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Theorem cats1un 14671
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 14566 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋)
2 wrdf 14469 . . . . 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 14570 . . . . . . . 8 (𝐴 ∈ Word 𝑋 → (♯‘(𝐴 ++ ⟨“𝐵”⟩)) = ((♯‘𝐴) + 1))
54oveq2d 7425 . . . . . . 7 (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = (0..^((♯‘𝐴) + 1)))
6 lencl 14483 . . . . . . . . 9 (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ ℕ0)
7 nn0uz 12864 . . . . . . . . 9 0 = (ℤ‘0)
86, 7eleqtrdi 2844 . . . . . . . 8 (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ (ℤ‘0))
9 fzosplitsn 13740 . . . . . . . 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 2773 . . . . . 6 (𝐴 ∈ Word 𝑋 → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
1211adantr 482 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
1312feq2d 6704 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩):(0..^(♯‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋 ↔ (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋))
143, 13mpbid 231 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶𝑋)
1514ffnd 6719 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
16 wrdf 14469 . . . . 5 (𝐴 ∈ Word 𝑋𝐴:(0..^(♯‘𝐴))⟶𝑋)
1716adantr 482 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴:(0..^(♯‘𝐴))⟶𝑋)
18 eqid 2733 . . . . . 6 {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}
19 fsng 7135 . . . . . 6 (((♯‘𝐴) ∈ ℕ0𝐵𝑋) → ({⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵} ↔ {⟨(♯‘𝐴), 𝐵⟩} = {⟨(♯‘𝐴), 𝐵⟩}))
2018, 19mpbiri 258 . . . . 5 (((♯‘𝐴) ∈ ℕ0𝐵𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
216, 20sylan 581 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵})
22 fzodisjsn 13670 . . . . 5 ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅
2322a1i 11 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅)
24 fun 6754 . . . 4 (((𝐴:(0..^(♯‘𝐴))⟶𝑋 ∧ {⟨(♯‘𝐴), 𝐵⟩}:{(♯‘𝐴)}⟶{𝐵}) ∧ ((0..^(♯‘𝐴)) ∩ {(♯‘𝐴)}) = ∅) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶(𝑋 ∪ {𝐵}))
2517, 21, 23, 24syl21anc 837 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}):((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})⟶(𝑋 ∪ {𝐵}))
2625ffnd 6719 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}) Fn ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}))
27 elun 4149 . . 3 (𝑥 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)}) ↔ (𝑥 ∈ (0..^(♯‘𝐴)) ∨ 𝑥 ∈ {(♯‘𝐴)}))
28 ccats1val1 14576 . . . . . 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 486 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ∈ (0..^(♯‘𝐴)))
31 fzonel 13646 . . . . . . . 8 ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))
32 nelne2 3041 . . . . . . . 8 ((𝑥 ∈ (0..^(♯‘𝐴)) ∧ ¬ (♯‘𝐴) ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
3330, 31, 32sylancl 587 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → 𝑥 ≠ (♯‘𝐴))
3433necomd 2997 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → (♯‘𝐴) ≠ 𝑥)
35 fvunsn 7177 . . . . . 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 2776 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(♯‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
38 fvexd 6907 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (♯‘𝐴) ∈ V)
39 simpr 486 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐵𝑋)
4017fdmd 6729 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → dom 𝐴 = (0..^(♯‘𝐴)))
4140eleq2d 2820 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((♯‘𝐴) ∈ dom 𝐴 ↔ (♯‘𝐴) ∈ (0..^(♯‘𝐴))))
4231, 41mtbiri 327 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ¬ (♯‘𝐴) ∈ dom 𝐴)
43 fsnunfv 7185 . . . . . . . 8 (((♯‘𝐴) ∈ V ∧ 𝐵𝑋 ∧ ¬ (♯‘𝐴) ∈ dom 𝐴) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
4438, 39, 42, 43syl3anc 1372 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)) = 𝐵)
45 simpl 484 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴 ∈ Word 𝑋)
46 s1cl 14552 . . . . . . . . . 10 (𝐵𝑋 → ⟨“𝐵”⟩ ∈ Word 𝑋)
4746adantl 483 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ⟨“𝐵”⟩ ∈ Word 𝑋)
48 s1len 14556 . . . . . . . . . . . 12 (♯‘⟨“𝐵”⟩) = 1
49 1nn 12223 . . . . . . . . . . . 12 1 ∈ ℕ
5048, 49eqeltri 2830 . . . . . . . . . . 11 (♯‘⟨“𝐵”⟩) ∈ ℕ
51 lbfzo0 13672 . . . . . . . . . . 11 (0 ∈ (0..^(♯‘⟨“𝐵”⟩)) ↔ (♯‘⟨“𝐵”⟩) ∈ ℕ)
5250, 51mpbir 230 . . . . . . . . . 10 0 ∈ (0..^(♯‘⟨“𝐵”⟩))
5352a1i 11 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 0 ∈ (0..^(♯‘⟨“𝐵”⟩)))
54 ccatval3 14529 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋 ∧ ⟨“𝐵”⟩ ∈ Word 𝑋 ∧ 0 ∈ (0..^(♯‘⟨“𝐵”⟩))) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
5545, 47, 53, 54syl3anc 1372 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = (⟨“𝐵”⟩‘0))
56 s1fv 14560 . . . . . . . . 9 (𝐵𝑋 → (⟨“𝐵”⟩‘0) = 𝐵)
5756adantl 483 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (⟨“𝐵”⟩‘0) = 𝐵)
5855, 57eqtrd 2773 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = 𝐵)
596adantr 482 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (♯‘𝐴) ∈ ℕ0)
6059nn0cnd 12534 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (♯‘𝐴) ∈ ℂ)
6160addlidd 11415 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0 + (♯‘𝐴)) = (♯‘𝐴))
6261fveq2d 6896 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (♯‘𝐴))) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
6344, 58, 623eqtr2rd 2780 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
64 elsni 4646 . . . . . . . 8 (𝑥 ∈ {(♯‘𝐴)} → 𝑥 = (♯‘𝐴))
6564fveq2d 6896 . . . . . . 7 (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)))
6664fveq2d 6896 . . . . . . 7 (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴)))
6765, 66eqeq12d 2749 . . . . . 6 (𝑥 ∈ {(♯‘𝐴)} → (((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥) ↔ ((𝐴 ++ ⟨“𝐵”⟩)‘(♯‘𝐴)) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘(♯‘𝐴))))
6863, 67syl5ibrcom 246 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝑥 ∈ {(♯‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥)))
6968imp 408 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ {(♯‘𝐴)}) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
7037, 69jaodan 957 . . 3 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ (𝑥 ∈ (0..^(♯‘𝐴)) ∨ 𝑥 ∈ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
7127, 70sylan2b 595 . 2 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ ((0..^(♯‘𝐴)) ∪ {(♯‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩})‘𝑥))
7215, 26, 71eqfnfvd 7036 1 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) = (𝐴 ∪ {⟨(♯‘𝐴), 𝐵⟩}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  cun 3947  cin 3948  c0 4323  {csn 4629  cop 4635  dom cdm 5677  wf 6540  cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  cn 12212  0cn0 12472  cuz 12822  ..^cfzo 13627  chash 14290  Word cword 14464   ++ cconcat 14520  ⟨“cs1 14545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-concat 14521  df-s1 14546
This theorem is referenced by:  s2prop  14858  s3tpop  14860  s4prop  14861  pgpfaclem1  19951  vdegp1ai  28793  vdegp1bi  28794  wwlksnext  29147
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