Theorem cats1un 13475

Description: Express a word with an extra symbol as the union of the word and the new value. (Contributed by Mario Carneiro, 28-Feb-2016.)

Assertion

Ref	Expression
cats1un	⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}))

Proof of Theorem cats1un

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatws1cl 13396	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋)
2		wrdf 13310	. . . . 5 ⊢ ((𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋 → (𝐴 ++ ⟨“𝐵”⟩):(0..^(#‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋)
3	1, 2	syl 17	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩):(0..^(#‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋)
4		ccatws1len 13398	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (#‘(𝐴 ++ ⟨“𝐵”⟩)) = ((#‘𝐴) + 1))
5	4	oveq2d 6666	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (0..^(#‘(𝐴 ++ ⟨“𝐵”⟩))) = (0..^((#‘𝐴) + 1)))
6		lencl 13324	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑋 → (#‘𝐴) ∈ ℕ0)
7	6	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (#‘𝐴) ∈ ℕ0)
8		nn0uz 11722	. . . . . . . 8 ⊢ ℕ0 = (ℤ≥‘0)
9	7, 8	syl6eleq 2711	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (#‘𝐴) ∈ (ℤ≥‘0))
10		fzosplitsn 12576	. . . . . . 7 ⊢ ((#‘𝐴) ∈ (ℤ≥‘0) → (0..^((#‘𝐴) + 1)) = ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
11	9, 10	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (0..^((#‘𝐴) + 1)) = ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
12	5, 11	eqtrd 2656	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (0..^(#‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
13	12	feq2d 6031	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩):(0..^(#‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋 ↔ (𝐴 ++ ⟨“𝐵”⟩):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶𝑋))
14	3, 13	mpbid 222	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶𝑋)
15		ffn 6045	. . 3 ⊢ ((𝐴 ++ ⟨“𝐵”⟩):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶𝑋 → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
16	14, 15	syl 17	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
17		wrdf 13310	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → 𝐴:(0..^(#‘𝐴))⟶𝑋)
18	17	adantr 481	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴:(0..^(#‘𝐴))⟶𝑋)
19		eqid 2622	. . . . . 6 ⊢ {⟨(#‘𝐴), 𝐵⟩} = {⟨(#‘𝐴), 𝐵⟩}
20		fsng 6404	. . . . . 6 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ 𝐵 ∈ 𝑋) → ({⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵} ↔ {⟨(#‘𝐴), 𝐵⟩} = {⟨(#‘𝐴), 𝐵⟩}))
21	19, 20	mpbiri 248	. . . . 5 ⊢ (((#‘𝐴) ∈ ℕ0 ∧ 𝐵 ∈ 𝑋) → {⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵})
22	6, 21	sylan 488	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → {⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵})
23		fzonel 12483	. . . . . 6 ⊢ ¬ (#‘𝐴) ∈ (0..^(#‘𝐴))
24	23	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ¬ (#‘𝐴) ∈ (0..^(#‘𝐴)))
25		disjsn 4246	. . . . 5 ⊢ (((0..^(#‘𝐴)) ∩ {(#‘𝐴)}) = ∅ ↔ ¬ (#‘𝐴) ∈ (0..^(#‘𝐴)))
26	24, 25	sylibr 224	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((0..^(#‘𝐴)) ∩ {(#‘𝐴)}) = ∅)
27		fun 6066	. . . 4 ⊢ (((𝐴:(0..^(#‘𝐴))⟶𝑋 ∧ {⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵}) ∧ ((0..^(#‘𝐴)) ∩ {(#‘𝐴)}) = ∅) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶(𝑋 ∪ {𝐵}))
28	18, 22, 26, 27	syl21anc 1325	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶(𝑋 ∪ {𝐵}))
29		ffn 6045	. . 3 ⊢ ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶(𝑋 ∪ {𝐵}) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
30	28, 29	syl 17	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
31		elun 3753	. . 3 ⊢ (𝑥 ∈ ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}) ↔ (𝑥 ∈ (0..^(#‘𝐴)) ∨ 𝑥 ∈ {(#‘𝐴)}))
32		ccats1val1 13403	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴‘𝑥))
33	32	3expa 1265	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴‘𝑥))
34		simpr 477	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → 𝑥 ∈ (0..^(#‘𝐴)))
35		nelne2 2891	. . . . . . . 8 ⊢ ((𝑥 ∈ (0..^(#‘𝐴)) ∧ ¬ (#‘𝐴) ∈ (0..^(#‘𝐴))) → 𝑥 ≠ (#‘𝐴))
36	34, 23, 35	sylancl 694	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → 𝑥 ≠ (#‘𝐴))
37	36	necomd 2849	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → (#‘𝐴) ≠ 𝑥)
38		fvunsn 6445	. . . . . 6 ⊢ ((#‘𝐴) ≠ 𝑥 → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) = (𝐴‘𝑥))
39	37, 38	syl 17	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) = (𝐴‘𝑥))
40	33, 39	eqtr4d 2659	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
41		fvexd 6203	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (#‘𝐴) ∈ V)
42		elex 3212	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ V)
43	42	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ V)
44		fdm 6051	. . . . . . . . . . 11 ⊢ (𝐴:(0..^(#‘𝐴))⟶𝑋 → dom 𝐴 = (0..^(#‘𝐴)))
45	18, 44	syl 17	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → dom 𝐴 = (0..^(#‘𝐴)))
46	45	eleq2d 2687	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((#‘𝐴) ∈ dom 𝐴 ↔ (#‘𝐴) ∈ (0..^(#‘𝐴))))
47	23, 46	mtbiri 317	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ¬ (#‘𝐴) ∈ dom 𝐴)
48		fsnunfv 6453	. . . . . . . 8 ⊢ (((#‘𝐴) ∈ V ∧ 𝐵 ∈ V ∧ ¬ (#‘𝐴) ∈ dom 𝐴) → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)) = 𝐵)
49	41, 43, 47, 48	syl3anc 1326	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)) = 𝐵)
50		simpl 473	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ Word 𝑋)
51		s1cl 13382	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → ⟨“𝐵”⟩ ∈ Word 𝑋)
52	51	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨“𝐵”⟩ ∈ Word 𝑋)
53		s1len 13385	. . . . . . . . . . . 12 ⊢ (#‘⟨“𝐵”⟩) = 1
54		1nn 11031	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
55	53, 54	eqeltri 2697	. . . . . . . . . . 11 ⊢ (#‘⟨“𝐵”⟩) ∈ ℕ
56	55	a1i 11	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (#‘⟨“𝐵”⟩) ∈ ℕ)
57		lbfzo0 12507	. . . . . . . . . 10 ⊢ (0 ∈ (0..^(#‘⟨“𝐵”⟩)) ↔ (#‘⟨“𝐵”⟩) ∈ ℕ)
58	56, 57	sylibr 224	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ∈ (0..^(#‘⟨“𝐵”⟩)))
59		ccatval3 13363	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ ⟨“𝐵”⟩ ∈ Word 𝑋 ∧ 0 ∈ (0..^(#‘⟨“𝐵”⟩))) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = (⟨“𝐵”⟩‘0))
60	50, 52, 58, 59	syl3anc 1326	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = (⟨“𝐵”⟩‘0))
61		s1fv 13390	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑋 → (⟨“𝐵”⟩‘0) = 𝐵)
62	61	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (⟨“𝐵”⟩‘0) = 𝐵)
63	60, 62	eqtrd 2656	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = 𝐵)
64	7	nn0cnd 11353	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (#‘𝐴) ∈ ℂ)
65	64	addid2d 10237	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (0 + (#‘𝐴)) = (#‘𝐴))
66	65	fveq2d 6195	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)))
67	49, 63, 66	3eqtr2rd 2663	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)))
68		elsni 4194	. . . . . . . 8 ⊢ (𝑥 ∈ {(#‘𝐴)} → 𝑥 = (#‘𝐴))
69	68	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 ∈ {(#‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)))
70	68	fveq2d 6195	. . . . . . 7 ⊢ (𝑥 ∈ {(#‘𝐴)} → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)))
71	69, 70	eqeq12d 2637	. . . . . 6 ⊢ (𝑥 ∈ {(#‘𝐴)} → (((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) ↔ ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴))))
72	67, 71	syl5ibrcom 237	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑥 ∈ {(#‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥)))
73	72	imp 445	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ {(#‘𝐴)}) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
74	40, 73	jaodan 826	. . 3 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑥 ∈ (0..^(#‘𝐴)) ∨ 𝑥 ∈ {(#‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
75	31, 74	sylan2b 492	. 2 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ ((0..^(#‘𝐴)) ∪ {(#‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
76	16, 30, 75	eqfnfvd 6314	1 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}))

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∪ cun 3572   ∩ cin 3573  ∅c0 3915  {csn 4177  ⟨cop 4183  dom cdm 5114   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  0cc0 9936  1c1 9937   + caddc 9939  ℕcn 11020  ℕ₀cn0 11292  ℤ_≥cuz 11687  ..^cfzo 12465  #chash 13117  Word cword 13291   ++ cconcat 13293  ⟨“cs1 13294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302

This theorem is referenced by:  s2prop  13652  s3tpop  13654  s4prop  13655  pgpfaclem1  18480  vdegp1ai  26432  vdegp1bi  26433  wwlksnext  26788