Theorem cats1un 14645

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 14541	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋)
2		wrdf 14443	. . . . 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 14545	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑋 → (♯‘(𝐴 ++ ⟨“𝐵”⟩)) = ((♯‘𝐴) + 1))
5	4	oveq2d 7369	. . . . . . 7 ⊢ (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = (0..^((♯‘𝐴) + 1)))
6		lencl 14458	. . . . . . . . 9 ⊢ (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ ℕ0)
7		nn0uz 12795	. . . . . . . . 9 ⊢ ℕ0 = (ℤ≥‘0)
8	6, 7	eleqtrdi 2838	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ (ℤ≥‘0))
9		fzosplitsn 13696	. . . . . . . 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 2764	. . . . . 6 ⊢ (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
12	11	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
13	12	feq2d 6640	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩):(0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋 ↔ (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋))
14	3, 13	mpbid 232	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋)
15	14	ffnd 6657	. 2 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
16		wrdf 14443	. . . . 5 ⊢ (𝐴 ∈ Word 𝑋 → 𝐴:(0..^(♯‘𝐴))⟶𝑋)
17	16	adantr 480	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴:(0..^(♯‘𝐴))⟶𝑋)
18		eqid 2729	. . . . . 6 ⊢ {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}
19		fsng 7075	. . . . . 6 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ 𝐵 ∈ 𝑋) → ({⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵} ↔ {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}))
20	18, 19	mpbiri 258	. . . . 5 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ 𝐵 ∈ 𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
21	6, 20	sylan 580	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
22		fzodisjsn 13618	. . . . 5 ⊢ ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅
23	22	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅)
24		fun 6690	. . . 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 837	. . 3 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶(𝑋 ∪ {𝐵}))
26	25	ffnd 6657	. 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 14551	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴‘𝑥))
29	28	adantlr 715	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴‘𝑥))
30		simpr 484	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ∈ (0..^(♯‘𝐴)))
31		fzonel 13594	. . . . . . . 8 ⊢ ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))
32		nelne2 3023	. . . . . . . 8 ⊢ ((𝑥 ∈ (0..^(♯‘𝐴)) ∧ ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
33	30, 31, 32	sylancl 586	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
34	33	necomd 2980	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → (♯‘𝐴) ≠ 𝑥)
35		fvunsn 7119	. . . . . 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 2767	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
38		fvexd 6841	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (♯‘𝐴) ∈ V)
39		simpr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
40	17	fdmd 6666	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → dom 𝐴 = (0..^(♯‘𝐴)))
41	40	eleq2d 2814	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((♯‘𝐴) ∈ dom 𝐴 ↔ (♯‘𝐴) ∈ (0..^(♯‘𝐴))))
42	31, 41	mtbiri 327	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ¬ (♯‘𝐴) ∈ dom 𝐴)
43		fsnunfv 7127	. . . . . . . 8 ⊢ (((♯‘𝐴) ∈ V ∧ 𝐵 ∈ 𝑋 ∧ ¬ (♯‘𝐴) ∈ dom 𝐴) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
44	38, 39, 42, 43	syl3anc 1373	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
45		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ Word 𝑋)
46		s1cl 14527	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝑋 → ⟨“𝐵”⟩ ∈ Word 𝑋)
47	46	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ⟨“𝐵”⟩ ∈ Word 𝑋)
48		s1len 14531	. . . . . . . . . . . 12 ⊢ (♯‘⟨“𝐵”⟩) = 1
49		1nn 12157	. . . . . . . . . . . 12 ⊢ 1 ∈ ℕ
50	48, 49	eqeltri 2824	. . . . . . . . . . 11 ⊢ (♯‘⟨“𝐵”⟩) ∈ ℕ
51		lbfzo0 13620	. . . . . . . . . . 11 ⊢ (0 ∈ (0..^(♯‘⟨“𝐵”⟩)) ↔ (♯‘⟨“𝐵”⟩) ∈ ℕ)
52	50, 51	mpbir 231	. . . . . . . . . 10 ⊢ 0 ∈ (0..^(♯‘⟨“𝐵”⟩))
53	52	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → 0 ∈ (0..^(♯‘⟨“𝐵”⟩)))
54		ccatval3 14504	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ ⟨“𝐵”⟩ ∈ Word 𝑋 ∧ 0 ∈ (0..^(♯‘⟨“𝐵”⟩))) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
55	45, 47, 53, 54	syl3anc 1373	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
56		s1fv 14535	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝑋 → (⟨“𝐵”⟩‘0) = 𝐵)
57	56	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (⟨“𝐵”⟩‘0) = 𝐵)
58	55, 57	eqtrd 2764	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = 𝐵)
59	6	adantr 480	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (♯‘𝐴) ∈ ℕ0)
60	59	nn0cnd 12465	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (♯‘𝐴) ∈ ℂ)
61	60	addlidd 11335	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (0 + (♯‘𝐴)) = (♯‘𝐴))
62	61	fveq2d 6830	. . . . . . 7 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
63	44, 58, 62	3eqtr2rd 2771	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
64		elsni 4596	. . . . . . . 8 ⊢ (𝑥 ∈ {(♯‘𝐴)} → 𝑥 = (♯‘𝐴))
65	64	fveq2d 6830	. . . . . . 7 ⊢ (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
66	64	fveq2d 6830	. . . . . . 7 ⊢ (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
67	65, 66	eqeq12d 2745	. . . . . 6 ⊢ (𝑥 ∈ {(♯‘𝐴)} → (((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) ↔ ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴))))
68	63, 67	syl5ibrcom 247	. . . . 5 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥)))
69	68	imp 406	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ {(♯‘𝐴)}) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
70	37, 69	jaodan 959	. . 3 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ (𝑥 ∈ (0..^(♯‘𝐴)) ∨ 𝑥 ∈ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
71	27, 70	sylan2b 594	. 2 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ 𝑥 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
72	15, 26, 71	eqfnfvd 6972	1 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  Vcvv 3438   ∪ cun 3903   ∩ cin 3904  ∅c0 4286  {csn 4579  ⟨cop 4585  dom cdm 5623  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  0cc0 11028  1c1 11029   + caddc 11031  ℕcn 12146  ℕ₀cn0 12402  ℤ_≥cuz 12753  ..^cfzo 13575  ♯chash 14255  Word cword 14438   ++ cconcat 14495  ⟨“cs1 14520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675  ax-cnex 11084  ax-resscn 11085  ax-1cn 11086  ax-icn 11087  ax-addcl 11088  ax-addrcl 11089  ax-mulcl 11090  ax-mulrcl 11091  ax-mulcom 11092  ax-addass 11093  ax-mulass 11094  ax-distr 11095  ax-i2m1 11096  ax-1ne0 11097  ax-1rid 11098  ax-rnegex 11099  ax-rrecex 11100  ax-cnre 11101  ax-pre-lttri 11102  ax-pre-lttrn 11103  ax-pre-ltadd 11104  ax-pre-mulgt0 11105

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-er 8632  df-en 8880  df-dom 8881  df-sdom 8882  df-fin 8883  df-card 9854  df-pnf 11170  df-mnf 11171  df-xr 11172  df-ltxr 11173  df-le 11174  df-sub 11367  df-neg 11368  df-nn 12147  df-n0 12403  df-z 12490  df-uz 12754  df-fz 13429  df-fzo 13576  df-hash 14256  df-word 14439  df-concat 14496  df-s1 14521

This theorem is referenced by:  s2prop  14832  s3tpop  14834  s4prop  14835  pgpfaclem1  19980  vdegp1ai  29500  vdegp1bi  29501  wwlksnext  29856