Theorem crctcshlem4

Description: Lemma for crctcsh . (Contributed by AV, 10-Mar-2021)

	Ref	Expression
Hypotheses	crctcsh.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	crctcsh.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	crctcsh.d	⊢ ( 𝜑 → 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 )
	crctcsh.n	⊢ 𝑁 = ( ♯ ‘ 𝐹 )
	crctcsh.s	⊢ ( 𝜑 → 𝑆 ∈ ( 0 ..^ 𝑁 ) )
	crctcsh.h	⊢ 𝐻 = ( 𝐹 cyclShift 𝑆 )
	crctcsh.q	⊢ 𝑄 = ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ ( 𝑁 − 𝑆 ) , ( 𝑃 ‘ ( 𝑥 + 𝑆 ) ) , ( 𝑃 ‘ ( ( 𝑥 + 𝑆 ) − 𝑁 ) ) ) )
Assertion	crctcshlem4	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝐻 = 𝐹 ∧ 𝑄 = 𝑃 ) )

Proof

Step	Hyp	Ref	Expression
1		crctcsh.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		crctcsh.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
3		crctcsh.d	⊢ ( 𝜑 → 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 )
4		crctcsh.n	⊢ 𝑁 = ( ♯ ‘ 𝐹 )
5		crctcsh.s	⊢ ( 𝜑 → 𝑆 ∈ ( 0 ..^ 𝑁 ) )
6		crctcsh.h	⊢ 𝐻 = ( 𝐹 cyclShift 𝑆 )
7		crctcsh.q	⊢ 𝑄 = ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ ( 𝑁 − 𝑆 ) , ( 𝑃 ‘ ( 𝑥 + 𝑆 ) ) , ( 𝑃 ‘ ( ( 𝑥 + 𝑆 ) − 𝑁 ) ) ) )
8		oveq2	⊢ ( 𝑆 = 0 → ( 𝐹 cyclShift 𝑆 ) = ( 𝐹 cyclShift 0 ) )
9		crctiswlk	⊢ ( 𝐹 ( Circuits ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
10	2	wlkf	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝐹 ∈ Word dom 𝐼 )
11		cshw0	⊢ ( 𝐹 ∈ Word dom 𝐼 → ( 𝐹 cyclShift 0 ) = 𝐹 )
12	3 9 10 11	4syl	⊢ ( 𝜑 → ( 𝐹 cyclShift 0 ) = 𝐹 )
13	8 12	sylan9eqr	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝐹 cyclShift 𝑆 ) = 𝐹 )
14	6 13	eqtrid	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → 𝐻 = 𝐹 )
15		oveq2	⊢ ( 𝑆 = 0 → ( 𝑁 − 𝑆 ) = ( 𝑁 − 0 ) )
16	1 2 3 4	crctcshlem1	⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
17	16	nn0cnd	⊢ ( 𝜑 → 𝑁 ∈ ℂ )
18	17	subid1d	⊢ ( 𝜑 → ( 𝑁 − 0 ) = 𝑁 )
19	15 18	sylan9eqr	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝑁 − 𝑆 ) = 𝑁 )
20	19	breq2d	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝑥 ≤ ( 𝑁 − 𝑆 ) ↔ 𝑥 ≤ 𝑁 ) )
21	20	adantr	⊢ ( ( ( 𝜑 ∧ 𝑆 = 0 ) ∧ 𝑥 ∈ ( 0 ... 𝑁 ) ) → ( 𝑥 ≤ ( 𝑁 − 𝑆 ) ↔ 𝑥 ≤ 𝑁 ) )
22		oveq2	⊢ ( 𝑆 = 0 → ( 𝑥 + 𝑆 ) = ( 𝑥 + 0 ) )
23	22	adantl	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝑥 + 𝑆 ) = ( 𝑥 + 0 ) )
24		elfzelz	⊢ ( 𝑥 ∈ ( 0 ... 𝑁 ) → 𝑥 ∈ ℤ )
25	24	zcnd	⊢ ( 𝑥 ∈ ( 0 ... 𝑁 ) → 𝑥 ∈ ℂ )
26	25	addridd	⊢ ( 𝑥 ∈ ( 0 ... 𝑁 ) → ( 𝑥 + 0 ) = 𝑥 )
27	23 26	sylan9eq	⊢ ( ( ( 𝜑 ∧ 𝑆 = 0 ) ∧ 𝑥 ∈ ( 0 ... 𝑁 ) ) → ( 𝑥 + 𝑆 ) = 𝑥 )
28	27	fveq2d	⊢ ( ( ( 𝜑 ∧ 𝑆 = 0 ) ∧ 𝑥 ∈ ( 0 ... 𝑁 ) ) → ( 𝑃 ‘ ( 𝑥 + 𝑆 ) ) = ( 𝑃 ‘ 𝑥 ) )
29	27	fvoveq1d	⊢ ( ( ( 𝜑 ∧ 𝑆 = 0 ) ∧ 𝑥 ∈ ( 0 ... 𝑁 ) ) → ( 𝑃 ‘ ( ( 𝑥 + 𝑆 ) − 𝑁 ) ) = ( 𝑃 ‘ ( 𝑥 − 𝑁 ) ) )
30	21 28 29	ifbieq12d	⊢ ( ( ( 𝜑 ∧ 𝑆 = 0 ) ∧ 𝑥 ∈ ( 0 ... 𝑁 ) ) → if ( 𝑥 ≤ ( 𝑁 − 𝑆 ) , ( 𝑃 ‘ ( 𝑥 + 𝑆 ) ) , ( 𝑃 ‘ ( ( 𝑥 + 𝑆 ) − 𝑁 ) ) ) = if ( 𝑥 ≤ 𝑁 , ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 − 𝑁 ) ) ) )
31	30	mpteq2dva	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ ( 𝑁 − 𝑆 ) , ( 𝑃 ‘ ( 𝑥 + 𝑆 ) ) , ( 𝑃 ‘ ( ( 𝑥 + 𝑆 ) − 𝑁 ) ) ) ) = ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ 𝑁 , ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 − 𝑁 ) ) ) ) )
32		elfzle2	⊢ ( 𝑥 ∈ ( 0 ... 𝑁 ) → 𝑥 ≤ 𝑁 )
33	32	adantl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 0 ... 𝑁 ) ) → 𝑥 ≤ 𝑁 )
34	33	iftrued	⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 0 ... 𝑁 ) ) → if ( 𝑥 ≤ 𝑁 , ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 − 𝑁 ) ) ) = ( 𝑃 ‘ 𝑥 ) )
35	34	mpteq2dva	⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ 𝑁 , ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 − 𝑁 ) ) ) ) = ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ ( 𝑃 ‘ 𝑥 ) ) )
36	1	wlkp	⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 )
37	3 9 36	3syl	⊢ ( 𝜑 → 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 )
38		ffn	⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 → 𝑃 Fn ( 0 ... ( ♯ ‘ 𝐹 ) ) )
39	4	eqcomi	⊢ ( ♯ ‘ 𝐹 ) = 𝑁
40	39	oveq2i	⊢ ( 0 ... ( ♯ ‘ 𝐹 ) ) = ( 0 ... 𝑁 )
41	40	fneq2i	⊢ ( 𝑃 Fn ( 0 ... ( ♯ ‘ 𝐹 ) ) ↔ 𝑃 Fn ( 0 ... 𝑁 ) )
42	38 41	sylib	⊢ ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 → 𝑃 Fn ( 0 ... 𝑁 ) )
43	42	adantl	⊢ ( ( 𝜑 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) → 𝑃 Fn ( 0 ... 𝑁 ) )
44		dffn5	⊢ ( 𝑃 Fn ( 0 ... 𝑁 ) ↔ 𝑃 = ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ ( 𝑃 ‘ 𝑥 ) ) )
45	43 44	sylib	⊢ ( ( 𝜑 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) → 𝑃 = ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ ( 𝑃 ‘ 𝑥 ) ) )
46	45	eqcomd	⊢ ( ( 𝜑 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ 𝑉 ) → ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ ( 𝑃 ‘ 𝑥 ) ) = 𝑃 )
47	37 46	mpdan	⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ ( 𝑃 ‘ 𝑥 ) ) = 𝑃 )
48	35 47	eqtrd	⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ 𝑁 , ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 − 𝑁 ) ) ) ) = 𝑃 )
49	48	adantr	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ 𝑁 , ( 𝑃 ‘ 𝑥 ) , ( 𝑃 ‘ ( 𝑥 − 𝑁 ) ) ) ) = 𝑃 )
50	31 49	eqtrd	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝑥 ∈ ( 0 ... 𝑁 ) ↦ if ( 𝑥 ≤ ( 𝑁 − 𝑆 ) , ( 𝑃 ‘ ( 𝑥 + 𝑆 ) ) , ( 𝑃 ‘ ( ( 𝑥 + 𝑆 ) − 𝑁 ) ) ) ) = 𝑃 )
51	7 50	eqtrid	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → 𝑄 = 𝑃 )
52	14 51	jca	⊢ ( ( 𝜑 ∧ 𝑆 = 0 ) → ( 𝐻 = 𝐹 ∧ 𝑄 = 𝑃 ) )