘ β†’ 𝐽 ∈ (0..^𝑁))
eucrct2eupth.e (πœ‘ β†’ (iEdgβ€˜π‘†) = (𝐼 β†Ύ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽}))))
eucrct2eupth.k 𝐾 = (𝐽 + 1)
eucrct2eupth.h 𝐻 = ((𝐹 cyclShift 𝐾) prefix (𝑁 βˆ’ 1))
eucrct2eupth.q 𝑄 = (π‘₯ ∈ (0..^𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁))))
Assertion
Ref Expression
eucrct2eupth (πœ‘ β†’ 𝐻(EulerPathsβ€˜π‘†)𝑄)
Distinct variable groups:   π‘₯,𝐹   π‘₯,𝐼   π‘₯,𝐽   π‘₯,𝐾   π‘₯,𝑁   π‘₯,𝑃   π‘₯,𝑉   πœ‘,π‘₯
Allowed substitution hints:   𝑄(π‘₯)   𝑆(π‘₯)   𝐺(π‘₯)   𝐻(π‘₯)

Proof of Theorem eucrct2eupth
StepHypRef Expression
1 eucrct2eupth1.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
2 eucrct2eupth1.i . . . 4 𝐼 = (iEdgβ€˜πΊ)
3 eucrct2eupth1.d . . . . . 6 (πœ‘ β†’ 𝐹(EulerPathsβ€˜πΊ)𝑃)
43adantl 483 . . . . 5 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝐹(EulerPathsβ€˜πΊ)𝑃)
5 eucrct2eupth.k . . . . . . . 8 𝐾 = (𝐽 + 1)
65eqcomi 2746 . . . . . . 7 (𝐽 + 1) = 𝐾
76oveq2i 7369 . . . . . 6 (𝐹 cyclShift (𝐽 + 1)) = (𝐹 cyclShift 𝐾)
8 oveq1 7365 . . . . . . . . 9 (𝐽 = (𝑁 βˆ’ 1) β†’ (𝐽 + 1) = ((𝑁 βˆ’ 1) + 1))
9 eucrct2eupth.j . . . . . . . . . 10 (πœ‘ β†’ 𝐽 ∈ (0..^𝑁))
10 elfzo0 13614 . . . . . . . . . . 11 (𝐽 ∈ (0..^𝑁) ↔ (𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁))
11 nncn 12162 . . . . . . . . . . . 12 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„‚)
12113ad2ant2 1135 . . . . . . . . . . 11 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ 𝑁 ∈ β„‚)
1310, 12sylbi 216 . . . . . . . . . 10 (𝐽 ∈ (0..^𝑁) β†’ 𝑁 ∈ β„‚)
14 npcan1 11581 . . . . . . . . . 10 (𝑁 ∈ β„‚ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
159, 13, 143syl 18 . . . . . . . . 9 (πœ‘ β†’ ((𝑁 βˆ’ 1) + 1) = 𝑁)
168, 15sylan9eq 2797 . . . . . . . 8 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐽 + 1) = 𝑁)
1716oveq2d 7374 . . . . . . 7 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 cyclShift (𝐽 + 1)) = (𝐹 cyclShift 𝑁))
18 eucrct2eupth.n . . . . . . . . . 10 (πœ‘ β†’ 𝑁 = (β™―β€˜πΉ))
1918oveq2d 7374 . . . . . . . . 9 (πœ‘ β†’ (𝐹 cyclShift 𝑁) = (𝐹 cyclShift (β™―β€˜πΉ)))
20 eucrct2eupth1.c . . . . . . . . . . 11 (πœ‘ β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)
21 crctiswlk 28747 . . . . . . . . . . . 12 (𝐹(Circuitsβ€˜πΊ)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃)
222wlkf 28565 . . . . . . . . . . . 12 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝐹 ∈ Word dom 𝐼)
2321, 22syl 17 . . . . . . . . . . 11 (𝐹(Circuitsβ€˜πΊ)𝑃 β†’ 𝐹 ∈ Word dom 𝐼)
2420, 23syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐹 ∈ Word dom 𝐼)
25 cshwn 14686 . . . . . . . . . 10 (𝐹 ∈ Word dom 𝐼 β†’ (𝐹 cyclShift (β™―β€˜πΉ)) = 𝐹)
2624, 25syl 17 . . . . . . . . 9 (πœ‘ β†’ (𝐹 cyclShift (β™―β€˜πΉ)) = 𝐹)
2719, 26eqtrd 2777 . . . . . . . 8 (πœ‘ β†’ (𝐹 cyclShift 𝑁) = 𝐹)
2827adantl 483 . . . . . . 7 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 cyclShift 𝑁) = 𝐹)
2917, 28eqtrd 2777 . . . . . 6 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 cyclShift (𝐽 + 1)) = 𝐹)
307, 29eqtr3id 2791 . . . . 5 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 cyclShift 𝐾) = 𝐹)
31 eqid 2737 . . . . . . . . . . . . . 14 (β™―β€˜πΉ) = (β™―β€˜πΉ)
321, 2, 20, 31crctcshlem1 28765 . . . . . . . . . . . . 13 (πœ‘ β†’ (β™―β€˜πΉ) ∈ β„•0)
33 fz0sn0fz1 13559 . . . . . . . . . . . . 13 ((β™―β€˜πΉ) ∈ β„•0 β†’ (0...(β™―β€˜πΉ)) = ({0} βˆͺ (1...(β™―β€˜πΉ))))
3432, 33syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ (0...(β™―β€˜πΉ)) = ({0} βˆͺ (1...(β™―β€˜πΉ))))
3534eleq2d 2824 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↔ π‘₯ ∈ ({0} βˆͺ (1...(β™―β€˜πΉ)))))
36 elun 4109 . . . . . . . . . . 11 (π‘₯ ∈ ({0} βˆͺ (1...(β™―β€˜πΉ))) ↔ (π‘₯ ∈ {0} ∨ π‘₯ ∈ (1...(β™―β€˜πΉ))))
3735, 36bitrdi 287 . . . . . . . . . 10 (πœ‘ β†’ (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↔ (π‘₯ ∈ {0} ∨ π‘₯ ∈ (1...(β™―β€˜πΉ)))))
38 elsni 4604 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ {0} β†’ π‘₯ = 0)
39 0le0 12255 . . . . . . . . . . . . . . . 16 0 ≀ 0
4038, 39eqbrtrdi 5145 . . . . . . . . . . . . . . 15 (π‘₯ ∈ {0} β†’ π‘₯ ≀ 0)
4140adantl 483 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ ∈ {0}) β†’ π‘₯ ≀ 0)
4241iftrued 4495 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ {0}) β†’ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))) = (π‘ƒβ€˜(π‘₯ + 𝑁)))
4318fveq2d 6847 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜(β™―β€˜πΉ)))
44 crctprop 28743 . . . . . . . . . . . . . . . . . 18 (𝐹(Circuitsβ€˜πΊ)𝑃 β†’ (𝐹(Trailsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
45 simpr 486 . . . . . . . . . . . . . . . . . . 19 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))
4645eqcomd 2743 . . . . . . . . . . . . . . . . . 18 ((𝐹(Trailsβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0))
4720, 44, 463syl 18 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜0))
4843, 47eqtrd 2777 . . . . . . . . . . . . . . . 16 (πœ‘ β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜0))
4948adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘₯ = 0) β†’ (π‘ƒβ€˜π‘) = (π‘ƒβ€˜0))
50 oveq1 7365 . . . . . . . . . . . . . . . . 17 (π‘₯ = 0 β†’ (π‘₯ + 𝑁) = (0 + 𝑁))
519, 13syl 17 . . . . . . . . . . . . . . . . . 18 (πœ‘ β†’ 𝑁 ∈ β„‚)
5251addid2d 11357 . . . . . . . . . . . . . . . . 17 (πœ‘ β†’ (0 + 𝑁) = 𝑁)
5350, 52sylan9eqr 2799 . . . . . . . . . . . . . . . 16 ((πœ‘ ∧ π‘₯ = 0) β†’ (π‘₯ + 𝑁) = 𝑁)
5453fveq2d 6847 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘₯ = 0) β†’ (π‘ƒβ€˜(π‘₯ + 𝑁)) = (π‘ƒβ€˜π‘))
55 fveq2 6843 . . . . . . . . . . . . . . . 16 (π‘₯ = 0 β†’ (π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜0))
5655adantl 483 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘₯ = 0) β†’ (π‘ƒβ€˜π‘₯) = (π‘ƒβ€˜0))
5749, 54, 563eqtr4d 2787 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ = 0) β†’ (π‘ƒβ€˜(π‘₯ + 𝑁)) = (π‘ƒβ€˜π‘₯))
5838, 57sylan2 594 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ {0}) β†’ (π‘ƒβ€˜(π‘₯ + 𝑁)) = (π‘ƒβ€˜π‘₯))
5942, 58eqtrd 2777 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ {0}) β†’ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))) = (π‘ƒβ€˜π‘₯))
6059ex 414 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯ ∈ {0} β†’ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))) = (π‘ƒβ€˜π‘₯)))
61 elfznn 13471 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ (1...(β™―β€˜πΉ)) β†’ π‘₯ ∈ β„•)
62 nnnle0 12187 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ β„• β†’ Β¬ π‘₯ ≀ 0)
6361, 62syl 17 . . . . . . . . . . . . . . 15 (π‘₯ ∈ (1...(β™―β€˜πΉ)) β†’ Β¬ π‘₯ ≀ 0)
6463adantl 483 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ ∈ (1...(β™―β€˜πΉ))) β†’ Β¬ π‘₯ ≀ 0)
6564iffalsed 4498 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ (1...(β™―β€˜πΉ))) β†’ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))) = (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁)))
6661nncnd 12170 . . . . . . . . . . . . . . . 16 (π‘₯ ∈ (1...(β™―β€˜πΉ)) β†’ π‘₯ ∈ β„‚)
6766adantl 483 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘₯ ∈ (1...(β™―β€˜πΉ))) β†’ π‘₯ ∈ β„‚)
6851adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ π‘₯ ∈ (1...(β™―β€˜πΉ))) β†’ 𝑁 ∈ β„‚)
6967, 68pncand 11514 . . . . . . . . . . . . . 14 ((πœ‘ ∧ π‘₯ ∈ (1...(β™―β€˜πΉ))) β†’ ((π‘₯ + 𝑁) βˆ’ 𝑁) = π‘₯)
7069fveq2d 6847 . . . . . . . . . . . . 13 ((πœ‘ ∧ π‘₯ ∈ (1...(β™―β€˜πΉ))) β†’ (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁)) = (π‘ƒβ€˜π‘₯))
7165, 70eqtrd 2777 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘₯ ∈ (1...(β™―β€˜πΉ))) β†’ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))) = (π‘ƒβ€˜π‘₯))
7271ex 414 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯ ∈ (1...(β™―β€˜πΉ)) β†’ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))) = (π‘ƒβ€˜π‘₯)))
7360, 72jaod 858 . . . . . . . . . 10 (πœ‘ β†’ ((π‘₯ ∈ {0} ∨ π‘₯ ∈ (1...(β™―β€˜πΉ))) β†’ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))) = (π‘ƒβ€˜π‘₯)))
7437, 73sylbid 239 . . . . . . . . 9 (πœ‘ β†’ (π‘₯ ∈ (0...(β™―β€˜πΉ)) β†’ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))) = (π‘ƒβ€˜π‘₯)))
7574imp 408 . . . . . . . 8 ((πœ‘ ∧ π‘₯ ∈ (0...(β™―β€˜πΉ))) β†’ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))) = (π‘ƒβ€˜π‘₯))
7675mpteq2dva 5206 . . . . . . 7 (πœ‘ β†’ (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁)))) = (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ (π‘ƒβ€˜π‘₯)))
7776adantl 483 . . . . . 6 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁)))) = (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ (π‘ƒβ€˜π‘₯)))
785oveq2i 7369 . . . . . . . . . 10 (𝑁 βˆ’ 𝐾) = (𝑁 βˆ’ (𝐽 + 1))
798oveq2d 7374 . . . . . . . . . . 11 (𝐽 = (𝑁 βˆ’ 1) β†’ (𝑁 βˆ’ (𝐽 + 1)) = (𝑁 βˆ’ ((𝑁 βˆ’ 1) + 1)))
8015oveq2d 7374 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑁 βˆ’ ((𝑁 βˆ’ 1) + 1)) = (𝑁 βˆ’ 𝑁))
8151subidd 11501 . . . . . . . . . . . 12 (πœ‘ β†’ (𝑁 βˆ’ 𝑁) = 0)
8280, 81eqtrd 2777 . . . . . . . . . . 11 (πœ‘ β†’ (𝑁 βˆ’ ((𝑁 βˆ’ 1) + 1)) = 0)
8379, 82sylan9eq 2797 . . . . . . . . . 10 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝑁 βˆ’ (𝐽 + 1)) = 0)
8478, 83eqtrid 2789 . . . . . . . . 9 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝑁 βˆ’ 𝐾) = 0)
8584breq2d 5118 . . . . . . . 8 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘₯ ≀ (𝑁 βˆ’ 𝐾) ↔ π‘₯ ≀ 0))
865oveq2i 7369 . . . . . . . . . 10 (π‘₯ + 𝐾) = (π‘₯ + (𝐽 + 1))
8786fveq2i 6846 . . . . . . . . 9 (π‘ƒβ€˜(π‘₯ + 𝐾)) = (π‘ƒβ€˜(π‘₯ + (𝐽 + 1)))
888oveq2d 7374 . . . . . . . . . . 11 (𝐽 = (𝑁 βˆ’ 1) β†’ (π‘₯ + (𝐽 + 1)) = (π‘₯ + ((𝑁 βˆ’ 1) + 1)))
8915oveq2d 7374 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯ + ((𝑁 βˆ’ 1) + 1)) = (π‘₯ + 𝑁))
9088, 89sylan9eq 2797 . . . . . . . . . 10 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘₯ + (𝐽 + 1)) = (π‘₯ + 𝑁))
9190fveq2d 6847 . . . . . . . . 9 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘ƒβ€˜(π‘₯ + (𝐽 + 1))) = (π‘ƒβ€˜(π‘₯ + 𝑁)))
9287, 91eqtrid 2789 . . . . . . . 8 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘ƒβ€˜(π‘₯ + 𝐾)) = (π‘ƒβ€˜(π‘₯ + 𝑁)))
9386oveq1i 7368 . . . . . . . . . 10 ((π‘₯ + 𝐾) βˆ’ 𝑁) = ((π‘₯ + (𝐽 + 1)) βˆ’ 𝑁)
9493fveq2i 6846 . . . . . . . . 9 (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)) = (π‘ƒβ€˜((π‘₯ + (𝐽 + 1)) βˆ’ 𝑁))
9588oveq1d 7373 . . . . . . . . . . 11 (𝐽 = (𝑁 βˆ’ 1) β†’ ((π‘₯ + (𝐽 + 1)) βˆ’ 𝑁) = ((π‘₯ + ((𝑁 βˆ’ 1) + 1)) βˆ’ 𝑁))
9689oveq1d 7373 . . . . . . . . . . 11 (πœ‘ β†’ ((π‘₯ + ((𝑁 βˆ’ 1) + 1)) βˆ’ 𝑁) = ((π‘₯ + 𝑁) βˆ’ 𝑁))
9795, 96sylan9eq 2797 . . . . . . . . . 10 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ ((π‘₯ + (𝐽 + 1)) βˆ’ 𝑁) = ((π‘₯ + 𝑁) βˆ’ 𝑁))
9897fveq2d 6847 . . . . . . . . 9 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘ƒβ€˜((π‘₯ + (𝐽 + 1)) βˆ’ 𝑁)) = (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁)))
9994, 98eqtrid 2789 . . . . . . . 8 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)) = (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁)))
10085, 92, 99ifbieq12d 4515 . . . . . . 7 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁))) = if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁))))
101100mpteq2dv 5208 . . . . . 6 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) = (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ 0, (π‘ƒβ€˜(π‘₯ + 𝑁)), (π‘ƒβ€˜((π‘₯ + 𝑁) βˆ’ 𝑁)))))
10220, 21syl 17 . . . . . . . . 9 (πœ‘ β†’ 𝐹(Walksβ€˜πΊ)𝑃)
1031wlkp 28567 . . . . . . . . 9 (𝐹(Walksβ€˜πΊ)𝑃 β†’ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰)
104 ffn 6669 . . . . . . . . 9 (𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ β†’ 𝑃 Fn (0...(β™―β€˜πΉ)))
105102, 103, 1043syl 18 . . . . . . . 8 (πœ‘ β†’ 𝑃 Fn (0...(β™―β€˜πΉ)))
106105adantl 483 . . . . . . 7 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝑃 Fn (0...(β™―β€˜πΉ)))
107 dffn5 6902 . . . . . . 7 (𝑃 Fn (0...(β™―β€˜πΉ)) ↔ 𝑃 = (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ (π‘ƒβ€˜π‘₯)))
108106, 107sylib 217 . . . . . 6 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝑃 = (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ (π‘ƒβ€˜π‘₯)))
10977, 101, 1083eqtr4d 2787 . . . . 5 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) = 𝑃)
1104, 30, 1093brtr4d 5138 . . . 4 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))))
11120adantl 483 . . . . 5 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)
112111, 30, 1093brtr4d 5138 . . . 4 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))))
113 eucrct2eupth1.s . . . 4 (Vtxβ€˜π‘†) = 𝑉
114 elfzolt3 13583 . . . . . . 7 (𝐽 ∈ (0..^𝑁) β†’ 0 < 𝑁)
1159, 114syl 17 . . . . . 6 (πœ‘ β†’ 0 < 𝑁)
116 elfzoelz 13573 . . . . . . . . . . 11 (𝐽 ∈ (0..^𝑁) β†’ 𝐽 ∈ β„€)
1179, 116syl 17 . . . . . . . . . 10 (πœ‘ β†’ 𝐽 ∈ β„€)
118117peano2zd 12611 . . . . . . . . 9 (πœ‘ β†’ (𝐽 + 1) ∈ β„€)
1195, 118eqeltrid 2842 . . . . . . . 8 (πœ‘ β†’ 𝐾 ∈ β„€)
120 cshwlen 14688 . . . . . . . . 9 ((𝐹 ∈ Word dom 𝐼 ∧ 𝐾 ∈ β„€) β†’ (β™―β€˜(𝐹 cyclShift 𝐾)) = (β™―β€˜πΉ))
121120eqcomd 2743 . . . . . . . 8 ((𝐹 ∈ Word dom 𝐼 ∧ 𝐾 ∈ β„€) β†’ (β™―β€˜πΉ) = (β™―β€˜(𝐹 cyclShift 𝐾)))
12224, 119, 121syl2anc 585 . . . . . . 7 (πœ‘ β†’ (β™―β€˜πΉ) = (β™―β€˜(𝐹 cyclShift 𝐾)))
12318, 122eqtrd 2777 . . . . . 6 (πœ‘ β†’ 𝑁 = (β™―β€˜(𝐹 cyclShift 𝐾)))
124115, 123breqtrd 5132 . . . . 5 (πœ‘ β†’ 0 < (β™―β€˜(𝐹 cyclShift 𝐾)))
125124adantl 483 . . . 4 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 0 < (β™―β€˜(𝐹 cyclShift 𝐾)))
126123adantl 483 . . . . 5 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝑁 = (β™―β€˜(𝐹 cyclShift 𝐾)))
127126oveq1d 7373 . . . 4 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝑁 βˆ’ 1) = ((β™―β€˜(𝐹 cyclShift 𝐾)) βˆ’ 1))
128 eucrct2eupth.e . . . . . 6 (πœ‘ β†’ (iEdgβ€˜π‘†) = (𝐼 β†Ύ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽}))))
129128adantl 483 . . . . 5 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (iEdgβ€˜π‘†) = (𝐼 β†Ύ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽}))))
13024, 18, 93jca 1129 . . . . . . . . 9 (πœ‘ β†’ (𝐹 ∈ Word dom 𝐼 ∧ 𝑁 = (β™―β€˜πΉ) ∧ 𝐽 ∈ (0..^𝑁)))
131130adantl 483 . . . . . . . 8 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 ∈ Word dom 𝐼 ∧ 𝑁 = (β™―β€˜πΉ) ∧ 𝐽 ∈ (0..^𝑁)))
132 cshimadifsn0 14720 . . . . . . . 8 ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 = (β™―β€˜πΉ) ∧ 𝐽 ∈ (0..^𝑁)) β†’ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) β€œ (0..^(𝑁 βˆ’ 1))))
133131, 132syl 17 . . . . . . 7 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) β€œ (0..^(𝑁 βˆ’ 1))))
1347imaeq1i 6011 . . . . . . 7 ((𝐹 cyclShift (𝐽 + 1)) β€œ (0..^(𝑁 βˆ’ 1))) = ((𝐹 cyclShift 𝐾) β€œ (0..^(𝑁 βˆ’ 1)))
135133, 134eqtrdi 2793 . . . . . 6 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽})) = ((𝐹 cyclShift 𝐾) β€œ (0..^(𝑁 βˆ’ 1))))
136135reseq2d 5938 . . . . 5 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐼 β†Ύ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽}))) = (𝐼 β†Ύ ((𝐹 cyclShift 𝐾) β€œ (0..^(𝑁 βˆ’ 1)))))
137129, 136eqtrd 2777 . . . 4 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (iEdgβ€˜π‘†) = (𝐼 β†Ύ ((𝐹 cyclShift 𝐾) β€œ (0..^(𝑁 βˆ’ 1)))))
138 eqid 2737 . . . 4 ((𝐹 cyclShift 𝐾) prefix (𝑁 βˆ’ 1)) = ((𝐹 cyclShift 𝐾) prefix (𝑁 βˆ’ 1))
139 eqid 2737 . . . 4 ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0...(𝑁 βˆ’ 1))) = ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0...(𝑁 βˆ’ 1)))
1401, 2, 110, 112, 113, 125, 127, 137, 138, 139eucrct2eupth1 29191 . . 3 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ ((𝐹 cyclShift 𝐾) prefix (𝑁 βˆ’ 1))(EulerPathsβ€˜π‘†)((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0...(𝑁 βˆ’ 1))))
141 eucrct2eupth.h . . . 4 𝐻 = ((𝐹 cyclShift 𝐾) prefix (𝑁 βˆ’ 1))
142141a1i 11 . . 3 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝐻 = ((𝐹 cyclShift 𝐾) prefix (𝑁 βˆ’ 1)))
143 eucrct2eupth.q . . . . 5 𝑄 = (π‘₯ ∈ (0..^𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁))))
144 fzossfz 13592 . . . . . . . 8 (0..^𝑁) βŠ† (0...𝑁)
14518oveq2d 7374 . . . . . . . 8 (πœ‘ β†’ (0...𝑁) = (0...(β™―β€˜πΉ)))
146144, 145sseqtrid 3997 . . . . . . 7 (πœ‘ β†’ (0..^𝑁) βŠ† (0...(β™―β€˜πΉ)))
147146resmptd 5995 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0..^𝑁)) = (π‘₯ ∈ (0..^𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))))
148 elfzoel2 13572 . . . . . . . 8 (𝐽 ∈ (0..^𝑁) β†’ 𝑁 ∈ β„€)
149 fzoval 13574 . . . . . . . 8 (𝑁 ∈ β„€ β†’ (0..^𝑁) = (0...(𝑁 βˆ’ 1)))
1509, 148, 1493syl 18 . . . . . . 7 (πœ‘ β†’ (0..^𝑁) = (0...(𝑁 βˆ’ 1)))
151150reseq2d 5938 . . . . . 6 (πœ‘ β†’ ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0..^𝑁)) = ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0...(𝑁 βˆ’ 1))))
152147, 151eqtr3d 2779 . . . . 5 (πœ‘ β†’ (π‘₯ ∈ (0..^𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) = ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0...(𝑁 βˆ’ 1))))
153143, 152eqtrid 2789 . . . 4 (πœ‘ β†’ 𝑄 = ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0...(𝑁 βˆ’ 1))))
154153adantl 483 . . 3 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝑄 = ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0...(𝑁 βˆ’ 1))))
155140, 142, 1543brtr4d 5138 . 2 ((𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝐻(EulerPathsβ€˜π‘†)𝑄)
15620adantl 483 . . . 4 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝐹(Circuitsβ€˜πΊ)𝑃)
157 peano2nn0 12454 . . . . . . . . . . . . 13 (𝐽 ∈ β„•0 β†’ (𝐽 + 1) ∈ β„•0)
1581573ad2ant1 1134 . . . . . . . . . . . 12 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ (𝐽 + 1) ∈ β„•0)
159158adantr 482 . . . . . . . . . . 11 (((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) ∧ Β¬ 𝐽 = (𝑁 βˆ’ 1)) β†’ (𝐽 + 1) ∈ β„•0)
160 simpl2 1193 . . . . . . . . . . 11 (((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) ∧ Β¬ 𝐽 = (𝑁 βˆ’ 1)) β†’ 𝑁 ∈ β„•)
161 1cnd 11151 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ 1 ∈ β„‚)
162 nn0cn 12424 . . . . . . . . . . . . . . . . 17 (𝐽 ∈ β„•0 β†’ 𝐽 ∈ β„‚)
1631623ad2ant1 1134 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ 𝐽 ∈ β„‚)
16412, 161, 163subadd2d 11532 . . . . . . . . . . . . . . 15 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ ((𝑁 βˆ’ 1) = 𝐽 ↔ (𝐽 + 1) = 𝑁))
165 eqcom 2744 . . . . . . . . . . . . . . 15 (𝐽 = (𝑁 βˆ’ 1) ↔ (𝑁 βˆ’ 1) = 𝐽)
166 eqcom 2744 . . . . . . . . . . . . . . 15 (𝑁 = (𝐽 + 1) ↔ (𝐽 + 1) = 𝑁)
167164, 165, 1663bitr4g 314 . . . . . . . . . . . . . 14 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ (𝐽 = (𝑁 βˆ’ 1) ↔ 𝑁 = (𝐽 + 1)))
168167necon3bbid 2982 . . . . . . . . . . . . 13 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ (Β¬ 𝐽 = (𝑁 βˆ’ 1) ↔ 𝑁 β‰  (𝐽 + 1)))
169157nn0red 12475 . . . . . . . . . . . . . . . 16 (𝐽 ∈ β„•0 β†’ (𝐽 + 1) ∈ ℝ)
1701693ad2ant1 1134 . . . . . . . . . . . . . . 15 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ (𝐽 + 1) ∈ ℝ)
171 nnre 12161 . . . . . . . . . . . . . . . 16 (𝑁 ∈ β„• β†’ 𝑁 ∈ ℝ)
1721713ad2ant2 1135 . . . . . . . . . . . . . . 15 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ 𝑁 ∈ ℝ)
173 nn0z 12525 . . . . . . . . . . . . . . . . 17 (𝐽 ∈ β„•0 β†’ 𝐽 ∈ β„€)
174 nnz 12521 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ β„• β†’ 𝑁 ∈ β„€)
175 zltp1le 12554 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝐽 < 𝑁 ↔ (𝐽 + 1) ≀ 𝑁))
176173, 174, 175syl2an 597 . . . . . . . . . . . . . . . 16 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„•) β†’ (𝐽 < 𝑁 ↔ (𝐽 + 1) ≀ 𝑁))
177176biimp3a 1470 . . . . . . . . . . . . . . 15 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ (𝐽 + 1) ≀ 𝑁)
178170, 172, 177leltned 11309 . . . . . . . . . . . . . 14 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ ((𝐽 + 1) < 𝑁 ↔ 𝑁 β‰  (𝐽 + 1)))
179178biimprd 248 . . . . . . . . . . . . 13 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ (𝑁 β‰  (𝐽 + 1) β†’ (𝐽 + 1) < 𝑁))
180168, 179sylbid 239 . . . . . . . . . . . 12 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ (Β¬ 𝐽 = (𝑁 βˆ’ 1) β†’ (𝐽 + 1) < 𝑁))
181180imp 408 . . . . . . . . . . 11 (((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) ∧ Β¬ 𝐽 = (𝑁 βˆ’ 1)) β†’ (𝐽 + 1) < 𝑁)
182159, 160, 1813jca 1129 . . . . . . . . . 10 (((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) ∧ Β¬ 𝐽 = (𝑁 βˆ’ 1)) β†’ ((𝐽 + 1) ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ (𝐽 + 1) < 𝑁))
183182ex 414 . . . . . . . . 9 ((𝐽 ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ 𝐽 < 𝑁) β†’ (Β¬ 𝐽 = (𝑁 βˆ’ 1) β†’ ((𝐽 + 1) ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ (𝐽 + 1) < 𝑁)))
18410, 183sylbi 216 . . . . . . . 8 (𝐽 ∈ (0..^𝑁) β†’ (Β¬ 𝐽 = (𝑁 βˆ’ 1) β†’ ((𝐽 + 1) ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ (𝐽 + 1) < 𝑁)))
185 elfzo0 13614 . . . . . . . 8 ((𝐽 + 1) ∈ (0..^𝑁) ↔ ((𝐽 + 1) ∈ β„•0 ∧ 𝑁 ∈ β„• ∧ (𝐽 + 1) < 𝑁))
186184, 185syl6ibr 252 . . . . . . 7 (𝐽 ∈ (0..^𝑁) β†’ (Β¬ 𝐽 = (𝑁 βˆ’ 1) β†’ (𝐽 + 1) ∈ (0..^𝑁)))
1879, 186syl 17 . . . . . 6 (πœ‘ β†’ (Β¬ 𝐽 = (𝑁 βˆ’ 1) β†’ (𝐽 + 1) ∈ (0..^𝑁)))
188187impcom 409 . . . . 5 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐽 + 1) ∈ (0..^𝑁))
1895a1i 11 . . . . 5 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝐾 = (𝐽 + 1))
19018eqcomd 2743 . . . . . . 7 (πœ‘ β†’ (β™―β€˜πΉ) = 𝑁)
191190oveq2d 7374 . . . . . 6 (πœ‘ β†’ (0..^(β™―β€˜πΉ)) = (0..^𝑁))
192191adantl 483 . . . . 5 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (0..^(β™―β€˜πΉ)) = (0..^𝑁))
193188, 189, 1923eltr4d 2853 . . . 4 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝐾 ∈ (0..^(β™―β€˜πΉ)))
194 eqid 2737 . . . 4 (𝐹 cyclShift 𝐾) = (𝐹 cyclShift 𝐾)
195 eqid 2737 . . . 4 (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) = (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ)))))
1963adantl 483 . . . 4 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝐹(EulerPathsβ€˜πΊ)𝑃)
1971, 2, 156, 31, 193, 194, 195, 196eucrctshift 29190 . . 3 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ)))))))
198 simprl 770 . . . . 5 (((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) ∧ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))) β†’ (𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))
199 simprr 772 . . . . 5 (((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) ∧ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))) β†’ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))
200124ad2antlr 726 . . . . 5 (((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) ∧ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))) β†’ 0 < (β™―β€˜(𝐹 cyclShift 𝐾)))
201123oveq1d 7373 . . . . . 6 (πœ‘ β†’ (𝑁 βˆ’ 1) = ((β™―β€˜(𝐹 cyclShift 𝐾)) βˆ’ 1))
202201ad2antlr 726 . . . . 5 (((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) ∧ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))) β†’ (𝑁 βˆ’ 1) = ((β™―β€˜(𝐹 cyclShift 𝐾)) βˆ’ 1))
203128adantl 483 . . . . . . 7 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (iEdgβ€˜π‘†) = (𝐼 β†Ύ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽}))))
204130adantl 483 . . . . . . . . . 10 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 ∈ Word dom 𝐼 ∧ 𝑁 = (β™―β€˜πΉ) ∧ 𝐽 ∈ (0..^𝑁)))
205204, 132syl 17 . . . . . . . . 9 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) β€œ (0..^(𝑁 βˆ’ 1))))
206205, 134eqtrdi 2793 . . . . . . . 8 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽})) = ((𝐹 cyclShift 𝐾) β€œ (0..^(𝑁 βˆ’ 1))))
207206reseq2d 5938 . . . . . . 7 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (𝐼 β†Ύ (𝐹 β€œ ((0..^𝑁) βˆ– {𝐽}))) = (𝐼 β†Ύ ((𝐹 cyclShift 𝐾) β€œ (0..^(𝑁 βˆ’ 1)))))
208203, 207eqtrd 2777 . . . . . 6 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (iEdgβ€˜π‘†) = (𝐼 β†Ύ ((𝐹 cyclShift 𝐾) β€œ (0..^(𝑁 βˆ’ 1)))))
209208adantr 482 . . . . 5 (((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) ∧ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))) β†’ (iEdgβ€˜π‘†) = (𝐼 β†Ύ ((𝐹 cyclShift 𝐾) β€œ (0..^(𝑁 βˆ’ 1)))))
210 eqid 2737 . . . . 5 ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) β†Ύ (0...(𝑁 βˆ’ 1))) = ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) β†Ύ (0...(𝑁 βˆ’ 1)))
2111, 2, 198, 199, 113, 200, 202, 209, 138, 210eucrct2eupth1 29191 . . . 4 (((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) ∧ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))) β†’ ((𝐹 cyclShift 𝐾) prefix (𝑁 βˆ’ 1))(EulerPathsβ€˜π‘†)((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) β†Ύ (0...(𝑁 βˆ’ 1))))
212141a1i 11 . . . 4 (((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) ∧ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))) β†’ 𝐻 = ((𝐹 cyclShift 𝐾) prefix (𝑁 βˆ’ 1)))
213190oveq1d 7373 . . . . . . . . . . . 12 (πœ‘ β†’ ((β™―β€˜πΉ) βˆ’ 𝐾) = (𝑁 βˆ’ 𝐾))
214213breq2d 5118 . . . . . . . . . . 11 (πœ‘ β†’ (π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾) ↔ π‘₯ ≀ (𝑁 βˆ’ 𝐾)))
215214adantl 483 . . . . . . . . . 10 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾) ↔ π‘₯ ≀ (𝑁 βˆ’ 𝐾)))
216190oveq2d 7374 . . . . . . . . . . . 12 (πœ‘ β†’ ((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ)) = ((π‘₯ + 𝐾) βˆ’ 𝑁))
217216fveq2d 6847 . . . . . . . . . . 11 (πœ‘ β†’ (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))) = (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))
218217adantl 483 . . . . . . . . . 10 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))) = (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))
219215, 218ifbieq2d 4513 . . . . . . . . 9 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ)))) = if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁))))
220219mpteq2dv 5208 . . . . . . . 8 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) = (π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))))
221150eqcomd 2743 . . . . . . . . 9 (πœ‘ β†’ (0...(𝑁 βˆ’ 1)) = (0..^𝑁))
222221adantl 483 . . . . . . . 8 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (0...(𝑁 βˆ’ 1)) = (0..^𝑁))
223220, 222reseq12d 5939 . . . . . . 7 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) β†Ύ (0...(𝑁 βˆ’ 1))) = ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0..^𝑁)))
22418adantl 483 . . . . . . . . . 10 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝑁 = (β™―β€˜πΉ))
225224oveq2d 7374 . . . . . . . . 9 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (0...𝑁) = (0...(β™―β€˜πΉ)))
226144, 225sseqtrid 3997 . . . . . . . 8 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ (0..^𝑁) βŠ† (0...(β™―β€˜πΉ)))
227226resmptd 5995 . . . . . . 7 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))) β†Ύ (0..^𝑁)) = (π‘₯ ∈ (0..^𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))))
228223, 227eqtrd 2777 . . . . . 6 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) β†Ύ (0...(𝑁 βˆ’ 1))) = (π‘₯ ∈ (0..^𝑁) ↦ if(π‘₯ ≀ (𝑁 βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ 𝑁)))))
229143, 228eqtr4id 2796 . . . . 5 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝑄 = ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) β†Ύ (0...(𝑁 βˆ’ 1))))
230229adantr 482 . . . 4 (((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) ∧ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))) β†’ 𝑄 = ((π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) β†Ύ (0...(𝑁 βˆ’ 1))))
231211, 212, 2303brtr4d 5138 . . 3 (((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) ∧ ((𝐹 cyclShift 𝐾)(EulerPathsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))) ∧ (𝐹 cyclShift 𝐾)(Circuitsβ€˜πΊ)(π‘₯ ∈ (0...(β™―β€˜πΉ)) ↦ if(π‘₯ ≀ ((β™―β€˜πΉ) βˆ’ 𝐾), (π‘ƒβ€˜(π‘₯ + 𝐾)), (π‘ƒβ€˜((π‘₯ + 𝐾) βˆ’ (β™―β€˜πΉ))))))) β†’ 𝐻(EulerPathsβ€˜π‘†)𝑄)
232197, 231mpdan 686 . 2 ((Β¬ 𝐽 = (𝑁 βˆ’ 1) ∧ πœ‘) β†’ 𝐻(EulerPathsβ€˜π‘†)𝑄)
233155, 232pm2.61ian 811 1 (πœ‘ β†’ 𝐻(EulerPathsβ€˜π‘†)𝑄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   βˆ– cdif 3908   βˆͺ cun 3909  ifcif 4487  {csn 4587   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5634   β†Ύ cres 5636   β€œ cima 5637   Fn wfn 6492  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  β„‚cc 11050  β„cr 11051  0cc0 11052  1c1 11053   + caddc 11055   < clt 11190   ≀ cle 11191   βˆ’ cmin 11386  β„•cn 12154  β„•0cn0 12414  β„€cz 12500  ...cfz 13425  ..^cfzo 13568  β™―chash 14231  Word cword 14403   prefix cpfx 14559   cyclShift ccsh 14677  Vtxcvtx 27950  iEdgciedg 27951  Walkscwlks 28547  Trailsctrls 28641  Circuitsccrcts 28735  EulerPathsceupth 29144
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129  ax-pre-sup 11130
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-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-map 8768  df-pm 8769  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9379  df-inf 9380  df-card 9876  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-div 11814  df-nn 12155  df-2 12217  df-n0 12415  df-z 12501  df-uz 12765  df-rp 12917  df-ico 13271  df-fz 13426  df-fzo 13569  df-fl 13698  df-mod 13776  df-hash 14232  df-word 14404  df-concat 14460  df-substr 14530  df-pfx 14560  df-csh 14678  df-wlks 28550  df-trls 28643  df-crcts 28737  df-eupth 29145
This theorem is referenced by: (None)
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