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Theorem cats1un 14071
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 13958 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋)
2 wrdf 13854 . . . . 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 13962 . . . . . . . 8 (𝐴 ∈ Word 𝑋 → (♯‘(𝐴 ++ ⟨“𝐵”⟩)) = ((♯‘𝐴) + 1))
54oveq2d 7161 . . . . . . 7 (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = (0..^((♯‘𝐴) + 1)))
6 lencl 13871 . . . . . . . . 9 (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ ℕ0)
7 nn0uz 12268 . . . . . . . . 9 0 = (ℤ‘0)
86, 7eleqtrdi 2920 . . . . . . . 8 (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ (ℤ‘0))
9 fzosplitsn 13133 . . . . . . . 8 ((♯‘𝐴) ∈ (ℤ‘0) → (0..^((♯‘𝐴) + 1)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
108, 9syl 17 . . . . . . 7 (𝐴 ∈ Word 𝑋 → (0..^((♯‘𝐴) + 1)) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
115, 10eqtrd 2853 . . . . . 6 (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
1211adantr 481 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
1312feq2d 6493 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩):(0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋 ↔ (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋))
143, 13mpbid 233 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋)
1514ffnd 6508 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
16 wrdf 13854 . . . . 5 (𝐴 ∈ Word 𝑋𝐴:(0..^(♯‘𝐴))⟶𝑋)
1716adantr 481 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴:(0..^(♯‘𝐴))⟶𝑋)
18 eqid 2818 . . . . . 6 {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}
19 fsng 6891 . . . . . 6 (((♯‘𝐴) ∈ ℕ0𝐵𝑋) → ({⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵} ↔ {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}))
2018, 19mpbiri 259 . . . . 5 (((♯‘𝐴) ∈ ℕ0𝐵𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
216, 20sylan 580 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
22 fzodisjsn 13063 . . . . 5 ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅
2322a1i 11 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅)
24 fun 6533 . . . 4 (((𝐴:(0..^(♯‘𝐴))⟶𝑋 ∧ {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵}) ∧ ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶(𝑋 ∪ {𝐵}))
2517, 21, 23, 24syl21anc 833 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶(𝑋 ∪ {𝐵}))
2625ffnd 6508 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}) Fn ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
27 elun 4122 . . 3 (𝑥 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}) ↔ (𝑥 ∈ (0..^(♯‘𝐴)) ∨ 𝑥 ∈ {(♯‘𝐴)}))
28 ccats1val1 13969 . . . . . 6 ((𝐴 ∈ Word 𝑋𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴𝑥))
2928adantlr 711 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴𝑥))
30 simpr 485 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ∈ (0..^(♯‘𝐴)))
31 fzonel 13039 . . . . . . . 8 ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))
32 nelne2 3112 . . . . . . . 8 ((𝑥 ∈ (0..^(♯‘𝐴)) ∧ ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
3330, 31, 32sylancl 586 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
3433necomd 3068 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → (♯‘𝐴) ≠ 𝑥)
35 fvunsn 6933 . . . . . 6 ((♯‘𝐴) ≠ 𝑥 → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) = (𝐴𝑥))
3634, 35syl 17 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) = (𝐴𝑥))
3729, 36eqtr4d 2856 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
38 fvexd 6678 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (♯‘𝐴) ∈ V)
39 simpr 485 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐵𝑋)
4017fdmd 6516 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → dom 𝐴 = (0..^(♯‘𝐴)))
4140eleq2d 2895 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((♯‘𝐴) ∈ dom 𝐴 ↔ (♯‘𝐴) ∈ (0..^(♯‘𝐴))))
4231, 41mtbiri 328 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ¬ (♯‘𝐴) ∈ dom 𝐴)
43 fsnunfv 6941 . . . . . . . 8 (((♯‘𝐴) ∈ V ∧ 𝐵𝑋 ∧ ¬ (♯‘𝐴) ∈ dom 𝐴) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
4438, 39, 42, 43syl3anc 1363 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
45 simpl 483 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴 ∈ Word 𝑋)
46 s1cl 13944 . . . . . . . . . 10 (𝐵𝑋 → ⟨“𝐵”⟩ ∈ Word 𝑋)
4746adantl 482 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ⟨“𝐵”⟩ ∈ Word 𝑋)
48 s1len 13948 . . . . . . . . . . . 12 (♯‘⟨“𝐵”⟩) = 1
49 1nn 11637 . . . . . . . . . . . 12 1 ∈ ℕ
5048, 49eqeltri 2906 . . . . . . . . . . 11 (♯‘⟨“𝐵”⟩) ∈ ℕ
51 lbfzo0 13065 . . . . . . . . . . 11 (0 ∈ (0..^(♯‘⟨“𝐵”⟩)) ↔ (♯‘⟨“𝐵”⟩) ∈ ℕ)
5250, 51mpbir 232 . . . . . . . . . 10 0 ∈ (0..^(♯‘⟨“𝐵”⟩))
5352a1i 11 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 0 ∈ (0..^(♯‘⟨“𝐵”⟩)))
54 ccatval3 13921 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋 ∧ ⟨“𝐵”⟩ ∈ Word 𝑋 ∧ 0 ∈ (0..^(♯‘⟨“𝐵”⟩))) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
5545, 47, 53, 54syl3anc 1363 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
56 s1fv 13952 . . . . . . . . 9 (𝐵𝑋 → (⟨“𝐵”⟩‘0) = 𝐵)
5756adantl 482 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (⟨“𝐵”⟩‘0) = 𝐵)
5855, 57eqtrd 2853 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = 𝐵)
596adantr 481 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (♯‘𝐴) ∈ ℕ0)
6059nn0cnd 11945 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (♯‘𝐴) ∈ ℂ)
6160addid2d 10829 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0 + (♯‘𝐴)) = (♯‘𝐴))
6261fveq2d 6667 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
6344, 58, 623eqtr2rd 2860 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
64 elsni 4574 . . . . . . . 8 (𝑥 ∈ {(♯‘𝐴)} → 𝑥 = (♯‘𝐴))
6564fveq2d 6667 . . . . . . 7 (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
6664fveq2d 6667 . . . . . . 7 (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
6765, 66eqeq12d 2834 . . . . . 6 (𝑥 ∈ {(♯‘𝐴)} → (((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) ↔ ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴))))
6863, 67syl5ibrcom 248 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥)))
6968imp 407 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ {(♯‘𝐴)}) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
7037, 69jaodan 951 . . 3 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ (𝑥 ∈ (0..^(♯‘𝐴)) ∨ 𝑥 ∈ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
7127, 70sylan2b 593 . 2 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
7215, 26, 71eqfnfvd 6797 1 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  Vcvv 3492  cun 3931  cin 3932  c0 4288  {csn 4557  cop 4563  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526   + caddc 10528  cn 11626  0cn0 11885  cuz 12231  ..^cfzo 13021  chash 13678  Word cword 13849   ++ cconcat 13910  ⟨“cs1 13937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-hash 13679  df-word 13850  df-concat 13911  df-s1 13938
This theorem is referenced by:  s2prop  14257  s3tpop  14259  s4prop  14260  pgpfaclem1  19132  vdegp1ai  27245  vdegp1bi  27246  wwlksnext  27598
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