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Theorem swrd2lsw 14848
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 484 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ π‘Š ∈ Word 𝑉)
2 lencl 14428 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„•0)
3 1z 12540 . . . . . . . . 9 1 ∈ β„€
4 nn0z 12531 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„€)
5 zltp1le 12560 . . . . . . . . 9 ((1 ∈ β„€ ∧ (β™―β€˜π‘Š) ∈ β„€) β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
63, 4, 5sylancr 588 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
7 1p1e2 12285 . . . . . . . . . . 11 (1 + 1) = 2
87a1i 11 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 + 1) = 2)
98breq1d 5120 . . . . . . . . 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 408 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ 2 ≀ (β™―β€˜π‘Š))
13 2nn0 12437 . . . . . . . . 9 2 ∈ β„•0
1413jctl 525 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0))
1514adantr 482 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0))
16 nn0sub 12470 . . . . . . 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 581 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0)
20 0red 11165 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ 0 ∈ ℝ)
21 1red 11163 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ 1 ∈ ℝ)
22 zre 12510 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ (β™―β€˜π‘Š) ∈ ℝ)
2320, 21, 223jca 1129 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„€ β†’ (0 ∈ ℝ ∧ 1 ∈ ℝ ∧ (β™―β€˜π‘Š) ∈ ℝ))
24 0lt1 11684 . . . . . . . . . . 11 0 < 1
25 lttr 11238 . . . . . . . . . . . 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 417 . . . . . . . . . . 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 12516 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„• ↔ ((β™―β€˜π‘Š) ∈ β„€ ∧ 0 < (β™―β€˜π‘Š)))
2928simplbi2 502 . . . . . . . . . 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 408 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
33 fzo0end 13671 . . . . . . 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 12430 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„‚)
36 2cn 12235 . . . . . . . . . . . 12 2 ∈ β„‚
3736a1i 11 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 2 ∈ β„‚)
38 1cnd 11157 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 1 ∈ β„‚)
3935, 37, 383jca 1129 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 1 ∈ β„‚))
40 1e2m1 12287 . . . . . . . . . . . . 13 1 = (2 βˆ’ 1)
4140a1i 11 . . . . . . . . . . . 12 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚ ∧ 1 ∈ β„‚) β†’ 1 = (2 βˆ’ 1))
4241oveq2d 7378 . . . . . . . . . . 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 11438 . . . . . . . . . . 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 2777 . . . . . . . . . 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 2743 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 1) = ((β™―β€˜π‘Š) βˆ’ 1))
4746eleq1d 2823 . . . . . . 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 482 . . . . . 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 257 . . . . 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 581 . . . 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 1129 . . 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 14836 . . 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 522 . . . . . 6 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚))
55 npcan 11417 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚) β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 2) = (β™―β€˜π‘Š))
5655eqcomd 2743 . . . . . 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 482 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) = (((β™―β€˜π‘Š) βˆ’ 2) + 2))
5958opeq2d 4842 . . 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 7378 . 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 2738 . . 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 14459 . . . . 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 2743 . . . . . . . . 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 12286 . . . . . . . . . . 11 (2 βˆ’ 1) = 1
6665a1i 11 . . . . . . . . . 10 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (2 βˆ’ 1) = 1)
6766oveq2d 7378 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) βˆ’ (2 βˆ’ 1)) = ((β™―β€˜π‘Š) βˆ’ 1))
6864, 67eqtrd 2777 . . . . . . . 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 2743 . . . . . 6 (π‘Š ∈ Word 𝑉 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((β™―β€˜π‘Š) βˆ’ 2) + 1))
7170fveq2d 6851 . . . . 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 2777 . . . 4 (π‘Š ∈ Word 𝑉 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 2) + 1)))
7372adantr 482 . . 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 14759 . 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 2787 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4597   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  β„cr 11057  0cc0 11058  1c1 11059   + caddc 11061   < clt 11196   ≀ cle 11197   βˆ’ cmin 11392  β„•cn 12160  2c2 12215  β„•0cn0 12420  β„€cz 12506  ..^cfzo 13574  β™―chash 14237  Word cword 14409  lastSclsw 14457   substr csubstr 14535  βŸ¨β€œcs2 14737
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-lsw 14458  df-concat 14466  df-s1 14491  df-substr 14536  df-s2 14744
This theorem is referenced by:  2swrd2eqwrdeq  14849
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