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Theorem swrd2lsw 14899
Description: Extract the last two symbols from a word. (Contributed by Alexander van der Vekens, 23-Sep-2018.)
Assertion
Ref Expression
swrd2lsw ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(lastSβ€˜π‘Š)β€βŸ©)
Proof of Theorem swrd2lsw
StepHypRef Expression
1 simpl 483 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ π‘Š ∈ Word 𝑉)
2 lencl 14479 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„•0)
3 1z 12588 . . . . . . . . 9 1 ∈ β„€
4 nn0z 12579 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„€)
5 zltp1le 12608 . . . . . . . . 9 ((1 ∈ β„€ ∧ (β™―β€˜π‘Š) ∈ β„€) β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
63, 4, 5sylancr 587 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
7 1p1e2 12333 . . . . . . . . . . 11 (1 + 1) = 2
87a1i 11 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 + 1) = 2)
98breq1d 5157 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((1 + 1) ≀ (β™―β€˜π‘Š) ↔ 2 ≀ (β™―β€˜π‘Š)))
109biimpd 228 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((1 + 1) ≀ (β™―β€˜π‘Š) β†’ 2 ≀ (β™―β€˜π‘Š)))
116, 10sylbid 239 . . . . . . 7 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 < (β™―β€˜π‘Š) β†’ 2 ≀ (β™―β€˜π‘Š)))
1211imp 407 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ 2 ≀ (β™―β€˜π‘Š))
13 2nn0 12485 . . . . . . . . 9 2 ∈ β„•0
1413jctl 524 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0))
1514adantr 481 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0))
16 nn0sub 12518 . . . . . . 7 ((2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0) β†’ (2 ≀ (β™―β€˜π‘Š) ↔ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0))
1715, 16syl 17 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (2 ≀ (β™―β€˜π‘Š) ↔ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0))
1812, 17mpbid 231 . . . . 5 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0)
192, 18sylan 580 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0)
20 0red 11213 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ 0 ∈ ℝ)
21 1red 11211 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ 1 ∈ ℝ)
22 zre 12558 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ (β™―β€˜π‘Š) ∈ ℝ)
2320, 21, 223jca 1128 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„€ β†’ (0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (β™―β€˜π‘Š) ∈ ℝ))
24 0lt1 11732 . . . . . . . . . . 11 0 < 1
25 lttr 11286 . . . . . . . . . . . 12 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (β™―β€˜π‘Š) ∈ ℝ) β†’ ((0 < 1 ∧ 1 < (β™―β€˜π‘Š)) β†’ 0 < (β™―β€˜π‘Š)))
2625expd 416 . . . . . . . . . . 11 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (β™―β€˜π‘Š) ∈ ℝ) β†’ (0 < 1 β†’ (1 < (β™―β€˜π‘Š) β†’ 0 < (β™―β€˜π‘Š))))
2723, 24, 26mpisyl 21 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„€ β†’ (1 < (β™―β€˜π‘Š) β†’ 0 < (β™―β€˜π‘Š)))
28 elnnz 12564 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„• ↔ ((β™―β€˜π‘Š) ∈ β„€ ∧ 0 < (β™―β€˜π‘Š)))
2928simplbi2 501 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„€ β†’ (0 < (β™―β€˜π‘Š) β†’ (β™―β€˜π‘Š) ∈ β„•))
3027, 29syld 47 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„€ β†’ (1 < (β™―β€˜π‘Š) β†’ (β™―β€˜π‘Š) ∈ β„•))
314, 30syl 17 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 < (β™―β€˜π‘Š) β†’ (β™―β€˜π‘Š) ∈ β„•))
3231imp 407 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
33 fzo0end 13720 . . . . . . 7 ((β™―β€˜π‘Š) ∈ β„• β†’ ((β™―β€˜π‘Š) βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š)))
3432, 33syl 17 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š)))
35 nn0cn 12478 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„‚)
36 2cn 12283 . . . . . . . . . . . 12 2 ∈ β„‚
3736a1i 11 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 2 ∈ β„‚)
38 1cnd 11205 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 1 ∈ β„‚)
3935, 37, 383jca 1128 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 1 ∈ β„‚))
40 1e2m1 12335 . . . . . . . . . . . . 13 1 = (2 βˆ’ 1)
4140a1i 11 . . . . . . . . . . . 12 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ 1 = (2 βˆ’ 1))
4241oveq2d 7421 . . . . . . . . . . 11 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((β™―β€˜π‘Š) βˆ’ 1) = ((β™―β€˜π‘Š) βˆ’ (2 βˆ’ 1)))
43 subsub 11486 . . . . . . . . . . 11 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((β™―β€˜π‘Š) βˆ’ (2 βˆ’ 1)) = (((β™―β€˜π‘Š) βˆ’ 2) + 1))
4442, 43eqtrd 2772 . . . . . . . . . 10 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((β™―β€˜π‘Š) βˆ’ 2) + 1))
4539, 44syl 17 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((β™―β€˜π‘Š) βˆ’ 2) + 1))
4645eqcomd 2738 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 1) = ((β™―β€˜π‘Š) βˆ’ 1))
4746eleq1d 2818 . . . . . . 7 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((((β™―β€˜π‘Š) βˆ’ 2) + 1) ∈ (0..^(β™―β€˜π‘Š)) ↔ ((β™―β€˜π‘Š) βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š))))
4847adantr 481 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((((β™―β€˜π‘Š) βˆ’ 2) + 1) ∈ (0..^(β™―β€˜π‘Š)) ↔ ((β™―β€˜π‘Š) βˆ’ 1) ∈ (0..^(β™―β€˜π‘Š))))
4934, 48mpbird 256 . . . . 5 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 1) ∈ (0..^(β™―β€˜π‘Š)))
502, 49sylan 580 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 1) ∈ (0..^(β™―β€˜π‘Š)))
511, 19, 503jca 1128 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0 ∧ (((β™―β€˜π‘Š) βˆ’ 2) + 1) ∈ (0..^(β™―β€˜π‘Š))))
52 swrds2 14887 . . 3 ((π‘Š ∈ Word 𝑉 ∧ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0 ∧ (((β™―β€˜π‘Š) βˆ’ 2) + 1) ∈ (0..^(β™―β€˜π‘Š))) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (((β™―β€˜π‘Š) βˆ’ 2) + 2)⟩) = βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 2) + 1))β€βŸ©)
5351, 52syl 17 . 2 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (((β™―β€˜π‘Š) βˆ’ 2) + 2)⟩) = βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 2) + 1))β€βŸ©)
5435, 36jctir 521 . . . . . 6 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚))
55 npcan 11465 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚) β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 2) = (β™―β€˜π‘Š))
5655eqcomd 2738 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚) β†’ (β™―β€˜π‘Š) = (((β™―β€˜π‘Š) βˆ’ 2) + 2))
572, 54, 563syl 18 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) = (((β™―β€˜π‘Š) βˆ’ 2) + 2))
5857adantr 481 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) = (((β™―β€˜π‘Š) βˆ’ 2) + 2))
5958opeq2d 4879 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩ = ⟨((β™―β€˜π‘Š) βˆ’ 2), (((β™―β€˜π‘Š) βˆ’ 2) + 2)⟩)
6059oveq2d 7421 . 2 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (((β™―β€˜π‘Š) βˆ’ 2) + 2)⟩))
61 eqidd 2733 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2)))
62 lsw 14510 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)))
6339, 43syl 17 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) βˆ’ (2 βˆ’ 1)) = (((β™―β€˜π‘Š) βˆ’ 2) + 1))
6463eqcomd 2738 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 1) = ((β™―β€˜π‘Š) βˆ’ (2 βˆ’ 1)))
65 2m1e1 12334 . . . . . . . . . . 11 (2 βˆ’ 1) = 1
6665a1i 11 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (2 βˆ’ 1) = 1)
6766oveq2d 7421 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) βˆ’ (2 βˆ’ 1)) = ((β™―β€˜π‘Š) βˆ’ 1))
6864, 67eqtrd 2772 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 1) = ((β™―β€˜π‘Š) βˆ’ 1))
692, 68syl 17 . . . . . . 7 (π‘Š ∈ Word 𝑉 β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 1) = ((β™―β€˜π‘Š) βˆ’ 1))
7069eqcomd 2738 . . . . . 6 (π‘Š ∈ Word 𝑉 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((β™―β€˜π‘Š) βˆ’ 2) + 1))
7170fveq2d 6892 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 1)) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 2) + 1)))
7262, 71eqtrd 2772 . . . 4 (π‘Š ∈ Word 𝑉 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 2) + 1)))
7372adantr 481 . . 3 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 2) + 1)))
7461, 73s2eqd 14810 . 2 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(lastSβ€˜π‘Š)β€βŸ© = βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 2) + 1))β€βŸ©)
7553, 60, 743eqtr4d 2782 1 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (π‘Š substr ⟨((β™―β€˜π‘Š) βˆ’ 2), (β™―β€˜π‘Š)⟩) = βŸ¨β€œ(π‘Šβ€˜((β™―β€˜π‘Š) βˆ’ 2))(lastSβ€˜π‘Š)β€βŸ©)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βŸ¨cop 4633   class class class wbr 5147  β€˜cfv 6540  (class class class)co 7405  β„‚cc 11104  β„cr 11105  0cc0 11106  1c1 11107   + caddc 11109   < clt 11244   ≀ cle 11245   βˆ’ cmin 11440  β„•cn 12208  2c2 12263  β„•0cn0 12468  β„€cz 12554  ..^cfzo 13623  β™―chash 14286  Word cword 14460  lastSclsw 14508   substr csubstr 14586  βŸ¨β€œcs2 14788
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-lsw 14509  df-concat 14517  df-s1 14542  df-substr 14587  df-s2 14795
This theorem is referenced by:  2swrd2eqwrdeq  14900
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