Theorem cats1un 14731

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 14627	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋)
2		wrdf 14528	. . . . 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 14631	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑋 → (♯‘(𝐴 ++ ⟨“𝐵”⟩)) = ((♯‘𝐴) + 1))
5	4	oveq2d 7408	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = (0..^((♯‘𝐴) + 1)))
6		lencl 14543	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ ℕ0)
7		nn0uz 12874	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0)
8	6, 7	eleqtrdi 2871	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ (ℤ≥‘0))
9		fzosplitsn 13779	. . . . . . . 8 ⊢ ((♯‘𝐴) ∈ (ℤ≥‘0) → (0..^((♯‘𝐴) + 1)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑋 → (0..^((♯‘𝐴) + 1)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
11	5, 10	eqtrd 2796	. . . . . 6 ⊢ (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
12	11	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
13	12	feq2d 6671	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩):(0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋 ↔ (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋))
14	3, 13	mpbid 234	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋)
15	14	ffnd 6688	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
16		wrdf 14528	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → 𝐴:(0..^(♯‘𝐴))⟶𝑋)
17	16	adantr 484	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴:(0..^(♯‘𝐴))⟶𝑋)
18		eqid 2761	. . . . . 6 ⊢ {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}
19		fsng 7115	. . . . . 6 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ 𝐵 ∈ 𝑋) → ({⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵} ↔ {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}))
20	18, 19	mpbiri 260	. . . . 5 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ 𝐵 ∈ 𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
21	6, 20	sylan 589	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
22		fzodisjsn 13700	. . . . 5 ⊢ ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅
23	22	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅)
24		fun 6722	. . . 4 ⊢ (((𝐴:(0..^(♯‘𝐴))⟶𝑋 ∧ {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵}) ∧ ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶(𝑋 ∪ {𝐵}))
25	17, 21, 23, 24	syl21anc 848	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶(𝑋 ∪ {𝐵}))
26	25	ffnd 6688	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}) Fn ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
27		elun 4106	. . 3 ⊢ (𝑥 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}) ↔ (𝑥 ∈ (0..^(♯‘𝐴)) ∨ 𝑥 ∈ {(♯‘𝐴)}))
28		ccats1val1 14637	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴‘𝑥))
29	28	adantlr 725	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴‘𝑥))
30		simpr 488	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ∈ (0..^(♯‘𝐴)))
31		fzonel 13676	. . . . . . . 8 ⊢ ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))
32		nelne2 3054	. . . . . . . 8 ⊢ ((𝑥 ∈ (0..^(♯‘𝐴)) ∧ ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
33	30, 31, 32	sylancl 595	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
34	33	necomd 3011	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → (♯‘𝐴) ≠ 𝑥)
35		fvunsn 7159	. . . . . 6 ⊢ ((♯‘𝐴) ≠ 𝑥 → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) = (𝐴‘𝑥))
36	34, 35	syl 17	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) = (𝐴‘𝑥))
37	29, 36	eqtr4d 2799	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
38		fvexd 6878	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (♯‘𝐴) ∈ V)
39		simpr 488	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
40	17	fdmd 6698	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → dom 𝐴 = (0..^(♯‘𝐴)))
41	40	eleq2d 2847	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((♯‘𝐴) ∈ dom 𝐴 ↔ (♯‘𝐴) ∈ (0..^(♯‘𝐴))))
42	31, 41	mtbiri 329	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ¬ (♯‘𝐴) ∈ dom 𝐴)
43		fsnunfv 7167	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ V ∧ 𝐵 ∈ 𝑋 ∧ ¬ (♯‘𝐴) ∈ dom 𝐴) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
44	38, 39, 42, 43	syl3anc 1389	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
45		simpl 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ Word 𝑋)
46		s1cl 14613	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → ⟨“𝐵”⟩ ∈ Word 𝑋)
47	46	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨“𝐵”⟩ ∈ Word 𝑋)
48		s1len 14617	. . . . . . . . . . . 12 ⊢ (♯‘⟨“𝐵”⟩) = 1
49		1nn 12218	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
50	48, 49	eqeltri 2857	. . . . . . . . . . 11 ⊢ (♯‘⟨“𝐵”⟩) ∈ ℕ
51		lbfzo0 13702	. . . . . . . . . . 11 ⊢ (0 ∈ (0..^(♯‘⟨“𝐵”⟩)) ↔ (♯‘⟨“𝐵”⟩) ∈ ℕ)
52	50, 51	mpbir 233	. . . . . . . . . 10 ⊢ 0 ∈ (0..^(♯‘⟨“𝐵”⟩))
53	52	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ∈ (0..^(♯‘⟨“𝐵”⟩)))
54		ccatval3 14589	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ ⟨“𝐵”⟩ ∈ Word 𝑋 ∧ 0 ∈ (0..^(♯‘⟨“𝐵”⟩))) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
55	45, 47, 53, 54	syl3anc 1389	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
56		s1fv 14621	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑋 → (⟨“𝐵”⟩‘0) = 𝐵)
57	56	adantl 485	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (⟨“𝐵”⟩‘0) = 𝐵)
58	55, 57	eqtrd 2796	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = 𝐵)
59	6	adantr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (♯‘𝐴) ∈ ℕ0)
60	59	nn0cnd 12541	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (♯‘𝐴) ∈ ℂ)
61	60	addlidd 11381	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (0 + (♯‘𝐴)) = (♯‘𝐴))
62	61	fveq2d 6867	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
63	44, 58, 62	3eqtr2rd 2803	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
64		elsni 4598	. . . . . . . 8 ⊢ (𝑥 ∈ {(♯‘𝐴)} → 𝑥 = (♯‘𝐴))
65	64	fveq2d 6867	. . . . . . 7 ⊢ (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
66	64	fveq2d 6867	. . . . . . 7 ⊢ (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
67	65, 66	eqeq12d 2777	. . . . . 6 ⊢ (𝑥 ∈ {(♯‘𝐴)} → (((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) ↔ ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴))))
68	63, 67	syl5ibrcom 249	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥)))
69	68	imp 410	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ {(♯‘𝐴)}) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
70	37, 69	jaodan 970	. . 3 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑥 ∈ (0..^(♯‘𝐴)) ∨ 𝑥 ∈ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
71	27, 70	sylan2b 603	. 2 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
72	15, 26, 71	eqfnfvd 7010	1 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1559   ∈ wcel 2141   ≠ wne 2956  Vcvv 3453   ∪ cun 3902   ∩ cin 3903  ∅c0 4285  {csn 4581  ⟨cop 4587  dom cdm 5645  ⟶wf 6513  ‘cfv 6517  (class class class)co 7392  0cc0 11070  1c1 11071   + caddc 11073  ℕcn 12207  ℕ₀cn0 12478  ℤ_≥cuz 12836  ..^cfzo 13656  ♯chash 14340  Word cword 14523   ++ cconcat 14580  ⟨“cs1 14606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-fzo 13657  df-hash 14341  df-word 14524  df-concat 14581  df-s1 14607

This theorem is referenced by:  s2prop  14917  s3tpop  14919  s4prop  14920  pgpfaclem1  20106  vdegp1ai  29683  vdegp1bi  29684  wwlksnext  30039