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Theorem swrd2lsw 14933
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 481 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ π‘Š ∈ Word 𝑉)
2 lencl 14513 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„•0)
3 1z 12620 . . . . . . . . 9 1 ∈ β„€
4 nn0z 12611 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„€)
5 zltp1le 12640 . . . . . . . . 9 ((1 ∈ β„€ ∧ (β™―β€˜π‘Š) ∈ β„€) β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
63, 4, 5sylancr 585 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
7 1p1e2 12365 . . . . . . . . . . 11 (1 + 1) = 2
87a1i 11 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 + 1) = 2)
98breq1d 5151 . . . . . . . . 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 405 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ 2 ≀ (β™―β€˜π‘Š))
13 2nn0 12517 . . . . . . . . 9 2 ∈ β„•0
1413jctl 522 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0))
1514adantr 479 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0))
16 nn0sub 12550 . . . . . . 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 578 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0)
20 0red 11245 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ 0 ∈ ℝ)
21 1red 11243 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ 1 ∈ ℝ)
22 zre 12590 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ (β™―β€˜π‘Š) ∈ ℝ)
2320, 21, 223jca 1125 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„€ β†’ (0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (β™―β€˜π‘Š) ∈ ℝ))
24 0lt1 11764 . . . . . . . . . . 11 0 < 1
25 lttr 11318 . . . . . . . . . . . 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 414 . . . . . . . . . . 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 12596 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„• ↔ ((β™―β€˜π‘Š) ∈ β„€ ∧ 0 < (β™―β€˜π‘Š)))
2928simplbi2 499 . . . . . . . . . 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 405 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
33 fzo0end 13754 . . . . . . 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 12510 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„‚)
36 2cn 12315 . . . . . . . . . . . 12 2 ∈ β„‚
3736a1i 11 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 2 ∈ β„‚)
38 1cnd 11237 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 1 ∈ β„‚)
3935, 37, 383jca 1125 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 1 ∈ β„‚))
40 1e2m1 12367 . . . . . . . . . . . . 13 1 = (2 βˆ’ 1)
4140a1i 11 . . . . . . . . . . . 12 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ 1 = (2 βˆ’ 1))
4241oveq2d 7430 . . . . . . . . . . 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 11518 . . . . . . . . . . 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 2765 . . . . . . . . . 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 2731 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 1) = ((β™―β€˜π‘Š) βˆ’ 1))
4746eleq1d 2810 . . . . . . 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 479 . . . . . 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 578 . . . 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 1125 . . 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 14921 . . 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 519 . . . . . 6 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚))
55 npcan 11497 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚) β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 2) = (β™―β€˜π‘Š))
5655eqcomd 2731 . . . . . 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 479 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) = (((β™―β€˜π‘Š) βˆ’ 2) + 2))
5958opeq2d 4874 . . 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 7430 . 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 2726 . . 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 14544 . . . . 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 2731 . . . . . . . . 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 12366 . . . . . . . . . . 11 (2 βˆ’ 1) = 1
6665a1i 11 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (2 βˆ’ 1) = 1)
6766oveq2d 7430 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) βˆ’ (2 βˆ’ 1)) = ((β™―β€˜π‘Š) βˆ’ 1))
6864, 67eqtrd 2765 . . . . . . . 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 2731 . . . . . 6 (π‘Š ∈ Word 𝑉 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((β™―β€˜π‘Š) βˆ’ 2) + 1))
7170fveq2d 6894 . . . . 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 2765 . . . 4 (π‘Š ∈ Word 𝑉 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 2) + 1)))
7372adantr 479 . . 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 14844 . 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 2775 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4628   class class class wbr 5141  β€˜cfv 6541  (class class class)co 7414  β„‚cc 11134  β„cr 11135  0cc0 11136  1c1 11137   + caddc 11139   < clt 11276   ≀ cle 11277   βˆ’ cmin 11472  β„•cn 12240  2c2 12295  β„•0cn0 12500  β„€cz 12586  ..^cfzo 13657  β™―chash 14319  Word cword 14494  lastSclsw 14542   substr csubstr 14620  βŸ¨β€œcs2 14822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4943  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-1o 8483  df-er 8721  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-card 9960  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-2 12303  df-n0 12501  df-z 12587  df-uz 12851  df-fz 13515  df-fzo 13658  df-hash 14320  df-word 14495  df-lsw 14543  df-concat 14551  df-s1 14576  df-substr 14621  df-s2 14829
This theorem is referenced by:  2swrd2eqwrdeq  14934
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