upgrimpths - Mathbox for Alexander van der Vekens

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Theorem upgrimpths 48222

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 29800	. . . . 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 48219	. . 3 ⊢ (𝜑 → 𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃))
11	1, 2, 3, 4, 5, 6, 7	upgrimpthslem1 48220	. . 3 ⊢ (𝜑 → Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
12		pthiswlk 29802	. . . . . . . . . . . 12 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
13	1	wlkf 29692	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
14	7, 12, 13	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
15		eqid 2737	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
16	15	wlkp 29694	. . . . . . . . . . . 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 48213	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻))
19	18	ffnd 6664	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐸)))
20	1, 2, 3, 4, 5, 6, 14	upgrimwlklem1 48210	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐸) = (♯‘𝐹))
21		wlkcl 29693	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
22	7, 12, 21	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
23	20, 22	eqeltrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐸) ∈ ℕ₀)
24		0elfz 13544	. . . . . . . . . 10 ⊢ ((♯‘𝐸) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐸)))
25	23, 24	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0...(♯‘𝐸)))
26		nn0fz0 13545	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
27	22, 26	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
28	20	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝜑 → (0...(♯‘𝐸)) = (0...(♯‘𝐹)))
29	27, 28	eleqtrrd 2840	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐸)))
30		fnimapr 6918	. . . . . . . . 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 1374	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) = {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))})
32	31	eleq2d 2823	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ↔ 𝑥 ∈ {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))}))
33		vex 3445	. . . . . . . 8 ⊢ 𝑥 ∈ V
34	33	elpr 4606	. . . . . . 7 ⊢ (𝑥 ∈ {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))} ↔ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
35	32, 34	bitrdi 287	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ↔ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))))
36	1, 2, 3, 4, 5, 6, 7	upgrimpthslem2 48221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0) ∧ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
37	36	simpld 494	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0))
38		eqeq2 2749	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘0) → (((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0)))
39	38	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘0) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘0)))
40	37, 39	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (𝑥 = ((𝑁 ∘ 𝑃)‘0) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
41	36	simprd 495	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))
42		eqeq2 2749	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → (((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
43	42	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → (¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥 ↔ ¬ ((𝑁 ∘ 𝑃)‘𝑦) = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))))
44	41, 43	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → (𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
45	40, 44	jaod 860	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ((𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
46	45	impancom 451	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑦 ∈ (1..^(♯‘𝐹)) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥))
47	46	imp 406	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) ∧ 𝑦 ∈ (1..^(♯‘𝐹))) → ¬ ((𝑁 ∘ 𝑃)‘𝑦) = 𝑥)
48	47	nrexdv 3132	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → ¬ ∃𝑦 ∈ (1..^(♯‘𝐹))((𝑁 ∘ 𝑃)‘𝑦) = 𝑥)
49	20	eqcomd 2743	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐹) = (♯‘𝐸))
50	49	oveq2d 7376	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...(♯‘𝐹)) = (0...(♯‘𝐸)))
51	50	feq2d 6647	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑁 ∘ 𝑃):(0...(♯‘𝐹))⟶(Vtx‘𝐻) ↔ (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻)))
52	18, 51	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐹))⟶(Vtx‘𝐻))
53	52	ffnd 6664	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐹)))
54	53	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐹)))
55		fzo0ss1 13609	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
56		fzossfz 13598	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
57	55, 56	sstri 3944	. . . . . . . . . 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 6908	. . . . . . . 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 325	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
61	60	ex 412	. . . . . 6 ⊢ (𝜑 → ((𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹))) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
62	35, 61	sylbid 240	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) → ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
63	62	ralrimiv 3128	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
64		disj 4403	. . . 4 ⊢ ((((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅ ↔ ∀𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
65	63, 64	sylibr 234	. . 3 ⊢ (𝜑 → (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))) = ∅)
66	20	oveq2d 7376	. . . . . . 7 ⊢ (𝜑 → (1..^(♯‘𝐸)) = (1..^(♯‘𝐹)))
67	66	reseq2d 5939	. . . . . 6 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
68	67	cnveqd 5825	. . . . 5 ⊢ (𝜑 → ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
69	68	funeqd 6515	. . . 4 ⊢ (𝜑 → (Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ↔ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹)))))
70		preq2 4692	. . . . . . . 8 ⊢ ((♯‘𝐸) = (♯‘𝐹) → {0, (♯‘𝐸)} = {0, (♯‘𝐹)})
71	70	imaeq2d 6020	. . . . . . 7 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) = ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}))
72		oveq2 7368	. . . . . . . 8 ⊢ ((♯‘𝐸) = (♯‘𝐹) → (1..^(♯‘𝐸)) = (1..^(♯‘𝐹)))
73	72	imaeq2d 6020	. . . . . . 7 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸))) = ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
74	71, 73	ineq12d 4174	. . . . . 6 ⊢ ((♯‘𝐸) = (♯‘𝐹) → (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
75	74	eqeq1d 2739	. . . . 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 1442	. . 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 1344	. 2 ⊢ (𝜑 → (𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅))
79		ispth 29798	. 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 234	1 ⊢ (𝜑 → 𝐸(Paths‘𝐻)(𝑁 ∘ 𝑃))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3061 ∩ cin 3901 ⊆ wss 3902 ∅c0 4286 {cpr 4583 class class class wbr 5099 ↦ cmpt 5180 ^◡ccnv 5624 dom cdm 5625 ↾ cres 5627 “ cima 5628 ∘ ccom 5629 Fun wfun 6487 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 0cc0 11030 1c1 11031 ℕ₀cn0 12405 ...cfz 13427 ..^cfzo 13574 ♯chash 14257 Word cword 14440 Vtxcvtx 29073 iEdgciedg 29074 USPGraphcuspgr 29225 Walkscwlks 29674 Trailsctrls 29766 Pathscpths 29787 GraphIso cgrim 48188

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ifp 1064 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-oadd 8403 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-dju 9817 df-card 9855 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-n0 12406 df-xnn0 12479 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-hash 14258 df-word 14441 df-edg 29125 df-uhgr 29135 df-upgr 29159 df-uspgr 29227 df-wlks 29677 df-trls 29768 df-pths 29791 df-grim 48191

This theorem is referenced by: upgrimcycls 48224

W3C validator