upgrimpths - Mathbox for Alexander van der Vekens

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Theorem upgrimpths 48536

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 29925	. . . . 5 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
9	7, 8	syl 17	. . . 4 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
10	1, 2, 3, 4, 5, 6, 9	upgrimtrls 48533	. . 3 ⊢ (𝜑 → 𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃))
11	1, 2, 3, 4, 5, 6, 7	upgrimpthslem1 48534	. . 3 ⊢ (𝜑 → Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
12		pthiswlk 29927	. . . . . . . . . . . 12 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
13	1	wlkf 29817	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
14	7, 12, 13	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
15		eqid 2764	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
16	15	wlkp 29819	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
17	7, 12, 16	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃:(0...(♯‘𝐹))⟶(Vtx‘𝐺))
18	1, 2, 3, 4, 5, 6, 14, 17	upgrimwlklem4 48527	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻))
19	18	ffnd 6694	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐸)))
20	1, 2, 3, 4, 5, 6, 14	upgrimwlklem1 48524	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐸) = (♯‘𝐹))
21		wlkcl 29818	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
22	7, 12, 21	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
23	20, 22	eqeltrd 2864	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐸) ∈ ℕ₀)
24		0elfz 13631	. . . . . . . . . 10 ⊢ ((♯‘𝐸) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐸)))
25	23, 24	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0...(♯‘𝐸)))
26		nn0fz0 13632	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
27	22, 26	sylib 220	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
28	20	oveq2d 7414	. . . . . . . . . 10 ⊢ (𝜑 → (0...(♯‘𝐸)) = (0...(♯‘𝐹)))
29	27, 28	eleqtrrd 2867	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐸)))
30		fnimapr 6952	. . . . . . . . 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 1392	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) = {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))})
32	31	eleq2d 2850	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ↔ 𝑥 ∈ {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))}))
33		vex 3460	. . . . . . . 8 ⊢ 𝑥 ∈ V
34	33	elpr 4609	. . . . . . 7 ⊢ (𝑥 ∈ {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))} ↔ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
35	32, 34	bitrdi 289	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ↔ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))))
36	1, 2, 3, 4, 5, 6, 7	upgrimpthslem2 48535	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
37	36	simpld 498	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0))
38		eqeq2 2776	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘0) → (((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0)))
39	38	notbid 320	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘0) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0)))
40	37, 39	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (𝑥 = ((𝑁 ∘ 𝑃)‘0) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
41	36	simprd 499	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))
42		eqeq2 2776	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → (((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
43	42	notbid 320	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
44	41, 43	syl5ibrcom 249	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
45	40, 44	jaod 870	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ((𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
46	45	impancom 455	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑦 ∈ (1..^(♯‘𝐹)) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
47	46	imp 410	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥)
48	47	nrexdv 3159	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → ¬ ∃𝑦 ∈ (1..^(♯‘𝐹))((𝑁 ∘ 𝑃)‘𝑦) = 𝑥)
49	20	eqcomd 2770	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐹) = (♯‘𝐸))
50	49	oveq2d 7414	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...(♯‘𝐹)) = (0...(♯‘𝐸)))
51	50	feq2d 6677	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 ∘ 𝑃):(0...(♯‘𝐹))⟶(Vtx‘𝐻) ↔ (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻)))
52	18, 51	mpbird 259	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐹))⟶(Vtx‘𝐻))
53	52	ffnd 6694	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐹)))
54	53	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐹)))
55		fzo0ss1 13697	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
56		fzossfz 13686	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
57	55, 56	sstri 3947	. . . . . . . . . 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 6942	. . . . . . . 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 327	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
61	60	ex 416	. . . . . 6 ⊢ (𝜑 → ((𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
62	35, 61	sylbid 242	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
63	62	ralrimiv 3155	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
64		disj 4406	. . . 4 ⊢ ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅ ↔ ∀𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
65	63, 64	sylibr 236	. . 3 ⊢ (𝜑 → (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅)
66	20	oveq2d 7414	. . . . . . 7 ⊢ (𝜑 → (1..^(♯‘𝐸)) = (1..^(♯‘𝐹)))
67	66	reseq2d 5967	. . . . . 6 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
68	67	cnveqd 5849	. . . . 5 ⊢ (𝜑 → ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
69	68	funeqd 6545	. . . 4 ⊢ (𝜑 → (Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ↔ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹)))))
70		preq2 4695	. . . . . . . 8 ⊢ ((♯‘𝐸) = (♯‘𝐹) → {0, (♯‘𝐸)} = {0, (♯‘𝐹)})
71	70	imaeq2d 6051	. . . . . . 7 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) = ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}))
72		oveq2 7406	. . . . . . . 8 ⊢ ((♯‘𝐸) = (♯‘𝐹) → (1..^(♯‘𝐸)) = (1..^(♯‘𝐹)))
73	72	imaeq2d 6051	. . . . . . 7 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸))) = ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
74	71, 73	ineq12d 4175	. . . . . 6 ⊢ ((♯‘𝐸) = (♯‘𝐹) → (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
75	74	eqeq1d 2766	. . . . 5 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅ ↔ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅))
76	20, 75	syl 17	. . . 4 ⊢ (𝜑 → ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅ ↔ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅))
77	69, 76	3anbi23d 1462	. . 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 1357	. 2 ⊢ (𝜑 → (𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅))
79		ispth 29923	. 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 236	1 ⊢ (𝜑 → 𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∨ wo 858 ∧ w3a 1099 = wceq 1562 ∈ wcel 2144 ∀wral 3078 ∃wrex 3088 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 {cpr 4586 class class class wbr 5102 ↦ cmpt 5183 ^◡ccnv 5648 dom cdm 5649 ↾ cres 5651 “ cima 5652 ∘ ccom 5653 Fun wfun 6517 Fn wfn 6518 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 0cc0 11075 1c1 11076 ℕ₀cn0 12483 ...cfz 13514 ..^cfzo 13661 ♯chash 14345 Word cword 14528 Vtxcvtx 29199 iEdgciedg 29200 USPGraphcuspgr 29351 Walkscwlks 29799 Trailsctrls 29891 Pathscpths 29912 GraphIso cgrim 48502

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-ifp 1075 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-oadd 8443 df-er 8680 df-map 8812 df-pm 8813 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-dju 9861 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-nn 12213 df-2 12282 df-n0 12484 df-xnn0 12557 df-z 12571 df-uz 12842 df-fz 13515 df-fzo 13662 df-hash 14346 df-word 14529 df-edg 29251 df-uhgr 29261 df-upgr 29285 df-uspgr 29353 df-wlks 29802 df-trls 29893 df-pths 29916 df-grim 48505

This theorem is referenced by: upgrimcycls 48538

W3C validator