crctcsh - Mathbox for Alexander van der Vekens

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Theorem crctcsh 41027

Description: Cyclically shifting the indices of a circuit ⟨𝐹, 𝑃⟩ results in a circuit ⟨𝐻, 𝑄⟩. (Contributed by AV, 10-Mar-2021.)

**Hypotheses**
Ref	Expression
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**
Ref	Expression
crctcsh	⊢ (𝜑 → 𝐻(CircuitS‘𝐺)𝑄)

Distinct variable groups: 𝑥,𝑁 𝑥,𝑃 𝑥,𝑆 𝜑,𝑥 𝑥,𝐹 𝑥,𝐼 𝑥,𝑉 𝑥,𝐻 Allowed substitution hints: 𝑄(𝑥) 𝐺(𝑥)

**Proof of Theorem crctcsh**
Step	Hyp	Ref	Expression
1		crctcsh.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		crctcsh.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
3		crctcsh.d	. . . 4 ⊢ (𝜑 → 𝐹(CircuitS‘𝐺)𝑃)
4		crctcsh.n	. . . 4 ⊢ 𝑁 = (#‘𝐹)
5		crctcsh.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))
6		crctcsh.h	. . . 4 ⊢ 𝐻 = (𝐹 cyclShift 𝑆)
7		crctcsh.q	. . . 4 ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
8	1, 2, 3, 4, 5, 6, 7	crctcshlem4 41023	. . 3 ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝐻 = 𝐹 ∧ 𝑄 = 𝑃))
9		breq12 4588	. . . . 5 ⊢ ((𝐻 = 𝐹 ∧ 𝑄 = 𝑃) → (𝐻(CircuitS‘𝐺)𝑄 ↔ 𝐹(CircuitS‘𝐺)𝑃))
10	3, 9	syl5ibrcom 236	. . . 4 ⊢ (𝜑 → ((𝐻 = 𝐹 ∧ 𝑄 = 𝑃) → 𝐻(CircuitS‘𝐺)𝑄))
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑆 = 0) → ((𝐻 = 𝐹 ∧ 𝑄 = 𝑃) → 𝐻(CircuitS‘𝐺)𝑄))
12	8, 11	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑆 = 0) → 𝐻(CircuitS‘𝐺)𝑄)
13	1, 2, 3, 4, 5, 6, 7	crctcshtrl 41026	. . . . 5 ⊢ (𝜑 → 𝐻(TrailS‘𝐺)𝑄)
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝐻(TrailS‘𝐺)𝑄)
15	7	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))))
16		breq1 4586	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑥 ≤ (𝑁 − 𝑆) ↔ 0 ≤ (𝑁 − 𝑆)))
17		oveq1 6556	. . . . . . . . 9 ⊢ (𝑥 = 0 → (𝑥 + 𝑆) = (0 + 𝑆))
18	17	fveq2d 6107	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑃‘(𝑥 + 𝑆)) = (𝑃‘(0 + 𝑆)))
19	17	oveq1d 6564	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((𝑥 + 𝑆) − 𝑁) = ((0 + 𝑆) − 𝑁))
20	19	fveq2d 6107	. . . . . . . 8 ⊢ (𝑥 = 0 → (𝑃‘((𝑥 + 𝑆) − 𝑁)) = (𝑃‘((0 + 𝑆) − 𝑁)))
21	16, 18, 20	ifbieq12d 4063	. . . . . . 7 ⊢ (𝑥 = 0 → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) = if(0 ≤ (𝑁 − 𝑆), (𝑃‘(0 + 𝑆)), (𝑃‘((0 + 𝑆) − 𝑁))))
22		elfzo0le 12379	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑆 ≤ 𝑁)
23	5, 22	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ≤ 𝑁)
24	1, 2, 3, 4	crctcshlem1 41020	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
25	24	nn0red 11229	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
26		elfzoelz 12339	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℤ)
27	5, 26	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℤ)
28	27	zred 11358	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ ℝ)
29	25, 28	subge0d 10496	. . . . . . . . . 10 ⊢ (𝜑 → (0 ≤ (𝑁 − 𝑆) ↔ 𝑆 ≤ 𝑁))
30	23, 29	mpbird 246	. . . . . . . . 9 ⊢ (𝜑 → 0 ≤ (𝑁 − 𝑆))
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 0 ≤ (𝑁 − 𝑆))
32	31	iftrued 4044	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → if(0 ≤ (𝑁 − 𝑆), (𝑃‘(0 + 𝑆)), (𝑃‘((0 + 𝑆) − 𝑁))) = (𝑃‘(0 + 𝑆)))
33	21, 32	sylan9eqr 2666	. . . . . 6 ⊢ (((𝜑 ∧ 𝑆 ≠ 0) ∧ 𝑥 = 0) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) = (𝑃‘(0 + 𝑆)))
34	3	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝐹(CircuitS‘𝐺)𝑃)
35	1, 2, 34, 4	crctcshlem1 41020	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝑁 ∈ ℕ₀)
36		0elfz 12305	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → 0 ∈ (0...𝑁))
37	35, 36	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 0 ∈ (0...𝑁))
38		fvex 6113	. . . . . . 7 ⊢ (𝑃‘(0 + 𝑆)) ∈ V
39	38	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝑃‘(0 + 𝑆)) ∈ V)
40	15, 33, 37, 39	fvmptd 6197	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝑄‘0) = (𝑃‘(0 + 𝑆)))
41		breq1 4586	. . . . . . . . 9 ⊢ (𝑥 = (#‘𝐻) → (𝑥 ≤ (𝑁 − 𝑆) ↔ (#‘𝐻) ≤ (𝑁 − 𝑆)))
42		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑥 = (#‘𝐻) → (𝑥 + 𝑆) = ((#‘𝐻) + 𝑆))
43	42	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑥 = (#‘𝐻) → (𝑃‘(𝑥 + 𝑆)) = (𝑃‘((#‘𝐻) + 𝑆)))
44	42	oveq1d 6564	. . . . . . . . . 10 ⊢ (𝑥 = (#‘𝐻) → ((𝑥 + 𝑆) − 𝑁) = (((#‘𝐻) + 𝑆) − 𝑁))
45	44	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑥 = (#‘𝐻) → (𝑃‘((𝑥 + 𝑆) − 𝑁)) = (𝑃‘(((#‘𝐻) + 𝑆) − 𝑁)))
46	41, 43, 45	ifbieq12d 4063	. . . . . . . 8 ⊢ (𝑥 = (#‘𝐻) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) = if((#‘𝐻) ≤ (𝑁 − 𝑆), (𝑃‘((#‘𝐻) + 𝑆)), (𝑃‘(((#‘𝐻) + 𝑆) − 𝑁))))
47		elfzoel2 12338	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
48		elfzonn0 12380	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (0..^𝑁) → 𝑆 ∈ ℕ₀)
49		simpr 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℕ₀) → 𝑆 ∈ ℕ₀)
50	49	anim1i 590	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℕ₀) ∧ 𝑆 ≠ 0) → (𝑆 ∈ ℕ₀ ∧ 𝑆 ≠ 0))
51		elnnne0 11183	. . . . . . . . . . . . . . . . 17 ⊢ (𝑆 ∈ ℕ ↔ (𝑆 ∈ ℕ₀ ∧ 𝑆 ≠ 0))
52	50, 51	sylibr 223	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℕ₀) ∧ 𝑆 ≠ 0) → 𝑆 ∈ ℕ)
53	52	nngt0d 10941	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℕ₀) ∧ 𝑆 ≠ 0) → 0 < 𝑆)
54		zre 11258	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
55		nn0re 11178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ ℕ₀ → 𝑆 ∈ ℝ)
56	54, 55	anim12ci 589	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℕ₀) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
57	56	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℕ₀) ∧ 𝑆 ≠ 0) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
58		ltsubpos 10399	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < 𝑆 ↔ (𝑁 − 𝑆) < 𝑁))
59	58	bicomd 212	. . . . . . . . . . . . . . . 16 ⊢ ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑁 − 𝑆) < 𝑁 ↔ 0 < 𝑆))
60	57, 59	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℕ₀) ∧ 𝑆 ≠ 0) → ((𝑁 − 𝑆) < 𝑁 ↔ 0 < 𝑆))
61	53, 60	mpbird 246	. . . . . . . . . . . . . 14 ⊢ (((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℕ₀) ∧ 𝑆 ≠ 0) → (𝑁 − 𝑆) < 𝑁)
62	61	ex 449	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℕ₀) → (𝑆 ≠ 0 → (𝑁 − 𝑆) < 𝑁))
63	47, 48, 62	syl2anc 691	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (0..^𝑁) → (𝑆 ≠ 0 → (𝑁 − 𝑆) < 𝑁))
64	5, 63	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 ≠ 0 → (𝑁 − 𝑆) < 𝑁))
65	64	imp 444	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝑁 − 𝑆) < 𝑁)
66	5	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝑆 ∈ (0..^𝑁))
67	1, 2, 34, 4, 66, 6	crctcshlem2 41021	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (#‘𝐻) = 𝑁)
68	67	breq1d 4593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → ((#‘𝐻) ≤ (𝑁 − 𝑆) ↔ 𝑁 ≤ (𝑁 − 𝑆)))
69	68	notbid 307	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (¬ (#‘𝐻) ≤ (𝑁 − 𝑆) ↔ ¬ 𝑁 ≤ (𝑁 − 𝑆)))
70	25, 28	resubcld 10337	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 𝑆) ∈ ℝ)
71	70, 25	jca 553	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑁 − 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ))
72	71	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → ((𝑁 − 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ))
73		ltnle 9996	. . . . . . . . . . . 12 ⊢ (((𝑁 − 𝑆) ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑁 − 𝑆) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 𝑆)))
74	72, 73	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → ((𝑁 − 𝑆) < 𝑁 ↔ ¬ 𝑁 ≤ (𝑁 − 𝑆)))
75	69, 74	bitr4d 270	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (¬ (#‘𝐻) ≤ (𝑁 − 𝑆) ↔ (𝑁 − 𝑆) < 𝑁))
76	65, 75	mpbird 246	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → ¬ (#‘𝐻) ≤ (𝑁 − 𝑆))
77	76	iffalsed 4047	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → if((#‘𝐻) ≤ (𝑁 − 𝑆), (𝑃‘((#‘𝐻) + 𝑆)), (𝑃‘(((#‘𝐻) + 𝑆) − 𝑁))) = (𝑃‘(((#‘𝐻) + 𝑆) − 𝑁)))
78	46, 77	sylan9eqr 2666	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑆 ≠ 0) ∧ 𝑥 = (#‘𝐻)) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) = (𝑃‘(((#‘𝐻) + 𝑆) − 𝑁)))
79	1, 2, 3, 4, 5, 6	crctcshlem2 41021	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (#‘𝐻) = 𝑁)
80	79, 24	eqeltrd 2688	. . . . . . . . . . . . 13 ⊢ (𝜑 → (#‘𝐻) ∈ ℕ₀)
81	80	nn0cnd 11230	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘𝐻) ∈ ℂ)
82	27	zcnd 11359	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ ℂ)
83	24	nn0cnd 11230	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℂ)
84	81, 82, 83	addsubd 10292	. . . . . . . . . . 11 ⊢ (𝜑 → (((#‘𝐻) + 𝑆) − 𝑁) = (((#‘𝐻) − 𝑁) + 𝑆))
85	79	oveq1d 6564	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((#‘𝐻) − 𝑁) = (𝑁 − 𝑁))
86	83	subidd 10259	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑁 − 𝑁) = 0)
87	85, 86	eqtrd 2644	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘𝐻) − 𝑁) = 0)
88	87	oveq1d 6564	. . . . . . . . . . 11 ⊢ (𝜑 → (((#‘𝐻) − 𝑁) + 𝑆) = (0 + 𝑆))
89	84, 88	eqtrd 2644	. . . . . . . . . 10 ⊢ (𝜑 → (((#‘𝐻) + 𝑆) − 𝑁) = (0 + 𝑆))
90	89	fveq2d 6107	. . . . . . . . 9 ⊢ (𝜑 → (𝑃‘(((#‘𝐻) + 𝑆) − 𝑁)) = (𝑃‘(0 + 𝑆)))
91	90	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝑃‘(((#‘𝐻) + 𝑆) − 𝑁)) = (𝑃‘(0 + 𝑆)))
92	91	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑆 ≠ 0) ∧ 𝑥 = (#‘𝐻)) → (𝑃‘(((#‘𝐻) + 𝑆) − 𝑁)) = (𝑃‘(0 + 𝑆)))
93	78, 92	eqtrd 2644	. . . . . 6 ⊢ (((𝜑 ∧ 𝑆 ≠ 0) ∧ 𝑥 = (#‘𝐻)) → if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) = (𝑃‘(0 + 𝑆)))
94	79	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (#‘𝐻) = 𝑁)
95		nn0fz0 12306	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ₀ ↔ 𝑁 ∈ (0...𝑁))
96	24, 95	sylib 207	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
97	96	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝑁 ∈ (0...𝑁))
98	94, 97	eqeltrd 2688	. . . . . 6 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (#‘𝐻) ∈ (0...𝑁))
99	15, 93, 98, 39	fvmptd 6197	. . . . 5 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝑄‘(#‘𝐻)) = (𝑃‘(0 + 𝑆)))
100	40, 99	eqtr4d 2647	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝑄‘0) = (𝑄‘(#‘𝐻)))
101	14, 100	jca 553	. . 3 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝐻(TrailS‘𝐺)𝑄 ∧ (𝑄‘0) = (𝑄‘(#‘𝐻))))
102	1, 2, 3, 4, 5, 6, 7	crctcshlem3 41022	. . . . 5 ⊢ (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
103	102	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
104		isCrct 40996	. . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V) → (𝐻(CircuitS‘𝐺)𝑄 ↔ (𝐻(TrailS‘𝐺)𝑄 ∧ (𝑄‘0) = (𝑄‘(#‘𝐻)))))
105	103, 104	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → (𝐻(CircuitS‘𝐺)𝑄 ↔ (𝐻(TrailS‘𝐺)𝑄 ∧ (𝑄‘0) = (𝑄‘(#‘𝐻)))))
106	101, 105	mpbird 246	. 2 ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝐻(CircuitS‘𝐺)𝑄)
107	12, 106	pm2.61dane 2869	1 ⊢ (𝜑 → 𝐻(CircuitS‘𝐺)𝑄)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ifcif 4036 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 + caddc 9818 < clt 9953 ≤ cle 9954 − cmin 10145 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ...cfz 12197 ..^cfzo 12334 #chash 12979 cyclShift ccsh 13385 Vtxcvtx 25673 iEdgciedg 25674 TrailSctrls 40899 CircuitSccrcts 40990

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-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

This theorem is referenced by: eucrctshift 41411

W3C validator