upgrimpths - Mathbox for Alexander van der Vekens

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Theorem upgrimpths 47902

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 29703	. . . . 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 47899	. . 3 ⊢ (𝜑 → 𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃))
11	1, 2, 3, 4, 5, 6, 7	upgrimpthslem1 47900	. . 3 ⊢ (𝜑 → Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
12		pthiswlk 29705	. . . . . . . . . . . 12 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
13	1	wlkf 29595	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼)
14	7, 12, 13	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼)
15		eqid 2729	. . . . . . . . . . . . 13 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
16	15	wlkp 29597	. . . . . . . . . . . 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 47893	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 ∘ 𝑃):(0...(♯‘𝐸))⟶(Vtx‘𝐻))
19	18	ffnd 6671	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 ∘ 𝑃) Fn (0...(♯‘𝐸)))
20	1, 2, 3, 4, 5, 6, 14	upgrimwlklem1 47890	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐸) = (♯‘𝐹))
21		wlkcl 29596	. . . . . . . . . . . 12 ⊢ (𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ₀)
22	7, 12, 21	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘𝐹) ∈ ℕ₀)
23	20, 22	eqeltrd 2828	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐸) ∈ ℕ₀)
24		0elfz 13561	. . . . . . . . . 10 ⊢ ((♯‘𝐸) ∈ ℕ₀ → 0 ∈ (0...(♯‘𝐸)))
25	23, 24	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ (0...(♯‘𝐸)))
26		nn0fz0 13562	. . . . . . . . . . 11 ⊢ ((♯‘𝐹) ∈ ℕ₀ ↔ (♯‘𝐹) ∈ (0...(♯‘𝐹)))
27	22, 26	sylib 218	. . . . . . . . . 10 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐹)))
28	20	oveq2d 7385	. . . . . . . . . 10 ⊢ (𝜑 → (0...(♯‘𝐸)) = (0...(♯‘𝐹)))
29	27, 28	eleqtrrd 2831	. . . . . . . . 9 ⊢ (𝜑 → (♯‘𝐹) ∈ (0...(♯‘𝐸)))
30		fnimapr 6926	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) = {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))})
32	31	eleq2d 2814	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ↔ 𝑥 ∈ {((𝑁 ∘ 𝑃)‘0), ((𝑁 ∘ 𝑃)‘(♯‘𝐹))}))
33		vex 3448	. . . . . . . 8 ⊢ 𝑥 ∈ V
34	33	elpr 4610	. . . . . . 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 47901	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . 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 859	. . . . . . . . . . 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 3128	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = ((𝑁 ∘ 𝑃)‘0) ∨ 𝑥 = ((𝑁 ∘ 𝑃)‘(♯‘𝐹)))) → ¬ ∃𝑦 ∈ (1..^(♯‘𝐹))((𝑁 ∘ 𝑃)‘𝑦) = 𝑥)
49	20	eqcomd 2735	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (♯‘𝐹) = (♯‘𝐸))
50	49	oveq2d 7385	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...(♯‘𝐹)) = (0...(♯‘𝐸)))
51	50	feq2d 6654	. . . . . . . . . . . 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 6671	. . . . . . . . . 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 13626	. . . . . . . . . . 11 ⊢ (1..^(♯‘𝐹)) ⊆ (0..^(♯‘𝐹))
56		fzossfz 13615	. . . . . . . . . . 11 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹))
57	55, 56	sstri 3953	. . . . . . . . . 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 6916	. . . . . . . 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 3124	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ¬ 𝑥 ∈ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹))))
64		disj 4409	. . . 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 7385	. . . . . . 7 ⊢ (𝜑 → (1..^(♯‘𝐸)) = (1..^(♯‘𝐹)))
67	66	reseq2d 5939	. . . . . 6 ⊢ (𝜑 → ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
68	67	cnveqd 5829	. . . . 5 ⊢ (𝜑 → ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) = ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹))))
69	68	funeqd 6522	. . . 4 ⊢ (𝜑 → (Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ↔ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐹)))))
70		preq2 4694	. . . . . . . 8 ⊢ ((♯‘𝐸) = (♯‘𝐹) → {0, (♯‘𝐸)} = {0, (♯‘𝐹)})
71	70	imaeq2d 6020	. . . . . . 7 ⊢ ((♯‘𝐸) = (♯‘𝐹) → ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) = ((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}))
72		oveq2 7377	. . . . . . . 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 4180	. . . . . 6 ⊢ ((♯‘𝐸) = (♯‘𝐹) → (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐹)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐹)))))
75	74	eqeq1d 2731	. . . . 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 1441	. . 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 1343	. 2 ⊢ (𝜑 → (𝐸(Trails‘𝐻)(𝑁 ∘ 𝑃) ∧ Fun ^◡((𝑁 ∘ 𝑃) ↾ (1..^(♯‘𝐸))) ∧ (((𝑁 ∘ 𝑃) “ {0, (♯‘𝐸)}) ∩ ((𝑁 ∘ 𝑃) “ (1..^(♯‘𝐸)))) = ∅))
79		ispth 29701	. 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 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 {cpr 4587 class class class wbr 5102 ↦ cmpt 5183 ^◡ccnv 5630 dom cdm 5631 ↾ cres 5633 “ cima 5634 ∘ ccom 5635 Fun wfun 6493 Fn wfn 6494 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 0cc0 11044 1c1 11045 ℕ₀cn0 12418 ...cfz 13444 ..^cfzo 13591 ♯chash 14271 Word cword 14454 Vtxcvtx 28976 iEdgciedg 28977 USPGraphcuspgr 29128 Walkscwlks 29577 Trailsctrls 29669 Pathscpths 29690 GraphIso cgrim 47868

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-map 8778 df-pm 8779 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-nn 12163 df-2 12225 df-n0 12419 df-xnn0 12492 df-z 12506 df-uz 12770 df-fz 13445 df-fzo 13592 df-hash 14272 df-word 14455 df-edg 29028 df-uhgr 29038 df-upgr 29062 df-uspgr 29130 df-wlks 29580 df-trls 29671 df-pths 29694 df-grim 47871

This theorem is referenced by: upgrimcycls 47904

W3C validator