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Theorem swrdco 13434
Description: Mapping of words commutes with the substring operation. (Contributed by AV, 11-Nov-2018.)
Assertion
Ref Expression
swrdco ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩)) = ((𝐹𝑊) substr ⟨𝑀, 𝑁⟩))

Proof of Theorem swrdco
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 ffn 5958 . . . 4 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
213ad2ant3 1077 . . 3 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → 𝐹 Fn 𝐴)
3 swrdvalfn 13278 . . . . 5 ((𝑊 ∈ Word 𝐴𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) → (𝑊 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
433expb 1258 . . . 4 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊)))) → (𝑊 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
543adant3 1074 . . 3 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → (𝑊 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
6 swrdrn 13281 . . . . 5 ((𝑊 ∈ Word 𝐴𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ 𝐴)
763expb 1258 . . . 4 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊)))) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ 𝐴)
873adant3 1074 . . 3 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ 𝐴)
9 fnco 5913 . . 3 ((𝐹 Fn 𝐴 ∧ (𝑊 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)) ∧ ran (𝑊 substr ⟨𝑀, 𝑁⟩) ⊆ 𝐴) → (𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩)) Fn (0..^(𝑁𝑀)))
102, 5, 8, 9syl3anc 1318 . 2 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩)) Fn (0..^(𝑁𝑀)))
11 wrdco 13428 . . . 4 ((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝐹𝑊) ∈ Word 𝐵)
12113adant2 1073 . . 3 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → (𝐹𝑊) ∈ Word 𝐵)
13 simp2l 1080 . . 3 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → 𝑀 ∈ (0...𝑁))
14 lenco 13429 . . . . . . . . . . . 12 ((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (#‘(𝐹𝑊)) = (#‘𝑊))
1514eqcomd 2616 . . . . . . . . . . 11 ((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (#‘𝑊) = (#‘(𝐹𝑊)))
1615oveq2d 6565 . . . . . . . . . 10 ((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (0...(#‘𝑊)) = (0...(#‘(𝐹𝑊))))
1716eleq2d 2673 . . . . . . . . 9 ((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝑁 ∈ (0...(#‘𝑊)) ↔ 𝑁 ∈ (0...(#‘(𝐹𝑊)))))
1817biimpd 218 . . . . . . . 8 ((𝑊 ∈ Word 𝐴𝐹:𝐴𝐵) → (𝑁 ∈ (0...(#‘𝑊)) → 𝑁 ∈ (0...(#‘(𝐹𝑊)))))
1918expcom 450 . . . . . . 7 (𝐹:𝐴𝐵 → (𝑊 ∈ Word 𝐴 → (𝑁 ∈ (0...(#‘𝑊)) → 𝑁 ∈ (0...(#‘(𝐹𝑊))))))
2019com13 86 . . . . . 6 (𝑁 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝐴 → (𝐹:𝐴𝐵𝑁 ∈ (0...(#‘(𝐹𝑊))))))
2120adantl 481 . . . . 5 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) → (𝑊 ∈ Word 𝐴 → (𝐹:𝐴𝐵𝑁 ∈ (0...(#‘(𝐹𝑊))))))
2221com12 32 . . . 4 (𝑊 ∈ Word 𝐴 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) → (𝐹:𝐴𝐵𝑁 ∈ (0...(#‘(𝐹𝑊))))))
23223imp 1249 . . 3 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → 𝑁 ∈ (0...(#‘(𝐹𝑊))))
24 swrdvalfn 13278 . . 3 (((𝐹𝑊) ∈ Word 𝐵𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐹𝑊)))) → ((𝐹𝑊) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
2512, 13, 23, 24syl3anc 1318 . 2 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → ((𝐹𝑊) substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)))
26 3anass 1035 . . . . . . 7 ((𝑊 ∈ Word 𝐴𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ↔ (𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊)))))
2726biimpri 217 . . . . . 6 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊)))) → (𝑊 ∈ Word 𝐴𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))))
28273adant3 1074 . . . . 5 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → (𝑊 ∈ Word 𝐴𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))))
29 swrdfv 13276 . . . . . 6 (((𝑊 ∈ Word 𝐴𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → ((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖) = (𝑊‘(𝑖 + 𝑀)))
3029fveq2d 6107 . . . . 5 (((𝑊 ∈ Word 𝐴𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → (𝐹‘((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖)) = (𝐹‘(𝑊‘(𝑖 + 𝑀))))
3128, 30sylan 487 . . . 4 (((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → (𝐹‘((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖)) = (𝐹‘(𝑊‘(𝑖 + 𝑀))))
32 wrdfn 13174 . . . . . . 7 (𝑊 ∈ Word 𝐴𝑊 Fn (0..^(#‘𝑊)))
33323ad2ant1 1075 . . . . . 6 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → 𝑊 Fn (0..^(#‘𝑊)))
3433adantr 480 . . . . 5 (((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → 𝑊 Fn (0..^(#‘𝑊)))
35 elfzodifsumelfzo 12401 . . . . . . 7 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) → (𝑖 ∈ (0..^(𝑁𝑀)) → (𝑖 + 𝑀) ∈ (0..^(#‘𝑊))))
36353ad2ant2 1076 . . . . . 6 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → (𝑖 ∈ (0..^(𝑁𝑀)) → (𝑖 + 𝑀) ∈ (0..^(#‘𝑊))))
3736imp 444 . . . . 5 (((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → (𝑖 + 𝑀) ∈ (0..^(#‘𝑊)))
38 fvco2 6183 . . . . 5 ((𝑊 Fn (0..^(#‘𝑊)) ∧ (𝑖 + 𝑀) ∈ (0..^(#‘𝑊))) → ((𝐹𝑊)‘(𝑖 + 𝑀)) = (𝐹‘(𝑊‘(𝑖 + 𝑀))))
3934, 37, 38syl2anc 691 . . . 4 (((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → ((𝐹𝑊)‘(𝑖 + 𝑀)) = (𝐹‘(𝑊‘(𝑖 + 𝑀))))
4031, 39eqtr4d 2647 . . 3 (((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → (𝐹‘((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖)) = ((𝐹𝑊)‘(𝑖 + 𝑀)))
41 fvco2 6183 . . . 4 (((𝑊 substr ⟨𝑀, 𝑁⟩) Fn (0..^(𝑁𝑀)) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → ((𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩))‘𝑖) = (𝐹‘((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖)))
425, 41sylan 487 . . 3 (((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → ((𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩))‘𝑖) = (𝐹‘((𝑊 substr ⟨𝑀, 𝑁⟩)‘𝑖)))
4314ancoms 468 . . . . . . . . . . . . . 14 ((𝐹:𝐴𝐵𝑊 ∈ Word 𝐴) → (#‘(𝐹𝑊)) = (#‘𝑊))
4443eqcomd 2616 . . . . . . . . . . . . 13 ((𝐹:𝐴𝐵𝑊 ∈ Word 𝐴) → (#‘𝑊) = (#‘(𝐹𝑊)))
4544oveq2d 6565 . . . . . . . . . . . 12 ((𝐹:𝐴𝐵𝑊 ∈ Word 𝐴) → (0...(#‘𝑊)) = (0...(#‘(𝐹𝑊))))
4645eleq2d 2673 . . . . . . . . . . 11 ((𝐹:𝐴𝐵𝑊 ∈ Word 𝐴) → (𝑁 ∈ (0...(#‘𝑊)) ↔ 𝑁 ∈ (0...(#‘(𝐹𝑊)))))
4746biimpd 218 . . . . . . . . . 10 ((𝐹:𝐴𝐵𝑊 ∈ Word 𝐴) → (𝑁 ∈ (0...(#‘𝑊)) → 𝑁 ∈ (0...(#‘(𝐹𝑊)))))
4847ex 449 . . . . . . . . 9 (𝐹:𝐴𝐵 → (𝑊 ∈ Word 𝐴 → (𝑁 ∈ (0...(#‘𝑊)) → 𝑁 ∈ (0...(#‘(𝐹𝑊))))))
4948com13 86 . . . . . . . 8 (𝑁 ∈ (0...(#‘𝑊)) → (𝑊 ∈ Word 𝐴 → (𝐹:𝐴𝐵𝑁 ∈ (0...(#‘(𝐹𝑊))))))
5049adantl 481 . . . . . . 7 ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) → (𝑊 ∈ Word 𝐴 → (𝐹:𝐴𝐵𝑁 ∈ (0...(#‘(𝐹𝑊))))))
5150com12 32 . . . . . 6 (𝑊 ∈ Word 𝐴 → ((𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) → (𝐹:𝐴𝐵𝑁 ∈ (0...(#‘(𝐹𝑊))))))
52513imp 1249 . . . . 5 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → 𝑁 ∈ (0...(#‘(𝐹𝑊))))
5312, 13, 523jca 1235 . . . 4 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → ((𝐹𝑊) ∈ Word 𝐵𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐹𝑊)))))
54 swrdfv 13276 . . . 4 ((((𝐹𝑊) ∈ Word 𝐵𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘(𝐹𝑊)))) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → (((𝐹𝑊) substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝐹𝑊)‘(𝑖 + 𝑀)))
5553, 54sylan 487 . . 3 (((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → (((𝐹𝑊) substr ⟨𝑀, 𝑁⟩)‘𝑖) = ((𝐹𝑊)‘(𝑖 + 𝑀)))
5640, 42, 553eqtr4d 2654 . 2 (((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) ∧ 𝑖 ∈ (0..^(𝑁𝑀))) → ((𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩))‘𝑖) = (((𝐹𝑊) substr ⟨𝑀, 𝑁⟩)‘𝑖))
5710, 25, 56eqfnfvd 6222 1 ((𝑊 ∈ Word 𝐴 ∧ (𝑀 ∈ (0...𝑁) ∧ 𝑁 ∈ (0...(#‘𝑊))) ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 substr ⟨𝑀, 𝑁⟩)) = ((𝐹𝑊) substr ⟨𝑀, 𝑁⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wss 3540  cop 4131  ran crn 5039  ccom 5042   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  0cc0 9815   + caddc 9818  cmin 10145  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   substr csubstr 13150
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-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-substr 13158
This theorem is referenced by:  pfxco  40301
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