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Theorem chnrev 18716

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 18697	. . 3 ⊢ (𝐵 ∈ ( < Chain 𝐴) → 𝐵 ∈ Word 𝐴)
3		revcl 14831	. . 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 13735	. . . . . . . . . . . . . . 15 ⊢ (0..^(♯‘(reverse‘𝐵))) ⊆ (0...(♯‘(reverse‘𝐵)))
7	6	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0..^(♯‘(reverse‘𝐵))) ⊆ (0...(♯‘(reverse‘𝐵))))
8		wrddm 14587	. . . . . . . . . . . . . . 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 14832	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ Word 𝐴 → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
11	2, 10	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
12	11	eqcomd 2766	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (♯‘𝐵) = (♯‘(reverse‘𝐵)))
13	12	oveq2d 7430	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0...(♯‘𝐵)) = (0...(♯‘(reverse‘𝐵))))
14	7, 9, 13	3sstr4d 3986	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) ⊆ (0...(♯‘𝐵)))
15	14	ssdifd 4092	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (dom (reverse‘𝐵) ∖ {0}) ⊆ ((0...(♯‘𝐵)) ∖ {0}))
16	15	sselda 3931	. . . . . . . . . . 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 14599	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Word 𝐴 → (♯‘𝐵) ∈ ℕ₀)
19	17, 18	syl 18	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∈ ℕ₀)
20		fz0dif1 13662	. . . . . . . . . . . 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 2862	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (1...(♯‘𝐵)))
23		ubmelfzo 13787	. . . . . . . . . 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 14587	. . . . . . . . . 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 2863	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ dom 𝐵)
28	19	nn0cnd 12592	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∈ ℂ)
29		eldifi 4078	. . . . . . . . . . 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 2846	. . . . . . . . . . . 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 13715	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0..^(♯‘(reverse‘𝐵))) → 𝑛 ∈ ℤ)
35	33, 34	syl 18	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ dom (reverse‘𝐵)) → 𝑛 ∈ ℤ)
36		zcn 12621	. . . . . . . . . 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 14596	. . . . . . . . . . . . 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 2761	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → dom (reverse‘𝐵) = dom (reverse‘𝐵))
44	42, 43	neleq12d 3066	. . . . . . . . . . . 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 3073	. . . . . . . . . . 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 3010	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ≠ 𝑛)
49	28, 37, 48	subne0d 11604	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ≠ 0)
50	27, 49	eldifsnd 4750	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ (dom 𝐵 ∖ {0}))
51	5, 50	chnltm1 18698	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 𝑛) − 1)) < (𝐵‘((♯‘𝐵) − 𝑛)))
52		1cnd 11227	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 1 ∈ ℂ)
53	28, 52, 37	sub32d 11626	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (((♯‘𝐵) − 1) − 𝑛) = (((♯‘𝐵) − 𝑛) − 1))
54	53	fveq2d 6883	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) = (𝐵‘(((♯‘𝐵) − 𝑛) − 1)))
55	28, 37, 52	nnncan2d 11629	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (((♯‘𝐵) − 1) − (𝑛 − 1)) = ((♯‘𝐵) − 𝑛))
56	55	fveq2d 6883	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))) = (𝐵‘((♯‘𝐵) − 𝑛)))
57	51, 54, 56	3brtr4d 5137	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) < (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
58		fvex 6892	. . . . . 6 ⊢ (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))) ∈ V
59		fvex 6892	. . . . . 6 ⊢ (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) ∈ V
60	58, 59	brcnv 5862	. . . . 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 2862	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (0..^(♯‘(reverse‘𝐵))))
64		elfzonn0 13764	. . . . . . . . 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 4753	. . . . . . . . 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 12543	. . . . . . . 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 12570	. . . . . . 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 13760	. . . . . . . 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 11598	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((𝑛 − 1) + 1) = 𝑛)
75	42	eqcomd 2766	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) = (♯‘(reverse‘𝐵)))
76	73, 74, 75	3brtr4d 5137	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((𝑛 − 1) + 1) ≤ (♯‘𝐵))
77		nn0p1elfzo 13759	. . . . . 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 14833	. . . . 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 7430	. . . . . . . . 9 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0..^(♯‘(reverse‘𝐵))) = (0..^(♯‘𝐵)))
82	31, 81	eqtrd 2795	. . . . . . . 8 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) = (0..^(♯‘𝐵)))
83	82	eleq2d 2846	. . . . . . 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 14833	. . . . 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 5137	. . 3 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘(𝑛 − 1))^◡ < ((reverse‘𝐵)‘𝑛))
89	88	ralrimiva 3154	. 2 ⊢ (𝐵 ∈ ( < Chain 𝐴) → ∀𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})((reverse‘𝐵)‘(𝑛 − 1))^◡ < ((reverse‘𝐵)‘𝑛))
90		ischn 18696	. 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 2955 ∉ wnel 3061 ∀wral 3076 ∖ cdif 3896 ⊆ wss 3899 {csn 4584 class class class wbr 5103 ^◡ccnv 5654 dom cdm 5655 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 0cc0 11125 1c1 11126 + caddc 11128 ≤ cle 11269 − cmin 11466 ℕcn 12258 ℕ₀cn0 12529 ℤcz 12616 ...cfz 13562 ..^cfzo 13710 ♯chash 14395 Word cword 14579 reversecreverse 14828 Chain cchn 18694

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 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 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 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-er 8697 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-n0 12530 df-z 12617 df-uz 12889 df-fz 13563 df-fzo 13711 df-hash 14396 df-word 14580 df-reverse 14829 df-chn 18695

This theorem is used by: (None)

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