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Theorem cats1un 13327
 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 13249 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) ∈ Word 𝑋)
2 wrdf 13165 . . . . 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 13251 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘(𝐴 ++ ⟨“𝐵”⟩)) = ((#‘𝐴) + 1))
54oveq2d 6565 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0..^(#‘(𝐴 ++ ⟨“𝐵”⟩))) = (0..^((#‘𝐴) + 1)))
6 lencl 13179 . . . . . . . . 9 (𝐴 ∈ Word 𝑋 → (#‘𝐴) ∈ ℕ0)
76adantr 480 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘𝐴) ∈ ℕ0)
8 nn0uz 11598 . . . . . . . 8 0 = (ℤ‘0)
97, 8syl6eleq 2698 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘𝐴) ∈ (ℤ‘0))
10 fzosplitsn 12442 . . . . . . 7 ((#‘𝐴) ∈ (ℤ‘0) → (0..^((#‘𝐴) + 1)) = ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
119, 10syl 17 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0..^((#‘𝐴) + 1)) = ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
125, 11eqtrd 2644 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0..^(#‘(𝐴 ++ ⟨“𝐵”⟩))) = ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
1312feq2d 5944 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩):(0..^(#‘(𝐴 ++ ⟨“𝐵”⟩)))⟶𝑋 ↔ (𝐴 ++ ⟨“𝐵”⟩):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶𝑋))
143, 13mpbid 221 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶𝑋)
15 ffn 5958 . . 3 ((𝐴 ++ ⟨“𝐵”⟩):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶𝑋 → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
1614, 15syl 17 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ++ ⟨“𝐵”⟩) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
17 wrdf 13165 . . . . 5 (𝐴 ∈ Word 𝑋𝐴:(0..^(#‘𝐴))⟶𝑋)
1817adantr 480 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴:(0..^(#‘𝐴))⟶𝑋)
19 eqid 2610 . . . . . 6 {⟨(#‘𝐴), 𝐵⟩} = {⟨(#‘𝐴), 𝐵⟩}
20 fsng 6310 . . . . . 6 (((#‘𝐴) ∈ ℕ0𝐵𝑋) → ({⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵} ↔ {⟨(#‘𝐴), 𝐵⟩} = {⟨(#‘𝐴), 𝐵⟩}))
2119, 20mpbiri 247 . . . . 5 (((#‘𝐴) ∈ ℕ0𝐵𝑋) → {⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵})
226, 21sylan 487 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → {⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵})
23 fzonel 12352 . . . . . 6 ¬ (#‘𝐴) ∈ (0..^(#‘𝐴))
2423a1i 11 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ¬ (#‘𝐴) ∈ (0..^(#‘𝐴)))
25 disjsn 4192 . . . . 5 (((0..^(#‘𝐴)) ∩ {(#‘𝐴)}) = ∅ ↔ ¬ (#‘𝐴) ∈ (0..^(#‘𝐴)))
2624, 25sylibr 223 . . . 4 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((0..^(#‘𝐴)) ∩ {(#‘𝐴)}) = ∅)
27 fun 5979 . . . 4 (((𝐴:(0..^(#‘𝐴))⟶𝑋 ∧ {⟨(#‘𝐴), 𝐵⟩}:{(#‘𝐴)}⟶{𝐵}) ∧ ((0..^(#‘𝐴)) ∩ {(#‘𝐴)}) = ∅) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶(𝑋 ∪ {𝐵}))
2818, 22, 26, 27syl21anc 1317 . . 3 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶(𝑋 ∪ {𝐵}))
29 ffn 5958 . . 3 ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}):((0..^(#‘𝐴)) ∪ {(#‘𝐴)})⟶(𝑋 ∪ {𝐵}) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
3028, 29syl 17 . 2 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩}) Fn ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}))
31 elun 3715 . . 3 (𝑥 ∈ ((0..^(#‘𝐴)) ∪ {(#‘𝐴)}) ↔ (𝑥 ∈ (0..^(#‘𝐴)) ∨ 𝑥 ∈ {(#‘𝐴)}))
32 ccats1val1 13255 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴𝑥))
33323expa 1257 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = (𝐴𝑥))
34 simpr 476 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → 𝑥 ∈ (0..^(#‘𝐴)))
35 nelne2 2879 . . . . . . . 8 ((𝑥 ∈ (0..^(#‘𝐴)) ∧ ¬ (#‘𝐴) ∈ (0..^(#‘𝐴))) → 𝑥 ≠ (#‘𝐴))
3634, 23, 35sylancl 693 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → 𝑥 ≠ (#‘𝐴))
3736necomd 2837 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → (#‘𝐴) ≠ 𝑥)
38 fvunsn 6350 . . . . . 6 ((#‘𝐴) ≠ 𝑥 → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) = (𝐴𝑥))
3937, 38syl 17 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) = (𝐴𝑥))
4033, 39eqtr4d 2647 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ (0..^(#‘𝐴))) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
41 fvex 6113 . . . . . . . . 9 (#‘𝐴) ∈ V
4241a1i 11 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘𝐴) ∈ V)
43 elex 3185 . . . . . . . . 9 (𝐵𝑋𝐵 ∈ V)
4443adantl 481 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐵 ∈ V)
45 fdm 5964 . . . . . . . . . . 11 (𝐴:(0..^(#‘𝐴))⟶𝑋 → dom 𝐴 = (0..^(#‘𝐴)))
4618, 45syl 17 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → dom 𝐴 = (0..^(#‘𝐴)))
4746eleq2d 2673 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((#‘𝐴) ∈ dom 𝐴 ↔ (#‘𝐴) ∈ (0..^(#‘𝐴))))
4823, 47mtbiri 316 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ¬ (#‘𝐴) ∈ dom 𝐴)
49 fsnunfv 6358 . . . . . . . 8 (((#‘𝐴) ∈ V ∧ 𝐵 ∈ V ∧ ¬ (#‘𝐴) ∈ dom 𝐴) → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)) = 𝐵)
5042, 44, 48, 49syl3anc 1318 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)) = 𝐵)
51 simpl 472 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 𝐴 ∈ Word 𝑋)
52 s1cl 13235 . . . . . . . . . 10 (𝐵𝑋 → ⟨“𝐵”⟩ ∈ Word 𝑋)
5352adantl 481 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ⟨“𝐵”⟩ ∈ Word 𝑋)
54 s1len 13238 . . . . . . . . . . . 12 (#‘⟨“𝐵”⟩) = 1
55 1nn 10908 . . . . . . . . . . . 12 1 ∈ ℕ
5654, 55eqeltri 2684 . . . . . . . . . . 11 (#‘⟨“𝐵”⟩) ∈ ℕ
5756a1i 11 . . . . . . . . . 10 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘⟨“𝐵”⟩) ∈ ℕ)
58 lbfzo0 12375 . . . . . . . . . 10 (0 ∈ (0..^(#‘⟨“𝐵”⟩)) ↔ (#‘⟨“𝐵”⟩) ∈ ℕ)
5957, 58sylibr 223 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → 0 ∈ (0..^(#‘⟨“𝐵”⟩)))
60 ccatval3 13216 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋 ∧ ⟨“𝐵”⟩ ∈ Word 𝑋 ∧ 0 ∈ (0..^(#‘⟨“𝐵”⟩))) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = (⟨“𝐵”⟩‘0))
6151, 53, 59, 60syl3anc 1318 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = (⟨“𝐵”⟩‘0))
62 s1fv 13243 . . . . . . . . 9 (𝐵𝑋 → (⟨“𝐵”⟩‘0) = 𝐵)
6362adantl 481 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (⟨“𝐵”⟩‘0) = 𝐵)
6461, 63eqtrd 2644 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = 𝐵)
657nn0cnd 11230 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (#‘𝐴) ∈ ℂ)
6665addid2d 10116 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (0 + (#‘𝐴)) = (#‘𝐴))
6766fveq2d 6107 . . . . . . 7 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(0 + (#‘𝐴))) = ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)))
6850, 64, 673eqtr2rd 2651 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵𝑋) → ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)))
69 elsni 4142 . . . . . . . 8 (𝑥 ∈ {(#‘𝐴)} → 𝑥 = (#‘𝐴))
7069fveq2d 6107 . . . . . . 7 (𝑥 ∈ {(#‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)))
7169fveq2d 6107 . . . . . . 7 (𝑥 ∈ {(#‘𝐴)} → ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴)))
7270, 71eqeq12d 2625 . . . . . 6 (𝑥 ∈ {(#‘𝐴)} → (((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥) ↔ ((𝐴 ++ ⟨“𝐵”⟩)‘(#‘𝐴)) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘(#‘𝐴))))
7368, 72syl5ibrcom 236 . . . . 5 ((𝐴 ∈ Word 𝑋𝐵𝑋) → (𝑥 ∈ {(#‘𝐴)} → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥)))
7473imp 444 . . . 4 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ {(#‘𝐴)}) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
7540, 74jaodan 822 . . 3 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ (𝑥 ∈ (0..^(#‘𝐴)) ∨ 𝑥 ∈ {(#‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
7631, 75sylan2b 491 . 2 (((𝐴 ∈ Word 𝑋𝐵𝑋) ∧ 𝑥 ∈ ((0..^(#‘𝐴)) ∪ {(#‘𝐴)})) → ((𝐴 ++ ⟨“𝐵”⟩)‘𝑥) = ((𝐴 ∪ {⟨(#‘𝐴), 𝐵⟩})‘𝑥))
7716, 30, 76eqfnfvd 6222 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 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∪ cun 3538   ∩ cin 3539  ∅c0 3874  {csn 4125  ⟨cop 4131  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕcn 10897  ℕ0cn0 11169  ℤ≥cuz 11563  ..^cfzo 12334  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157 This theorem is referenced by:  s2prop  13502  s3tpop  13504  s4prop  13505  pgpfaclem1  18303  wwlknext  26252  vdegp1ai  26511  vdegp1bi  26512  vdegp1ai-av  40752  vdegp1bi-av  40753  wwlksnext  41099
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