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Theorem cats1un 13521
 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 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}))

Proof of Theorem cats1un
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ccatws1cl 13433 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋)
2 wrdf 13342 . . . . 5 ((𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋 → (𝐴 ++ ⟨“𝐵”⟩):(0..^(#‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋)
31, 2syl 17 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩):(0..^(#‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋)
4 ccatws1len 13437 . . . . . . . 8 (𝐴 ∈ Word 𝑋 → (#‘(𝐴 ++ ⟨“𝐵”⟩)) = ((#‘𝐴) + 1))
54adantr 480 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘(𝐴 ++ ⟨“𝐵”⟩)) = ((#‘𝐴) + 1))
65oveq2d 6706 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0..^(#‘(𝐴 ++ ⟨“𝐵”⟩))) = (0..^((#‘𝐴) + 1)))
7 lencl 13356 . . . . . . . . 9 (𝐴 ∈ Word 𝑋 → (#‘𝐴) ∈ ℕ0)
87adantr 480 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘𝐴) ∈ ℕ0)
9 nn0uz 11760 . . . . . . . 8 0 = (ℤ‘0)
108, 9syl6eleq 2740 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘𝐴) ∈ (ℤ‘0))
11 fzosplitsn 12616 . . . . . . 7 ((#‘𝐴) ∈ (ℤ‘0) → (0..^((#‘𝐴) + 1)) = ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
1210, 11syl 17 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0..^((#‘𝐴) + 1)) = ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
136, 12eqtrd 2685 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0..^(#‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
1413feq2d 6069 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩):(0..^(#‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋 ↔ (𝐴 ++ ⟨“𝐵”⟩):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶𝑋))
153, 14mpbid 222 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶𝑋)
1615ffnd 6084 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
17 wrdf 13342 . . . . 5 (𝐴 ∈ Word 𝑋𝐴:(0..^(#‘𝐴))⟶𝑋)
1817adantr 480 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴:(0..^(#‘𝐴))⟶𝑋)
19 eqid 2651 . . . . . 6 {⟨(#‘𝐴), 𝐵⟩} = {⟨(#‘𝐴), 𝐵⟩}
20 fsng 6444 . . . . . 6 (((#‘𝐴) ∈ ℕ0𝐵𝑋) → ({⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵} ↔ {⟨(#‘𝐴), 𝐵⟩} = {⟨(#‘𝐴), 𝐵⟩}))
2119, 20mpbiri 248 . . . . 5 (((#‘𝐴) ∈ ℕ0𝐵𝑋) → {⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵})
227, 21sylan 487 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → {⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵})
23 fzonel 12522 . . . . . 6 ¬ (#‘𝐴) ∈ (0..^(#‘𝐴))
2423a1i 11 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ¬ (#‘𝐴) ∈ (0..^(#‘𝐴)))
25 disjsn 4278 . . . . 5 (((0..^(#‘𝐴)) ∩ {(#‘𝐴)}) = ∅ ↔ ¬ (#‘𝐴) ∈ (0..^(#‘𝐴)))
2624, 25sylibr 224 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((0..^(#‘𝐴)) ∩ {(#‘𝐴)}) = ∅)
27 fun 6104 . . . 4 (((𝐴:(0..^(#‘𝐴))⟶𝑋 ∧ {⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵}) ∧ ((0..^(#‘𝐴)) ∩ {(#‘𝐴)}) = ∅) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶(𝑋 ∪ {𝐵}))
2818, 22, 26, 27syl21anc 1365 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶(𝑋 ∪ {𝐵}))
2928ffnd 6084 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
30 elun 3786 . . 3 (𝑥 ∈ ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}) ↔ (𝑥 ∈ (0..^(#‘𝐴)) ∨ 𝑥 ∈ {(#‘𝐴)}))
31 ccats1val1 13446 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴𝑥))
32313expa 1284 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴𝑥))
33 simpr 476 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → 𝑥 ∈ (0..^(#‘𝐴)))
34 nelne2 2920 . . . . . . . 8 ((𝑥 ∈ (0..^(#‘𝐴)) ∧ ¬ (#‘𝐴) ∈ (0..^(#‘𝐴))) → 𝑥 ≠ (#‘𝐴))
3533, 23, 34sylancl 695 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → 𝑥 ≠ (#‘𝐴))
3635necomd 2878 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → (#‘𝐴) ≠ 𝑥)
37 fvunsn 6486 . . . . . 6 ((#‘𝐴) ≠ 𝑥 → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) = (𝐴𝑥))
3836, 37syl 17 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) = (𝐴𝑥))
3932, 38eqtr4d 2688 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
40 fvexd 6241 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘𝐴) ∈ V)
41 elex 3243 . . . . . . . . 9 (𝐵𝑋𝐵 ∈ V)
4241adantl 481 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐵 ∈ V)
43 fdm 6089 . . . . . . . . . . 11 (𝐴:(0..^(#‘𝐴))⟶𝑋 → dom 𝐴 = (0..^(#‘𝐴)))
4418, 43syl 17 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → dom 𝐴 = (0..^(#‘𝐴)))
4544eleq2d 2716 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((#‘𝐴) ∈ dom 𝐴 ↔ (#‘𝐴) ∈ (0..^(#‘𝐴))))
4623, 45mtbiri 316 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ¬ (#‘𝐴) ∈ dom 𝐴)
47 fsnunfv 6494 . . . . . . . 8 (((#‘𝐴) ∈ V ∧ 𝐵 ∈ V ∧ ¬ (#‘𝐴) ∈ dom 𝐴) → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)) = 𝐵)
4840, 42, 46, 47syl3anc 1366 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)) = 𝐵)
49 simpl 472 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴 ∈ Word 𝑋)
50 s1cl 13418 . . . . . . . . . 10 (𝐵𝑋 → ⟨“𝐵”⟩ ∈ Word 𝑋)
5150adantl 481 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ⟨“𝐵”⟩ ∈ Word 𝑋)
52 s1len 13422 . . . . . . . . . . . 12 (#‘⟨“𝐵”⟩) = 1
53 1nn 11069 . . . . . . . . . . . 12 1 ∈ ℕ
5452, 53eqeltri 2726 . . . . . . . . . . 11 (#‘⟨“𝐵”⟩) ∈ ℕ
5554a1i 11 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘⟨“𝐵”⟩) ∈ ℕ)
56 lbfzo0 12547 . . . . . . . . . 10 (0 ∈ (0..^(#‘⟨“𝐵”⟩)) ↔ (#‘⟨“𝐵”⟩) ∈ ℕ)
5755, 56sylibr 224 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 0 ∈ (0..^(#‘⟨“𝐵”⟩)))
58 ccatval3 13397 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋 ∧ ⟨“𝐵”⟩ ∈ Word 𝑋 ∧ 0 ∈ (0..^(#‘⟨“𝐵”⟩))) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = (⟨“𝐵”⟩‘0))
5949, 51, 57, 58syl3anc 1366 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = (⟨“𝐵”⟩‘0))
60 s1fv 13427 . . . . . . . . 9 (𝐵𝑋 → (⟨“𝐵”⟩‘0) = 𝐵)
6160adantl 481 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (⟨“𝐵”⟩‘0) = 𝐵)
6259, 61eqtrd 2685 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = 𝐵)
638nn0cnd 11391 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘𝐴) ∈ ℂ)
6463addid2d 10275 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0 + (#‘𝐴)) = (#‘𝐴))
6564fveq2d 6233 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)))
6648, 62, 653eqtr2rd 2692 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)))
67 elsni 4227 . . . . . . . 8 (𝑥 ∈ {(#‘𝐴)} → 𝑥 = (#‘𝐴))
6867fveq2d 6233 . . . . . . 7 (𝑥 ∈ {(#‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)))
6967fveq2d 6233 . . . . . . 7 (𝑥 ∈ {(#‘𝐴)} → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)))
7068, 69eqeq12d 2666 . . . . . 6 (𝑥 ∈ {(#‘𝐴)} → (((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) ↔ ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴))))
7166, 70syl5ibrcom 237 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝑥 ∈ {(#‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥)))
7271imp 444 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ {(#‘𝐴)}) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
7339, 72jaodan 843 . . 3 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ (𝑥 ∈ (0..^(#‘𝐴)) ∨ 𝑥 ∈ {(#‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
7430, 73sylan2b 491 . 2 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ ((0..^(#‘𝐴)) ∪ {(#‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
7516, 29, 74eqfnfvd 6354 1 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  Vcvv 3231   ∪ cun 3605   ∩ cin 3606  ∅c0 3948  {csn 4210  ⟨cop 4216  dom cdm 5143  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  ℕcn 11058  ℕ0cn0 11330  ℤ≥cuz 11725  ..^cfzo 12504  #chash 13157  Word cword 13323   ++ cconcat 13325  ⟨“cs1 13326 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334 This theorem is referenced by:  s2prop  13698  s3tpop  13700  s4prop  13701  pgpfaclem1  18526  vdegp1ai  26488  vdegp1bi  26489  wwlksnext  26856
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