upgrimpths - Mathbox for Alexander van der Vekens

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Theorem upgrimpths 48597

Description: Graph isomorphisms between simple pseudographs map paths onto paths. (Contributed by AV, 31-Oct-2025.)

**Hypotheses**
Ref	Expression
upgrimwlk.i	⊢ 𝐼 = (iEdg‘𝐺)
upgrimwlk.j	⊢ 𝐽 = (iEdg‘𝐻)
upgrimwlk.g	⊢ (𝜑 → 𝐺 ∈ USPGraph)
upgrimwlk.h	⊢ (𝜑 → 𝐻 ∈ USPGraph)
upgrimwlk.n	⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))
upgrimwlk.e	⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (^◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))
upgrimpths.p	⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)

**Assertion**
Ref	Expression
upgrimpths	⊢ (𝜑 → 𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐺 𝑥,𝐼 𝑥,𝐽 𝑥,𝑃 𝜑,𝑥 𝑥,𝐸 𝑥,𝑁 Allowed substitution hint: 𝐻(𝑥)

Proof of Theorem upgrimpths Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgrimwlk.i	. . . 4 ⊢ 𝐼 = (iEdg‘𝐺)
2		upgrimwlk.j	. . . 4 ⊢ 𝐽 = (iEdg‘𝐻)
3		upgrimwlk.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ USPGraph)
4		upgrimwlk.h	. . . 4 ⊢ (𝜑 → 𝐻 ∈ USPGraph)
5		upgrimwlk.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (𝐺 GraphIso 𝐻))
6		upgrimwlk.e	. . . 4 ⊢ 𝐸 = (𝑥 ∈ dom 𝐹 ↦ (^◡𝐽‘(𝑁 “ (𝐼‘(𝐹‘𝑥)))))
7		upgrimpths.p	. . . . 5 ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)
8		pthistrl 30013	. . . . 5 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
9	7, 8	syl 18	. . . 4 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
10	1, 2, 3, 4, 5, 6, 9	upgrimtrls 48594	. . 3 ⊢ (𝜑 → 𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃))
11	1, 2, 3, 4, 5, 6, 7	upgrimpthslem1 48595	. . 3 ⊢ (𝜑 → Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
12		pthiswlk 30015	. . . . . . . . . . . 12 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
13	1	wlkf 29905	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
14	7, 12, 13	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
15		eqid 2769	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
16	15	wlkp 29907	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
17	7, 12, 16	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
18	1, 2, 3, 4, 5, 6, 14, 17	upgrimwlklem4 48588	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻))
19	18	ffnd 6707	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐸)))
20	1, 2, 3, 4, 5, 6, 14	upgrimwlklem1 48585	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐸) = (♯‘𝐹))
21		wlkcl 29906	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
22	7, 12, 21	3syl 19	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
23	20, 22	eqeltrd 2869	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐸) ∈ ℕ₀)
24		0elfz 13652	. . . . . . . . . 10 ⊢ ((♯‘𝐸) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐸)))
25	23, 24	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0...(♯‘𝐸)))
26		nn0fz0 13653	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
27	22, 26	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
28	20	oveq2d 7427	. . . . . . . . . 10 ⊢ (𝜑 → (0...(♯‘𝐸)) = (0...(♯‘𝐹)))
29	27, 28	eleqtrrd 2872	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐸)))
30		fnimapr 6965	. . . . . . . . 9 ⊢ (((𝑁 ∘ 𝑃) Fn (0...(♯‘𝐸)) ∧ 0 ∈ (0...(♯‘𝐸)) ∧ (♯‘𝐹) ∈ (0...(♯‘𝐸))) → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) = {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))})
31	19, 25, 29, 30	syl3anc 1396	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) = {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))})
32	31	eleq2d 2855	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ↔ 𝑥 ∈ {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))}))
33		vex 3467	. . . . . . . 8 ⊢ 𝑥 ∈ V
34	33	elpr 4619	. . . . . . 7 ⊢ (𝑥 ∈ {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))} ↔ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
35	32, 34	bitrdi 290	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ↔ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))))
36	1, 2, 3, 4, 5, 6, 7	upgrimpthslem2 48596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
37	36	simpld 499	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0))
38		eqeq2 2781	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘0) → (((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0)))
39	38	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘0) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0)))
40	37, 39	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (𝑥 = ((𝑁 ∘ 𝑃)‘0) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
41	36	simprd 500	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))
42		eqeq2 2781	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → (((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
43	42	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
44	41, 43	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
45	40, 44	jaod 872	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ((𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
46	45	impancom 456	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑦 ∈ (1..^(♯‘𝐹)) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
47	46	imp 411	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥)
48	47	nrexdv 3166	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → ¬ ∃𝑦 ∈ (1..^(♯‘𝐹))((𝑁 ∘ 𝑃)‘𝑦) = 𝑥)
49	20	eqcomd 2775	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐹) = (♯‘𝐸))
50	49	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...(♯‘𝐹)) = (0...(♯‘𝐸)))
51	50	feq2d 6690	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 ∘ 𝑃):(0...(♯‘𝐹))⟶(Vtx‘𝐻) ↔ (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻)))
52	18, 51	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐹))⟶(Vtx‘𝐻))
53	52	ffnd 6707	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐹)))
54	53	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐹)))
55		fzo0ss1 13718	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
56		fzossfz 13707	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
57	55, 56	sstri 3954	. . . . . . . . . 10 ⊢ (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
58	57	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (1..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)))
59	54, 58	fvelimabd 6955	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))) ↔ ∃𝑦 ∈ (1..^(♯‘𝐹))((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
60	48, 59	mtbird 328	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
61	60	ex 417	. . . . . 6 ⊢ (𝜑 → ((𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
62	35, 61	sylbid 243	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
63	62	ralrimiv 3162	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
64		disj 4416	. . . 4 ⊢ ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅ ↔ ∀𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
65	63, 64	sylibr 237	. . 3 ⊢ (𝜑 → (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅)
66	20	oveq2d 7427	. . . . . . 7 ⊢ (𝜑 → (1..^(♯‘𝐸)) = (1..^(♯‘𝐹)))
67	66	reseq2d 5979	. . . . . 6 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
68	67	cnveqd 5862	. . . . 5 ⊢ (𝜑 → ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
69	68	funeqd 6559	. . . 4 ⊢ (𝜑 → (Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ↔ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹)))))
70		preq2 4705	. . . . . . . 8 ⊢ ((♯‘𝐸) = (♯‘𝐹) → {0, (♯‘𝐸)} = {0, (♯‘𝐹)})
71	70	imaeq2d 6063	. . . . . . 7 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) = ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}))
72		oveq2 7419	. . . . . . . 8 ⊢ ((♯‘𝐸) = (♯‘𝐹) → (1..^(♯‘𝐸)) = (1..^(♯‘𝐹)))
73	72	imaeq2d 6063	. . . . . . 7 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸))) = ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
74	71, 73	ineq12d 4182	. . . . . 6 ⊢ ((♯‘𝐸) = (♯‘𝐹) → (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
75	74	eqeq1d 2771	. . . . 5 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅ ↔ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅))
76	20, 75	syl 18	. . . 4 ⊢ (𝜑 → ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅ ↔ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅))
77	69, 76	3anbi23d 1465	. . 3 ⊢ (𝜑 → ((𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅) ↔ (𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅)))
78	10, 11, 65, 77	mpbir3and 1359	. 2 ⊢ (𝜑 → (𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅))
79		ispth 30011	. 2 ⊢ (𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃) ↔ (𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅))
80	78, 79	sylibr 237	1 ⊢ (𝜑 → 𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ∀wral 3085 ∃wrex 3095 ∩ cin 3912 ⊆ wss 3913 ∅c0 4294 {cpr 4596 class class class wbr 5113 ↦ cmpt 5196 ^◡ccnv 5661 dom cdm 5662 ↾ cres 5664 “ cima 5665 ∘ ccom 5666 Fun wfun 6531 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 0cc0 11100 1c1 11101 ℕ₀cn0 12504 ...cfz 13535 ..^cfzo 13682 ♯chash 14366 Word cword 14550 Vtxcvtx 29287 iEdgciedg 29288 USPGraphcuspgr 29439 Walkscwlks 29887 Trailsctrls 29979 Pathscpths 30000 GraphIso cgrim 48563

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1077 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-er 8694 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-dju 9887 df-card 9925 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-nn 12234 df-2 12303 df-n0 12505 df-xnn0 12578 df-z 12592 df-uz 12863 df-fz 13536 df-fzo 13683 df-hash 14367 df-word 14551 df-edg 29339 df-uhgr 29349 df-upgr 29373 df-uspgr 29441 df-wlks 29890 df-trls 29981 df-pths 30004 df-grim 48566

This theorem is referenced by: upgrimcycls 48599

W3C validator