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Theorem chnrev 18669

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 22	. . . 4 ⊢ (𝐵 ∈ ( < Chain 𝐴) → 𝐵 ∈ ( < Chain 𝐴))
2	1	chnwrd 18650	. . 3 ⊢ (𝐵 ∈ ( < Chain 𝐴) → 𝐵 ∈ Word 𝐴)
3		revcl 14784	. . 3 ⊢ (𝐵 ∈ Word 𝐴 → (reverse‘𝐵) ∈ Word 𝐴)
4	2, 3	syl 17	. 2 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (reverse‘𝐵) ∈ Word 𝐴)
5		simpl 486	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝐵 ∈ ( < Chain 𝐴))
6		fzossfz 13694	. . . . . . . . . . . . . . 15 ⊢ (0..^(♯‘(reverse‘𝐵))) ⊆ (0...(♯‘(reverse‘𝐵)))
7	6	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0..^(♯‘(reverse‘𝐵))) ⊆ (0...(♯‘(reverse‘𝐵))))
8		wrddm 14544	. . . . . . . . . . . . . . 15 ⊢ ((reverse‘𝐵) ∈ Word 𝐴 → dom (reverse‘𝐵) = (0..^(♯‘(reverse‘𝐵))))
9	4, 8	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) = (0..^(♯‘(reverse‘𝐵))))
10		revlen 14785	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ Word 𝐴 → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
11	2, 10	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
12	11	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (♯‘𝐵) = (♯‘(reverse‘𝐵)))
13	12	oveq2d 7412	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0...(♯‘𝐵)) = (0...(♯‘(reverse‘𝐵))))
14	7, 9, 13	3sstr4d 3992	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) ⊆ (0...(♯‘𝐵)))
15	14	ssdifd 4099	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (dom (reverse‘𝐵) ∖ {0}) ⊆ ((0...(♯‘𝐵)) ∖ {0}))
16	15	sselda 3937	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ ((0...(♯‘𝐵)) ∖ {0}))
17	2	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝐵 ∈ Word 𝐴)
18		lencl 14556	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Word 𝐴 → (♯‘𝐵) ∈ ℕ₀)
19	17, 18	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∈ ℕ₀)
20		fz0dif1 13621	. . . . . . . . . . . 12 ⊢ ((♯‘𝐵) ∈ ℕ₀ → ((0...(♯‘𝐵)) ∖ {0}) = (1...(♯‘𝐵)))
21	19, 20	syl 17	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((0...(♯‘𝐵)) ∖ {0}) = (1...(♯‘𝐵)))
22	16, 21	eleqtrd 2865	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (1...(♯‘𝐵)))
23		ubmelfzo 13746	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(♯‘𝐵)) → ((♯‘𝐵) − 𝑛) ∈ (0..^(♯‘𝐵)))
24	22, 23	syl 17	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ (0..^(♯‘𝐵)))
25		wrddm 14544	. . . . . . . . . 10 ⊢ (𝐵 ∈ Word 𝐴 → dom 𝐵 = (0..^(♯‘𝐵)))
26	17, 25	syl 17	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → dom 𝐵 = (0..^(♯‘𝐵)))
27	24, 26	eleqtrrd 2866	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ dom 𝐵)
28	19	nn0cnd 12554	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∈ ℂ)
29		eldifi 4085	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (dom (reverse‘𝐵) ∖ {0}) → 𝑛 ∈ dom (reverse‘𝐵))
30	29	anim2i 626	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ dom (reverse‘𝐵)))
31	2, 3, 8	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) = (0..^(♯‘(reverse‘𝐵))))
32	31	eleq2d 2849	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (𝑛 ∈ dom (reverse‘𝐵) ↔ 𝑛 ∈ (0..^(♯‘(reverse‘𝐵)))))
33	32	biimpa 480	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ dom (reverse‘𝐵)) → 𝑛 ∈ (0..^(♯‘(reverse‘𝐵))))
34		elfzoelz 13674	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0..^(♯‘(reverse‘𝐵))) → 𝑛 ∈ ℤ)
35	33, 34	syl 17	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ dom (reverse‘𝐵)) → 𝑛 ∈ ℤ)
36		zcn 12583	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ)
37	30, 35, 36	3syl 18	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ ℂ)
38	29	adantl 485	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ dom (reverse‘𝐵))
39	17, 3	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (reverse‘𝐵) ∈ Word 𝐴)
40		wrdlndm 14553	. . . . . . . . . . . . 13 ⊢ ((reverse‘𝐵) ∈ Word 𝐴 → (♯‘(reverse‘𝐵)) ∉ dom (reverse‘𝐵))
41	39, 40	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘(reverse‘𝐵)) ∉ dom (reverse‘𝐵))
42	17, 10	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
43		eqidd 2764	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → dom (reverse‘𝐵) = dom (reverse‘𝐵))
44	42, 43	neleq12d 3067	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘(reverse‘𝐵)) ∉ dom (reverse‘𝐵) ↔ (♯‘𝐵) ∉ dom (reverse‘𝐵)))
45	41, 44	mpbid 234	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∉ dom (reverse‘𝐵))
46		elnelne2 3074	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ dom (reverse‘𝐵) ∧ (♯‘𝐵) ∉ dom (reverse‘𝐵)) → 𝑛 ≠ (♯‘𝐵))
47	38, 45, 46	syl2anc 593	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ≠ (♯‘𝐵))
48	47	necomd 3013	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ≠ 𝑛)
49	28, 37, 48	subne0d 11562	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ≠ 0)
50	27, 49	eldifsnd 4748	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ (dom 𝐵 ∖ {0}))
51	5, 50	chnltm1 18651	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 𝑛) − 1)) < (𝐵‘((♯‘𝐵) − 𝑛)))
52		1cnd 11186	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 1 ∈ ℂ)
53	28, 52, 37	sub32d 11585	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (((♯‘𝐵) − 1) − 𝑛) = (((♯‘𝐵) − 𝑛) − 1))
54	53	fveq2d 6871	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) = (𝐵‘(((♯‘𝐵) − 𝑛) − 1)))
55	28, 37, 52	nnncan2d 11588	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (((♯‘𝐵) − 1) − (𝑛 − 1)) = ((♯‘𝐵) − 𝑛))
56	55	fveq2d 6871	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))) = (𝐵‘((♯‘𝐵) − 𝑛)))
57	51, 54, 56	3brtr4d 5133	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) < (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
58		fvex 6880	. . . . . 6 ⊢ (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))) ∈ V
59		fvex 6880	. . . . . 6 ⊢ (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) ∈ V
60	58, 59	brcnv 5855	. . . . 5 ⊢ ((𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1)))^◡ < (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) ↔ (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) < (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
61	57, 60	sylibr 236	. . . 4 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1)))^◡ < (𝐵‘(((♯‘𝐵) − 1) − 𝑛)))
62	39, 8	syl 17	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → dom (reverse‘𝐵) = (0..^(♯‘(reverse‘𝐵))))
63	38, 62	eleqtrd 2865	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (0..^(♯‘(reverse‘𝐵))))
64		elfzonn0 13723	. . . . . . . . 9 ⊢ (𝑛 ∈ (0..^(♯‘(reverse‘𝐵))) → 𝑛 ∈ ℕ₀)
65	63, 64	syl 17	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ ℕ₀)
66		eldifsni 4751	. . . . . . . . 9 ⊢ (𝑛 ∈ (dom (reverse‘𝐵) ∖ {0}) → 𝑛 ≠ 0)
67	66	adantl 485	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ≠ 0)
68		elnnne0 12505	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ₀ ∧ 𝑛 ≠ 0))
69	65, 67, 68	sylanbrc 592	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ ℕ)
70		nnm1nn0 12532	. . . . . . 7 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
71	69, 70	syl 17	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝑛 − 1) ∈ ℕ₀)
72		elfzo0le 13719	. . . . . . . 8 ⊢ (𝑛 ∈ (0..^(♯‘(reverse‘𝐵))) → 𝑛 ≤ (♯‘(reverse‘𝐵)))
73	63, 72	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ≤ (♯‘(reverse‘𝐵)))
74	37, 52	npcand 11557	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((𝑛 − 1) + 1) = 𝑛)
75	42	eqcomd 2769	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) = (♯‘(reverse‘𝐵)))
76	73, 74, 75	3brtr4d 5133	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((𝑛 − 1) + 1) ≤ (♯‘𝐵))
77		nn0p1elfzo 13718	. . . . . 6 ⊢ (((𝑛 − 1) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀ ∧ ((𝑛 − 1) + 1) ≤ (♯‘𝐵)) → (𝑛 − 1) ∈ (0..^(♯‘𝐵)))
78	71, 19, 76, 77	syl3anc 1392	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝑛 − 1) ∈ (0..^(♯‘𝐵)))
79		revfv 14786	. . . . 5 ⊢ ((𝐵 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ (0..^(♯‘𝐵))) → ((reverse‘𝐵)‘(𝑛 − 1)) = (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
80	17, 78, 79	syl2anc 593	. . . 4 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘(𝑛 − 1)) = (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
81	11	oveq2d 7412	. . . . . . . . 9 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0..^(♯‘(reverse‘𝐵))) = (0..^(♯‘𝐵)))
82	31, 81	eqtrd 2798	. . . . . . . 8 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) = (0..^(♯‘𝐵)))
83	82	eleq2d 2849	. . . . . . 7 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (𝑛 ∈ dom (reverse‘𝐵) ↔ 𝑛 ∈ (0..^(♯‘𝐵))))
84	29, 83	imbitrid 246	. . . . . 6 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (𝑛 ∈ (dom (reverse‘𝐵) ∖ {0}) → 𝑛 ∈ (0..^(♯‘𝐵))))
85	84	imp 410	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (0..^(♯‘𝐵)))
86		revfv 14786	. . . . 5 ⊢ ((𝐵 ∈ Word 𝐴 ∧ 𝑛 ∈ (0..^(♯‘𝐵))) → ((reverse‘𝐵)‘𝑛) = (𝐵‘(((♯‘𝐵) − 1) − 𝑛)))
87	17, 85, 86	syl2anc 593	. . . 4 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘𝑛) = (𝐵‘(((♯‘𝐵) − 1) − 𝑛)))
88	61, 80, 87	3brtr4d 5133	. . 3 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘(𝑛 − 1))^◡ < ((reverse‘𝐵)‘𝑛))
89	88	ralrimiva 3155	. 2 ⊢ (𝐵 ∈ ( < Chain 𝐴) → ∀𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})((reverse‘𝐵)‘(𝑛 − 1))^◡ < ((reverse‘𝐵)‘𝑛))
90		ischn 18649	. 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 592	1 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (reverse‘𝐵) ∈ (^◡ < Chain 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 ∉ wnel 3062 ∀wral 3077 ∖ cdif 3902 ⊆ wss 3905 {csn 4583 class class class wbr 5101 ^◡ccnv 5647 dom cdm 5648 ‘cfv 6521 (class class class)co 7396 ℂcc 11082 0cc0 11084 1c1 11085 + caddc 11087 ≤ cle 11228 − cmin 11425 ℕcn 12220 ℕ₀cn0 12491 ℤcz 12578 ...cfz 13522 ..^cfzo 13669 ♯chash 14353 Word cword 14536 reversecreverse 14781 Chain cchn 18647

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7847 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-er 8678 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-card 9909 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-nn 12221 df-n0 12492 df-z 12579 df-uz 12850 df-fz 13523 df-fzo 13670 df-hash 14354 df-word 14537 df-reverse 14782 df-chn 18648

This theorem is referenced by: (None)

W3C validator