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Theorem lswccatn0lsw 14545
Description: The last symbol of a word concatenated with a nonempty word is the last symbol of the nonempty word. (Contributed by AV, 22-Oct-2018.) (Proof shortened by AV, 1-May-2020.)
Assertion
Ref Expression
lswccatn0lsw ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (lastSβ€˜(𝐴 ++ 𝐡)) = (lastSβ€˜π΅))

Proof of Theorem lswccatn0lsw
StepHypRef Expression
1 ccatlen 14529 . . . . . . 7 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ (β™―β€˜(𝐴 ++ 𝐡)) = ((β™―β€˜π΄) + (β™―β€˜π΅)))
21oveq1d 7426 . . . . . 6 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ ((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) = (((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ 1))
323adant3 1130 . . . . 5 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ ((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) = (((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ 1))
4 lencl 14487 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 β†’ (β™―β€˜π΄) ∈ β„•0)
54nn0zd 12588 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 β†’ (β™―β€˜π΄) ∈ β„€)
6 lennncl 14488 . . . . . . . . 9 ((𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (β™―β€˜π΅) ∈ β„•)
7 simpl 481 . . . . . . . . . 10 (((β™―β€˜π΄) ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (β™―β€˜π΄) ∈ β„€)
8 nnz 12583 . . . . . . . . . . 11 ((β™―β€˜π΅) ∈ β„• β†’ (β™―β€˜π΅) ∈ β„€)
9 zaddcl 12606 . . . . . . . . . . 11 (((β™―β€˜π΄) ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„€) β†’ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„€)
108, 9sylan2 591 . . . . . . . . . 10 (((β™―β€˜π΄) ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„•) β†’ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„€)
11 zre 12566 . . . . . . . . . . 11 ((β™―β€˜π΄) ∈ β„€ β†’ (β™―β€˜π΄) ∈ ℝ)
12 nnrp 12989 . . . . . . . . . . 11 ((β™―β€˜π΅) ∈ β„• β†’ (β™―β€˜π΅) ∈ ℝ+)
13 ltaddrp 13015 . . . . . . . . . . 11 (((β™―β€˜π΄) ∈ ℝ ∧ (β™―β€˜π΅) ∈ ℝ+) β†’ (β™―β€˜π΄) < ((β™―β€˜π΄) + (β™―β€˜π΅)))
1411, 12, 13syl2an 594 . . . . . . . . . 10 (((β™―β€˜π΄) ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (β™―β€˜π΄) < ((β™―β€˜π΄) + (β™―β€˜π΅)))
157, 10, 143jca 1126 . . . . . . . . 9 (((β™―β€˜π΄) ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„•) β†’ ((β™―β€˜π΄) ∈ β„€ ∧ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„€ ∧ (β™―β€˜π΄) < ((β™―β€˜π΄) + (β™―β€˜π΅))))
165, 6, 15syl2an 594 . . . . . . . 8 ((𝐴 ∈ Word 𝑉 ∧ (𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…)) β†’ ((β™―β€˜π΄) ∈ β„€ ∧ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„€ ∧ (β™―β€˜π΄) < ((β™―β€˜π΄) + (β™―β€˜π΅))))
17163impb 1113 . . . . . . 7 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ ((β™―β€˜π΄) ∈ β„€ ∧ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„€ ∧ (β™―β€˜π΄) < ((β™―β€˜π΄) + (β™―β€˜π΅))))
18 fzolb 13642 . . . . . . 7 ((β™―β€˜π΄) ∈ ((β™―β€˜π΄)..^((β™―β€˜π΄) + (β™―β€˜π΅))) ↔ ((β™―β€˜π΄) ∈ β„€ ∧ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„€ ∧ (β™―β€˜π΄) < ((β™―β€˜π΄) + (β™―β€˜π΅))))
1917, 18sylibr 233 . . . . . 6 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (β™―β€˜π΄) ∈ ((β™―β€˜π΄)..^((β™―β€˜π΄) + (β™―β€˜π΅))))
20 fzoend 13727 . . . . . 6 ((β™―β€˜π΄) ∈ ((β™―β€˜π΄)..^((β™―β€˜π΄) + (β™―β€˜π΅))) β†’ (((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ 1) ∈ ((β™―β€˜π΄)..^((β™―β€˜π΄) + (β™―β€˜π΅))))
2119, 20syl 17 . . . . 5 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ 1) ∈ ((β™―β€˜π΄)..^((β™―β€˜π΄) + (β™―β€˜π΅))))
223, 21eqeltrd 2831 . . . 4 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ ((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) ∈ ((β™―β€˜π΄)..^((β™―β€˜π΄) + (β™―β€˜π΅))))
23 ccatval2 14532 . . . 4 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ ((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) ∈ ((β™―β€˜π΄)..^((β™―β€˜π΄) + (β™―β€˜π΅)))) β†’ ((𝐴 ++ 𝐡)β€˜((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1)) = (π΅β€˜(((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) βˆ’ (β™―β€˜π΄))))
2422, 23syld3an3 1407 . . 3 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ ((𝐴 ++ 𝐡)β€˜((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1)) = (π΅β€˜(((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) βˆ’ (β™―β€˜π΄))))
252oveq1d 7426 . . . . . 6 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ (((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) βˆ’ (β™―β€˜π΄)) = ((((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ 1) βˆ’ (β™―β€˜π΄)))
264nn0cnd 12538 . . . . . . 7 (𝐴 ∈ Word 𝑉 β†’ (β™―β€˜π΄) ∈ β„‚)
27 lencl 14487 . . . . . . . 8 (𝐡 ∈ Word 𝑉 β†’ (β™―β€˜π΅) ∈ β„•0)
2827nn0cnd 12538 . . . . . . 7 (𝐡 ∈ Word 𝑉 β†’ (β™―β€˜π΅) ∈ β„‚)
29 addcl 11194 . . . . . . . . 9 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„‚)
30 1cnd 11213 . . . . . . . . 9 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ 1 ∈ β„‚)
31 simpl 481 . . . . . . . . 9 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ (β™―β€˜π΄) ∈ β„‚)
3229, 30, 31sub32d 11607 . . . . . . . 8 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ ((((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ 1) βˆ’ (β™―β€˜π΄)) = ((((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ (β™―β€˜π΄)) βˆ’ 1))
33 pncan2 11471 . . . . . . . . 9 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ (((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ (β™―β€˜π΄)) = (β™―β€˜π΅))
3433oveq1d 7426 . . . . . . . 8 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ ((((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ (β™―β€˜π΄)) βˆ’ 1) = ((β™―β€˜π΅) βˆ’ 1))
3532, 34eqtrd 2770 . . . . . . 7 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ ((((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ 1) βˆ’ (β™―β€˜π΄)) = ((β™―β€˜π΅) βˆ’ 1))
3626, 28, 35syl2an 594 . . . . . 6 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ ((((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ 1) βˆ’ (β™―β€˜π΄)) = ((β™―β€˜π΅) βˆ’ 1))
3725, 36eqtrd 2770 . . . . 5 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ (((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) βˆ’ (β™―β€˜π΄)) = ((β™―β€˜π΅) βˆ’ 1))
38373adant3 1130 . . . 4 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) βˆ’ (β™―β€˜π΄)) = ((β™―β€˜π΅) βˆ’ 1))
3938fveq2d 6894 . . 3 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (π΅β€˜(((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) βˆ’ (β™―β€˜π΄))) = (π΅β€˜((β™―β€˜π΅) βˆ’ 1)))
4024, 39eqtrd 2770 . 2 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ ((𝐴 ++ 𝐡)β€˜((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1)) = (π΅β€˜((β™―β€˜π΅) βˆ’ 1)))
41 ovex 7444 . . 3 (𝐴 ++ 𝐡) ∈ V
42 lsw 14518 . . 3 ((𝐴 ++ 𝐡) ∈ V β†’ (lastSβ€˜(𝐴 ++ 𝐡)) = ((𝐴 ++ 𝐡)β€˜((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1)))
4341, 42mp1i 13 . 2 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (lastSβ€˜(𝐴 ++ 𝐡)) = ((𝐴 ++ 𝐡)β€˜((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1)))
44 lsw 14518 . . 3 (𝐡 ∈ Word 𝑉 β†’ (lastSβ€˜π΅) = (π΅β€˜((β™―β€˜π΅) βˆ’ 1)))
45443ad2ant2 1132 . 2 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (lastSβ€˜π΅) = (π΅β€˜((β™―β€˜π΅) βˆ’ 1)))
4640, 43, 453eqtr4d 2780 1 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (lastSβ€˜(𝐴 ++ 𝐡)) = (lastSβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  Vcvv 3472  βˆ…c0 4321   class class class wbr 5147  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  β„cr 11111  1c1 11113   + caddc 11115   < clt 11252   βˆ’ cmin 11448  β„•cn 12216  β„€cz 12562  β„+crp 12978  ..^cfzo 13631  β™―chash 14294  Word cword 14468  lastSclsw 14516   ++ cconcat 14524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  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 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  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 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  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-n0 12477  df-z 12563  df-uz 12827  df-rp 12979  df-fz 13489  df-fzo 13632  df-hash 14295  df-word 14469  df-lsw 14517  df-concat 14525
This theorem is referenced by:  lswccats1  14588  clwwlkccat  29510
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