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Theorem swrd2lsw 14909
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 482 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ π‘Š ∈ Word 𝑉)
2 lencl 14489 . . . . 5 (π‘Š ∈ Word 𝑉 β†’ (β™―β€˜π‘Š) ∈ β„•0)
3 1z 12596 . . . . . . . . 9 1 ∈ β„€
4 nn0z 12587 . . . . . . . . 9 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„€)
5 zltp1le 12616 . . . . . . . . 9 ((1 ∈ β„€ ∧ (β™―β€˜π‘Š) ∈ β„€) β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
63, 4, 5sylancr 586 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (1 < (β™―β€˜π‘Š) ↔ (1 + 1) ≀ (β™―β€˜π‘Š)))
7 1p1e2 12341 . . . . . . . . . . 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 406 . . . . . 6 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ 2 ≀ (β™―β€˜π‘Š))
13 2nn0 12493 . . . . . . . . 9 2 ∈ β„•0
1413jctl 523 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0))
1514adantr 480 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (2 ∈ β„•0 ∧ (β™―β€˜π‘Š) ∈ β„•0))
16 nn0sub 12526 . . . . . . 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 579 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ ((β™―β€˜π‘Š) βˆ’ 2) ∈ β„•0)
20 0red 11221 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ 0 ∈ ℝ)
21 1red 11219 . . . . . . . . . . . 12 ((β™―β€˜π‘Š) ∈ β„€ β†’ 1 ∈ ℝ)
22 zre 12566 . . . . . . . . . . . 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 11740 . . . . . . . . . . 11 0 < 1
25 lttr 11294 . . . . . . . . . . . 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 415 . . . . . . . . . . 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 12572 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„• ↔ ((β™―β€˜π‘Š) ∈ β„€ ∧ 0 < (β™―β€˜π‘Š)))
2928simplbi2 500 . . . . . . . . . 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 406 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„•0 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) ∈ β„•)
33 fzo0end 13730 . . . . . . 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 12486 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (β™―β€˜π‘Š) ∈ β„‚)
36 2cn 12291 . . . . . . . . . . . 12 2 ∈ β„‚
3736a1i 11 . . . . . . . . . . 11 ((β™―β€˜π‘Š) ∈ β„•0 β†’ 2 ∈ β„‚)
38 1cnd 11213 . . . . . . . . . . 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 12343 . . . . . . . . . . . . 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 11494 . . . . . . . . . . 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 2766 . . . . . . . . . 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 2732 . . . . . . . 8 ((β™―β€˜π‘Š) ∈ β„•0 β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 1) = ((β™―β€˜π‘Š) βˆ’ 1))
4746eleq1d 2812 . . . . . . 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 480 . . . . . 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 579 . . . 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 14897 . . 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 520 . . . . . 6 ((β™―β€˜π‘Š) ∈ β„•0 β†’ ((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚))
55 npcan 11473 . . . . . . 7 (((β™―β€˜π‘Š) ∈ β„‚ ∧ 2 ∈ β„‚) β†’ (((β™―β€˜π‘Š) βˆ’ 2) + 2) = (β™―β€˜π‘Š))
5655eqcomd 2732 . . . . . 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 480 . . . 4 ((π‘Š ∈ Word 𝑉 ∧ 1 < (β™―β€˜π‘Š)) β†’ (β™―β€˜π‘Š) = (((β™―β€˜π‘Š) βˆ’ 2) + 2))
5958opeq2d 4875 . . 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 2727 . . 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 14520 . . . . 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 2732 . . . . . . . . 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 12342 . . . . . . . . . . 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 2766 . . . . . . . 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 2732 . . . . . 6 (π‘Š ∈ Word 𝑉 β†’ ((β™―β€˜π‘Š) βˆ’ 1) = (((β™―β€˜π‘Š) βˆ’ 2) + 1))
7170fveq2d 6889 . . . . 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 2766 . . . 4 (π‘Š ∈ Word 𝑉 β†’ (lastSβ€˜π‘Š) = (π‘Šβ€˜(((β™―β€˜π‘Š) βˆ’ 2) + 1)))
7372adantr 480 . . 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 14820 . 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 2776 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4629   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  β„‚cc 11110  β„cr 11111  0cc0 11112  1c1 11113   + caddc 11115   < clt 11252   ≀ cle 11253   βˆ’ cmin 11448  β„•cn 12216  2c2 12271  β„•0cn0 12476  β„€cz 12562  ..^cfzo 13633  β™―chash 14295  Word cword 14470  lastSclsw 14518   substr csubstr 14596  βŸ¨β€œcs2 14798
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-hash 14296  df-word 14471  df-lsw 14519  df-concat 14527  df-s1 14552  df-substr 14597  df-s2 14805
This theorem is referenced by:  2swrd2eqwrdeq  14910
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