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Theorem lswccatn0lsw 14486
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 14470 . . . . . . 7 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ (β™―β€˜(𝐴 ++ 𝐡)) = ((β™―β€˜π΄) + (β™―β€˜π΅)))
21oveq1d 7377 . . . . . 6 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ ((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) = (((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ 1))
323adant3 1133 . . . . 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 14428 . . . . . . . . . 10 (𝐴 ∈ Word 𝑉 β†’ (β™―β€˜π΄) ∈ β„•0)
54nn0zd 12532 . . . . . . . . 9 (𝐴 ∈ Word 𝑉 β†’ (β™―β€˜π΄) ∈ β„€)
6 lennncl 14429 . . . . . . . . 9 ((𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (β™―β€˜π΅) ∈ β„•)
7 simpl 484 . . . . . . . . . 10 (((β™―β€˜π΄) ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (β™―β€˜π΄) ∈ β„€)
8 nnz 12527 . . . . . . . . . . 11 ((β™―β€˜π΅) ∈ β„• β†’ (β™―β€˜π΅) ∈ β„€)
9 zaddcl 12550 . . . . . . . . . . 11 (((β™―β€˜π΄) ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„€) β†’ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„€)
108, 9sylan2 594 . . . . . . . . . 10 (((β™―β€˜π΄) ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„•) β†’ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„€)
11 zre 12510 . . . . . . . . . . 11 ((β™―β€˜π΄) ∈ β„€ β†’ (β™―β€˜π΄) ∈ ℝ)
12 nnrp 12933 . . . . . . . . . . 11 ((β™―β€˜π΅) ∈ β„• β†’ (β™―β€˜π΅) ∈ ℝ+)
13 ltaddrp 12959 . . . . . . . . . . 11 (((β™―β€˜π΄) ∈ ℝ ∧ (β™―β€˜π΅) ∈ ℝ+) β†’ (β™―β€˜π΄) < ((β™―β€˜π΄) + (β™―β€˜π΅)))
1411, 12, 13syl2an 597 . . . . . . . . . 10 (((β™―β€˜π΄) ∈ β„€ ∧ (β™―β€˜π΅) ∈ β„•) β†’ (β™―β€˜π΄) < ((β™―β€˜π΄) + (β™―β€˜π΅)))
157, 10, 143jca 1129 . . . . . . . . 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 597 . . . . . . . 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 1116 . . . . . . 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 13585 . . . . . . 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 13670 . . . . . 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 2838 . . . 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 14473 . . . 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 1410 . . 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 7377 . . . . . 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 12482 . . . . . . 7 (𝐴 ∈ Word 𝑉 β†’ (β™―β€˜π΄) ∈ β„‚)
27 lencl 14428 . . . . . . . 8 (𝐡 ∈ Word 𝑉 β†’ (β™―β€˜π΅) ∈ β„•0)
2827nn0cnd 12482 . . . . . . 7 (𝐡 ∈ Word 𝑉 β†’ (β™―β€˜π΅) ∈ β„‚)
29 addcl 11140 . . . . . . . . 9 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ ((β™―β€˜π΄) + (β™―β€˜π΅)) ∈ β„‚)
30 1cnd 11157 . . . . . . . . 9 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ 1 ∈ β„‚)
31 simpl 484 . . . . . . . . 9 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ (β™―β€˜π΄) ∈ β„‚)
3229, 30, 31sub32d 11551 . . . . . . . 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 11415 . . . . . . . . 9 (((β™―β€˜π΄) ∈ β„‚ ∧ (β™―β€˜π΅) ∈ β„‚) β†’ (((β™―β€˜π΄) + (β™―β€˜π΅)) βˆ’ (β™―β€˜π΄)) = (β™―β€˜π΅))
3433oveq1d 7377 . . . . . . . 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 2777 . . . . . . 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 597 . . . . . 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 2777 . . . . 5 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉) β†’ (((β™―β€˜(𝐴 ++ 𝐡)) βˆ’ 1) βˆ’ (β™―β€˜π΄)) = ((β™―β€˜π΅) βˆ’ 1))
38373adant3 1133 . . . 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 6851 . . 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 2777 . 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 7395 . . 3 (𝐴 ++ 𝐡) ∈ V
42 lsw 14459 . . 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 14459 . . 3 (𝐡 ∈ Word 𝑉 β†’ (lastSβ€˜π΅) = (π΅β€˜((β™―β€˜π΅) βˆ’ 1)))
45443ad2ant2 1135 . 2 ((𝐴 ∈ Word 𝑉 ∧ 𝐡 ∈ Word 𝑉 ∧ 𝐡 β‰  βˆ…) β†’ (lastSβ€˜π΅) = (π΅β€˜((β™―β€˜π΅) βˆ’ 1)))
4640, 43, 453eqtr4d 2787 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  Vcvv 3448  βˆ…c0 4287   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  β„‚cc 11056  β„cr 11057  1c1 11059   + caddc 11061   < clt 11196   βˆ’ cmin 11392  β„•cn 12160  β„€cz 12506  β„+crp 12922  ..^cfzo 13574  β™―chash 14237  Word cword 14409  lastSclsw 14457   ++ cconcat 14465
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-n0 12421  df-z 12507  df-uz 12771  df-rp 12923  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-lsw 14458  df-concat 14466
This theorem is referenced by:  lswccats1  14529  clwwlkccat  28976
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