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Theorem cats1un 14130
 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 14017 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋)
2 wrdf 13918 . . . . 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 14021 . . . . . . . 8 (𝐴 ∈ Word 𝑋 → (♯‘(𝐴 ++ ⟨“𝐵”⟩)) = ((♯‘𝐴) + 1))
54oveq2d 7166 . . . . . . 7 (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = (0..^((♯‘𝐴) + 1)))
6 lencl 13932 . . . . . . . . 9 (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ ℕ0)
7 nn0uz 12320 . . . . . . . . 9 0 = (ℤ‘0)
86, 7eleqtrdi 2862 . . . . . . . 8 (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ (ℤ‘0))
9 fzosplitsn 13194 . . . . . . . 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 2793 . . . . . 6 (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
1211adantr 484 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
1312feq2d 6484 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩):(0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋 ↔ (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋))
143, 13mpbid 235 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋)
1514ffnd 6499 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
16 wrdf 13918 . . . . 5 (𝐴 ∈ Word 𝑋𝐴:(0..^(♯‘𝐴))⟶𝑋)
1716adantr 484 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴:(0..^(♯‘𝐴))⟶𝑋)
18 eqid 2758 . . . . . 6 {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}
19 fsng 6890 . . . . . 6 (((♯‘𝐴) ∈ ℕ0𝐵𝑋) → ({⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵} ↔ {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}))
2018, 19mpbiri 261 . . . . 5 (((♯‘𝐴) ∈ ℕ0𝐵𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
216, 20sylan 583 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
22 fzodisjsn 13124 . . . . 5 ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅
2322a1i 11 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅)
24 fun 6525 . . . 4 (((𝐴:(0..^(♯‘𝐴))⟶𝑋 ∧ {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵}) ∧ ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶(𝑋 ∪ {𝐵}))
2517, 21, 23, 24syl21anc 836 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶(𝑋 ∪ {𝐵}))
2625ffnd 6499 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}) Fn ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
27 elun 4054 . . 3 (𝑥 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}) ↔ (𝑥 ∈ (0..^(♯‘𝐴)) ∨ 𝑥 ∈ {(♯‘𝐴)}))
28 ccats1val1 14028 . . . . . 6 ((𝐴 ∈ Word 𝑋𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴𝑥))
2928adantlr 714 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴𝑥))
30 simpr 488 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ∈ (0..^(♯‘𝐴)))
31 fzonel 13100 . . . . . . . 8 ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))
32 nelne2 3048 . . . . . . . 8 ((𝑥 ∈ (0..^(♯‘𝐴)) ∧ ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
3330, 31, 32sylancl 589 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
3433necomd 3006 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → (♯‘𝐴) ≠ 𝑥)
35 fvunsn 6932 . . . . . 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 2796 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
38 fvexd 6673 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (♯‘𝐴) ∈ V)
39 simpr 488 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐵𝑋)
4017fdmd 6508 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → dom 𝐴 = (0..^(♯‘𝐴)))
4140eleq2d 2837 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((♯‘𝐴) ∈ dom 𝐴 ↔ (♯‘𝐴) ∈ (0..^(♯‘𝐴))))
4231, 41mtbiri 330 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ¬ (♯‘𝐴) ∈ dom 𝐴)
43 fsnunfv 6940 . . . . . . . 8 (((♯‘𝐴) ∈ V ∧ 𝐵𝑋 ∧ ¬ (♯‘𝐴) ∈ dom 𝐴) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
4438, 39, 42, 43syl3anc 1368 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
45 simpl 486 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴 ∈ Word 𝑋)
46 s1cl 14003 . . . . . . . . . 10 (𝐵𝑋 → ⟨“𝐵”⟩ ∈ Word 𝑋)
4746adantl 485 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ⟨“𝐵”⟩ ∈ Word 𝑋)
48 s1len 14007 . . . . . . . . . . . 12 (♯‘⟨“𝐵”⟩) = 1
49 1nn 11685 . . . . . . . . . . . 12 1 ∈ ℕ
5048, 49eqeltri 2848 . . . . . . . . . . 11 (♯‘⟨“𝐵”⟩) ∈ ℕ
51 lbfzo0 13126 . . . . . . . . . . 11 (0 ∈ (0..^(♯‘⟨“𝐵”⟩)) ↔ (♯‘⟨“𝐵”⟩) ∈ ℕ)
5250, 51mpbir 234 . . . . . . . . . 10 0 ∈ (0..^(♯‘⟨“𝐵”⟩))
5352a1i 11 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 0 ∈ (0..^(♯‘⟨“𝐵”⟩)))
54 ccatval3 13980 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋 ∧ ⟨“𝐵”⟩ ∈ Word 𝑋 ∧ 0 ∈ (0..^(♯‘⟨“𝐵”⟩))) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
5545, 47, 53, 54syl3anc 1368 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
56 s1fv 14011 . . . . . . . . 9 (𝐵𝑋 → (⟨“𝐵”⟩‘0) = 𝐵)
5756adantl 485 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (⟨“𝐵”⟩‘0) = 𝐵)
5855, 57eqtrd 2793 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = 𝐵)
596adantr 484 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (♯‘𝐴) ∈ ℕ0)
6059nn0cnd 11996 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (♯‘𝐴) ∈ ℂ)
6160addid2d 10879 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0 + (♯‘𝐴)) = (♯‘𝐴))
6261fveq2d 6662 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
6344, 58, 623eqtr2rd 2800 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
64 elsni 4539 . . . . . . . 8 (𝑥 ∈ {(♯‘𝐴)} → 𝑥 = (♯‘𝐴))
6564fveq2d 6662 . . . . . . 7 (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
6664fveq2d 6662 . . . . . . 7 (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
6765, 66eqeq12d 2774 . . . . . 6 (𝑥 ∈ {(♯‘𝐴)} → (((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) ↔ ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴))))
6863, 67syl5ibrcom 250 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥)))
6968imp 410 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ {(♯‘𝐴)}) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
7037, 69jaodan 955 . . 3 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ (𝑥 ∈ (0..^(♯‘𝐴)) ∨ 𝑥 ∈ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
7127, 70sylan2b 596 . 2 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
7215, 26, 71eqfnfvd 6796 1 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  Vcvv 3409   ∪ cun 3856   ∩ cin 3857  ∅c0 4225  {csn 4522  ⟨cop 4528  dom cdm 5524  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150  0cc0 10575  1c1 10576   + caddc 10578  ℕcn 11674  ℕ0cn0 11934  ℤ≥cuz 12282  ..^cfzo 13082  ♯chash 13740  Word cword 13913   ++ cconcat 13969  ⟨“cs1 13996 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-n0 11935  df-z 12021  df-uz 12283  df-fz 12940  df-fzo 13083  df-hash 13741  df-word 13914  df-concat 13970  df-s1 13997 This theorem is referenced by:  s2prop  14316  s3tpop  14318  s4prop  14319  pgpfaclem1  19271  vdegp1ai  27425  vdegp1bi  27426  wwlksnext  27778
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