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 18554

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 18535	. . 3 ⊢ (𝐵 ∈ ( < Chain 𝐴) → 𝐵 ∈ Word 𝐴)
3		revcl 14688	. . 3 ⊢ (𝐵 ∈ Word 𝐴 → (reverse‘𝐵) ∈ Word 𝐴)
4	2, 3	syl 17	. 2 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (reverse‘𝐵) ∈ Word 𝐴)
5		simpl 482	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝐵 ∈ ( < Chain 𝐴))
6		fzossfz 13598	. . . . . . . . . . . . . . 15 ⊢ (0..^(♯‘(reverse‘𝐵))) ⊆ (0...(♯‘(reverse‘𝐵)))
7	6	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0..^(♯‘(reverse‘𝐵))) ⊆ (0...(♯‘(reverse‘𝐵))))
8		wrddm 14448	. . . . . . . . . . . . . . 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 14689	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ∈ Word 𝐴 → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
11	2, 10	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (♯‘(reverse‘𝐵)) = (♯‘𝐵))
12	11	eqcomd 2743	. . . . . . . . . . . . . . 15 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (♯‘𝐵) = (♯‘(reverse‘𝐵)))
13	12	oveq2d 7376	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0...(♯‘𝐵)) = (0...(♯‘(reverse‘𝐵))))
14	7, 9, 13	3sstr4d 3990	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) ⊆ (0...(♯‘𝐵)))
15	14	ssdifd 4098	. . . . . . . . . . . 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 480	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝐵 ∈ Word 𝐴)
18		lencl 14460	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ Word 𝐴 → (♯‘𝐵) ∈ ℕ₀)
19	17, 18	syl 17	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∈ ℕ₀)
20		fz0dif1 13526	. . . . . . . . . . . 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 2839	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (1...(♯‘𝐵)))
23		ubmelfzo 13650	. . . . . . . . . 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 14448	. . . . . . . . . 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 2840	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ dom 𝐵)
28	19	nn0cnd 12468	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∈ ℂ)
29		eldifi 4084	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (dom (reverse‘𝐵) ∖ {0}) → 𝑛 ∈ dom (reverse‘𝐵))
30	29	anim2i 618	. . . . . . . . . 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 2823	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (𝑛 ∈ dom (reverse‘𝐵) ↔ 𝑛 ∈ (0..^(♯‘(reverse‘𝐵)))))
33	32	biimpa 476	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ dom (reverse‘𝐵)) → 𝑛 ∈ (0..^(♯‘(reverse‘𝐵))))
34		elfzoelz 13579	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (0..^(♯‘(reverse‘𝐵))) → 𝑛 ∈ ℤ)
35	33, 34	syl 17	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ dom (reverse‘𝐵)) → 𝑛 ∈ ℤ)
36		zcn 12497	. . . . . . . . . 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 481	. . . . . . . . . . 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 14457	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → dom (reverse‘𝐵) = dom (reverse‘𝐵))
44	42, 43	neleq12d 3042	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘(reverse‘𝐵)) ∉ dom (reverse‘𝐵) ↔ (♯‘𝐵) ∉ dom (reverse‘𝐵)))
45	41, 44	mpbid 232	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ∉ dom (reverse‘𝐵))
46		elnelne2 3049	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ dom (reverse‘𝐵) ∧ (♯‘𝐵) ∉ dom (reverse‘𝐵)) → 𝑛 ≠ (♯‘𝐵))
47	38, 45, 46	syl2anc 585	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ≠ (♯‘𝐵))
48	47	necomd 2988	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) ≠ 𝑛)
49	28, 37, 48	subne0d 11505	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ≠ 0)
50	27, 49	eldifsnd 4744	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((♯‘𝐵) − 𝑛) ∈ (dom 𝐵 ∖ {0}))
51	5, 50	chnltm1 18536	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 𝑛) − 1)) < (𝐵‘((♯‘𝐵) − 𝑛)))
52		1cnd 11131	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 1 ∈ ℂ)
53	28, 52, 37	sub32d 11528	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (((♯‘𝐵) − 1) − 𝑛) = (((♯‘𝐵) − 𝑛) − 1))
54	53	fveq2d 6839	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) = (𝐵‘(((♯‘𝐵) − 𝑛) − 1)))
55	28, 37, 52	nnncan2d 11531	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (((♯‘𝐵) − 1) − (𝑛 − 1)) = ((♯‘𝐵) − 𝑛))
56	55	fveq2d 6839	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))) = (𝐵‘((♯‘𝐵) − 𝑛)))
57	51, 54, 56	3brtr4d 5131	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) < (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
58		fvex 6848	. . . . . 6 ⊢ (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))) ∈ V
59		fvex 6848	. . . . . 6 ⊢ (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) ∈ V
60	58, 59	brcnv 5832	. . . . 5 ⊢ ((𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1)))^◡ < (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) ↔ (𝐵‘(((♯‘𝐵) − 1) − 𝑛)) < (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
61	57, 60	sylibr 234	. . . 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 2839	. . . . . . . . 9 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (0..^(♯‘(reverse‘𝐵))))
64		elfzonn0 13627	. . . . . . . . 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 4747	. . . . . . . . 9 ⊢ (𝑛 ∈ (dom (reverse‘𝐵) ∖ {0}) → 𝑛 ≠ 0)
67	66	adantl 481	. . . . . . . 8 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ≠ 0)
68		elnnne0 12419	. . . . . . . 8 ⊢ (𝑛 ∈ ℕ ↔ (𝑛 ∈ ℕ₀ ∧ 𝑛 ≠ 0))
69	65, 67, 68	sylanbrc 584	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ ℕ)
70		nnm1nn0 12446	. . . . . . 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 13623	. . . . . . . 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 11500	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((𝑛 − 1) + 1) = 𝑛)
75	42	eqcomd 2743	. . . . . . 7 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (♯‘𝐵) = (♯‘(reverse‘𝐵)))
76	73, 74, 75	3brtr4d 5131	. . . . . 6 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((𝑛 − 1) + 1) ≤ (♯‘𝐵))
77		nn0p1elfzo 13622	. . . . . 6 ⊢ (((𝑛 − 1) ∈ ℕ₀ ∧ (♯‘𝐵) ∈ ℕ₀ ∧ ((𝑛 − 1) + 1) ≤ (♯‘𝐵)) → (𝑛 − 1) ∈ (0..^(♯‘𝐵)))
78	71, 19, 76, 77	syl3anc 1374	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → (𝑛 − 1) ∈ (0..^(♯‘𝐵)))
79		revfv 14690	. . . . 5 ⊢ ((𝐵 ∈ Word 𝐴 ∧ (𝑛 − 1) ∈ (0..^(♯‘𝐵))) → ((reverse‘𝐵)‘(𝑛 − 1)) = (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
80	17, 78, 79	syl2anc 585	. . . 4 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘(𝑛 − 1)) = (𝐵‘(((♯‘𝐵) − 1) − (𝑛 − 1))))
81	11	oveq2d 7376	. . . . . . . . 9 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (0..^(♯‘(reverse‘𝐵))) = (0..^(♯‘𝐵)))
82	31, 81	eqtrd 2772	. . . . . . . 8 ⊢ (𝐵 ∈ ( < Chain 𝐴) → dom (reverse‘𝐵) = (0..^(♯‘𝐵)))
83	82	eleq2d 2823	. . . . . . 7 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (𝑛 ∈ dom (reverse‘𝐵) ↔ 𝑛 ∈ (0..^(♯‘𝐵))))
84	29, 83	imbitrid 244	. . . . . 6 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (𝑛 ∈ (dom (reverse‘𝐵) ∖ {0}) → 𝑛 ∈ (0..^(♯‘𝐵))))
85	84	imp 406	. . . . 5 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → 𝑛 ∈ (0..^(♯‘𝐵)))
86		revfv 14690	. . . . 5 ⊢ ((𝐵 ∈ Word 𝐴 ∧ 𝑛 ∈ (0..^(♯‘𝐵))) → ((reverse‘𝐵)‘𝑛) = (𝐵‘(((♯‘𝐵) − 1) − 𝑛)))
87	17, 85, 86	syl2anc 585	. . . 4 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘𝑛) = (𝐵‘(((♯‘𝐵) − 1) − 𝑛)))
88	61, 80, 87	3brtr4d 5131	. . 3 ⊢ ((𝐵 ∈ ( < Chain 𝐴) ∧ 𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})) → ((reverse‘𝐵)‘(𝑛 − 1))^◡ < ((reverse‘𝐵)‘𝑛))
89	88	ralrimiva 3129	. 2 ⊢ (𝐵 ∈ ( < Chain 𝐴) → ∀𝑛 ∈ (dom (reverse‘𝐵) ∖ {0})((reverse‘𝐵)‘(𝑛 − 1))^◡ < ((reverse‘𝐵)‘𝑛))
90		ischn 18534	. 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 584	1 ⊢ (𝐵 ∈ ( < Chain 𝐴) → (reverse‘𝐵) ∈ (^◡ < Chain 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 ∉ wnel 3037 ∀wral 3052 ∖ cdif 3899 ⊆ wss 3902 {csn 4581 class class class wbr 5099 ^◡ccnv 5624 dom cdm 5625 ‘cfv 6493 (class class class)co 7360 ℂcc 11028 0cc0 11030 1c1 11031 + caddc 11033 ≤ cle 11171 − cmin 11368 ℕcn 12149 ℕ₀cn0 12405 ℤcz 12492 ...cfz 13427 ..^cfzo 13574 ♯chash 14257 Word cword 14440 reversecreverse 14685 Chain cchn 18532

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-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-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 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-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-fzo 13575 df-hash 14258 df-word 14441 df-reverse 14686 df-chn 18533

This theorem is referenced by: (None)

W3C validator