chnrev - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > chnrev		Structured version Visualization version GIF version

Theorem chnrev 18719

Description: Reverse of a chain is chain under the converse relation and same domain. (Contributed by Ender Ting, 20-Jan-2026.)

**Assertion**
Ref	Expression
chnrev	⊢ (𝐵 ∈ ( < Chain 𝐴) → (reverse‘𝐵) ∈ (^◡ < Chain 𝐴))

Proof of Theorem chnrev Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . . 4 ⊢ (𝐵 ∈ ( < Chain 𝐴) → 𝐵 ∈ ( < Chain 𝐴))
2	1	chnwrd 18700	. . 3 ⊢ (𝐵 ∈ ( < Chain 𝐴) → 𝐵 ∈ Word 𝐴)
3		revcl 14832	. . 3 ⊢ (𝐵 ∈ Word 𝐴 → (reverse‘𝐵) ∈ Word 𝐴)
4	2, 3	syl 18	. 2 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (reverse‘𝐵) ∈ Word 𝐴)
5		simpl 488	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝐵 ∈ ( < Chain 𝐴))
6		fzossfz 13736	. . . . . . . . . . . . . . 15 ⊢ (0..^(♯‘(reverse‘𝐵))) ⊆ (0...(♯‘(reverse‘𝐵)))
7	6	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0..^(♯‘(reverse‘𝐵))) ⊆ (0...(♯‘(reverse‘𝐵))))
8		wrddm 14588	. . . . . . . . . . . . . . 15 ⊢ ((reverse‘𝐵) ∈ Word 𝐴 → dom (reverse‘𝐵) = (0..^(♯‘(reverse‘𝐵))))
9	4, 8	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) = (0..^(♯‘(reverse‘𝐵))))
10		revlen 14833	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ Word 𝐴 → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
11	2, 10	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
12	11	eqcomd 2768	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (♯‘𝐵) = (♯‘(reverse‘𝐵)))
13	12	oveq2d 7432	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0...(♯‘𝐵)) = (0...(♯‘(reverse‘𝐵))))
14	7, 9, 13	3sstr4d 3989	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) ⊆ (0...(♯‘𝐵)))
15	14	ssdifd 4095	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (dom (reverse‘𝐵) ∖ {0}) ⊆ ((0...(♯‘𝐵)) ∖ {0}))
16	15	sselda 3934	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ ((0...(♯‘𝐵)) ∖ {0}))
17	2	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝐵 ∈ Word 𝐴)
18		lencl 14600	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Word 𝐴 → (♯‘𝐵) ∈ ℕ₀)
19	17, 18	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∈ ℕ₀)
20		fz0dif1 13663	. . . . . . . . . . . 12 ⊢ ((♯‘𝐵) ∈ ℕ₀ → ((0...(♯‘𝐵)) ∖ {0}) = (1...(♯‘𝐵)))
21	19, 20	syl 18	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((0...(♯‘𝐵)) ∖ {0}) = (1...(♯‘𝐵)))
22	16, 21	eleqtrd 2864	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (1...(♯‘𝐵)))
23		ubmelfzo 13788	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(♯‘𝐵)) → ((♯‘𝐵) − 𝑛) ∈ (0..^(♯‘𝐵)))
24	22, 23	syl 18	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ (0..^(♯‘𝐵)))
25		wrddm 14588	. . . . . . . . . 10 ⊢ (𝐵 ∈ Word 𝐴 → dom 𝐵 = (0..^(♯‘𝐵)))
26	17, 25	syl 18	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → dom 𝐵 = (0..^(♯‘𝐵)))
27	24, 26	eleqtrrd 2865	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ dom 𝐵)
28	19	nn0cnd 12594	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∈ ℂ)
29		eldifi 4081	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (dom (reverse‘𝐵) ∖ {0}) → 𝑛 ∈ dom (reverse‘𝐵))
30	29	anim2i 629	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ dom (reverse‘𝐵)))
31	2, 3, 8	3syl 19	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) = (0..^(♯‘(reverse‘𝐵))))
32	31	eleq2d 2848	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (𝑛 ∈ dom (reverse‘𝐵) ↔ 𝑛 ∈ (0..^(♯‘(reverse‘𝐵)))))
33	32	biimpa 482	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ dom (reverse‘𝐵)) → 𝑛 ∈ (0..^(♯‘(reverse‘𝐵))))
34		elfzoelz 13716	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0..^(♯‘(reverse‘𝐵))) → 𝑛 ∈ ℤ)
35	33, 34	syl 18	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ dom (reverse‘𝐵)) → 𝑛 ∈ ℤ)
36		zcn 12623	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
37	30, 35, 36	3syl 19	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ ℂ)
38	29	adantl 487	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ dom (reverse‘𝐵))
39	17, 3	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (reverse‘𝐵) ∈ Word 𝐴)
40		wrdlndm 14597	. . . . . . . . . . . . 13 ⊢ ((reverse‘𝐵) ∈ Word 𝐴 → (♯‘(reverse‘𝐵)) ∉ dom (reverse‘𝐵))
41	39, 40	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘(reverse‘𝐵)) ∉ dom (reverse‘𝐵))
42	17, 10	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
43		eqidd 2763	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → dom (reverse‘𝐵) = dom (reverse‘𝐵))
44	42, 43	neleq12d 3068	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘(reverse‘𝐵)) ∉ dom (reverse‘𝐵) ↔ (♯‘𝐵) ∉ dom (reverse‘𝐵)))
45	41, 44	mpbid 235	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∉ dom (reverse‘𝐵))
46		elnelne2 3075	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ dom (reverse‘𝐵) ∧ (♯‘𝐵) ∉ dom (reverse‘𝐵)) → 𝑛 ≠ (♯‘𝐵))
47	38, 45, 46	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ≠ (♯‘𝐵))
48	47	necomd 3012	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ≠ 𝑛)
49	28, 37, 48	subne0d 11605	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ≠ 0)
50	27, 49	eldifsnd 4753	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ (dom 𝐵 ∖ {0}))
51	5, 50	chnltm1 18701	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 𝑛) − 1)) < (𝐵‘((♯‘𝐵) − 𝑛)))
52		1cnd 11229	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 1 ∈ ℂ)
53	28, 52, 37	sub32d 11628	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (((♯‘𝐵) − 1) − 𝑛) = (((♯‘𝐵) − 𝑛) − 1))
54	53	fveq2d 6886	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) = (𝐵‘(((♯‘𝐵) − 𝑛) − 1)))
55	28, 37, 52	nnncan2d 11631	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (((♯‘𝐵) − 1) − (𝑛 − 1)) = ((♯‘𝐵) − 𝑛))
56	55	fveq2d 6886	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))) = (𝐵‘((♯‘𝐵) − 𝑛)))
57	51, 54, 56	3brtr4d 5141	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) < (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
58		fvex 6895	. . . . . 6 ⊢ (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))) ∈ V
59		fvex 6895	. . . . . 6 ⊢ (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) ∈ V
60	58, 59	brcnv 5866	. . . . 5 ⊢ ((𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1)))^◡ < (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) ↔ (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) < (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
61	57, 60	sylibr 237	. . . 4 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1)))^◡ < (𝐵‘(((♯‘𝐵) − 1) − 𝑛)))
62	39, 8	syl 18	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → dom (reverse‘𝐵) = (0..^(♯‘(reverse‘𝐵))))
63	38, 62	eleqtrd 2864	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (0..^(♯‘(reverse‘𝐵))))
64		elfzonn0 13765	. . . . . . . . 9 ⊢ (𝑛 ∈ (0..^(♯‘(reverse‘𝐵))) → 𝑛 ∈ ℕ₀)
65	63, 64	syl 18	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ ℕ₀)
66		eldifsni 4756	. . . . . . . . 9 ⊢ (𝑛 ∈ (dom (reverse‘𝐵) ∖ {0}) → 𝑛 ≠ 0)
67	66	adantl 487	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ≠ 0)
68		elnnne0 12545	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ₀ ∧ 𝑛 ≠ 0))
69	65, 67, 68	sylanbrc 595	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ ℕ)
70		nnm1nn0 12572	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
71	69, 70	syl 18	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝑛 − 1) ∈ ℕ₀)
72		elfzo0le 13761	. . . . . . . 8 ⊢ (𝑛 ∈ (0..^(♯‘(reverse‘𝐵))) → 𝑛 ≤ (♯‘(reverse‘𝐵)))
73	63, 72	syl 18	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ≤ (♯‘(reverse‘𝐵)))
74	37, 52	npcand 11600	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((𝑛 − 1) + 1) = 𝑛)
75	42	eqcomd 2768	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) = (♯‘(reverse‘𝐵)))
76	73, 74, 75	3brtr4d 5141	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((𝑛 − 1) + 1) ≤ (♯‘𝐵))
77		nn0p1elfzo 13760	. . . . . 6 ⊢ (((𝑛 − 1) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀ ∧ ((𝑛 − 1) + 1) ≤ (♯‘𝐵)) → (𝑛 − 1) ∈ (0..^(♯‘𝐵)))
78	71, 19, 76, 77	syl3anc 1398	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝑛 − 1) ∈ (0..^(♯‘𝐵)))
79		revfv 14834	. . . . 5 ⊢ ((𝐵 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ (0..^(♯‘𝐵))) → ((reverse‘𝐵)‘(𝑛 − 1)) = (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
80	17, 78, 79	syl2anc 596	. . . 4 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘(𝑛 − 1)) = (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
81	11	oveq2d 7432	. . . . . . . . 9 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0..^(♯‘(reverse‘𝐵))) = (0..^(♯‘𝐵)))
82	31, 81	eqtrd 2797	. . . . . . . 8 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) = (0..^(♯‘𝐵)))
83	82	eleq2d 2848	. . . . . . 7 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (𝑛 ∈ dom (reverse‘𝐵) ↔ 𝑛 ∈ (0..^(♯‘𝐵))))
84	29, 83	imbitrid 247	. . . . . 6 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (𝑛 ∈ (dom (reverse‘𝐵) ∖ {0}) → 𝑛 ∈ (0..^(♯‘𝐵))))
85	84	imp 412	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (0..^(♯‘𝐵)))
86		revfv 14834	. . . . 5 ⊢ ((𝐵 ∈ Word 𝐴 ∧ 𝑛 ∈ (0..^(♯‘𝐵))) → ((reverse‘𝐵)‘𝑛) = (𝐵‘(((♯‘𝐵) − 1) − 𝑛)))
87	17, 85, 86	syl2anc 596	. . . 4 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘𝑛) = (𝐵‘(((♯‘𝐵) − 1) − 𝑛)))
88	61, 80, 87	3brtr4d 5141	. . 3 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘(𝑛 − 1))^◡ < ((reverse‘𝐵)‘𝑛))
89	88	ralrimiva 3156	. 2 ⊢ (𝐵 ∈ ( < Chain 𝐴) → ∀𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})((reverse‘𝐵)‘(𝑛 − 1))^◡ < ((reverse‘𝐵)‘𝑛))
90		ischn 18699	. 2 ⊢ ((reverse‘𝐵) ∈ (^◡ < Chain 𝐴) ↔ ((reverse‘𝐵) ∈ Word 𝐴 ∧ ∀𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})((reverse‘𝐵)‘(𝑛 − 1))^◡ < ((reverse‘𝐵)‘𝑛)))
91	4, 89, 90	sylanbrc 595	1 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (reverse‘𝐵) ∈ (^◡ < Chain 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ∉ wnel 3063 ∀wral 3078 ∖ cdif 3899 ⊆ wss 3902 {csn 4587 class class class wbr 5107 ^◡ccnv 5658 dom cdm 5659 ‘cfv 6537 (class class class)co 7416 ℂcc 11125 0cc0 11127 1c1 11128 + caddc 11130 ≤ cle 11271 − cmin 11468 ℕcn 12260 ℕ₀cn0 12531 ℤcz 12618 ...cfz 13563 ..^cfzo 13711 ♯chash 14396 Word cword 14580 reversecreverse 14829 Chain cchn 18697

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-card 9947 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-n0 12532 df-z 12619 df-uz 12891 df-fz 13564 df-fzo 13712 df-hash 14397 df-word 14581 df-reverse 14830 df-chn 18698

This theorem is used by: (None)

W3C validator