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Theorem s4prop 12636
Description: A length 4 word is a union of two unordered pairs of ordered pairs. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
Assertion
Ref Expression
s4prop  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <" A B C D ">  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  D >. } ) )

Proof of Theorem s4prop
StepHypRef Expression
1 df-s4 12588 . 2  |-  <" A B C D ">  =  ( <" A B C "> concat  <" D "> )
2 simpl 457 . . . . . . 7  |-  ( ( A  e.  S  /\  B  e.  S )  ->  A  e.  S )
32adantr 465 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  A  e.  S )
4 simpr 461 . . . . . . 7  |-  ( ( A  e.  S  /\  B  e.  S )  ->  B  e.  S )
54adantr 465 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  B  e.  S )
6 simpl 457 . . . . . . 7  |-  ( ( C  e.  S  /\  D  e.  S )  ->  C  e.  S )
76adantl 466 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  C  e.  S )
83, 5, 7s3cld 12608 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <" A B C ">  e. Word  S
)
9 simpr 461 . . . . . 6  |-  ( ( C  e.  S  /\  D  e.  S )  ->  D  e.  S )
109adantl 466 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  D  e.  S )
11 cats1un 12481 . . . . 5  |-  ( (
<" A B C ">  e. Word  S  /\  D  e.  S
)  ->  ( <" A B C "> concat 
<" D "> )  =  ( <" A B C ">  u.  { <. ( # `
 <" A B C "> ) ,  D >. } ) )
128, 10, 11syl2anc 661 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <" A B C "> concat  <" D "> )  =  (
<" A B C ">  u.  { <. ( # `  <" A B C "> ) ,  D >. } ) )
13 df-s3 12587 . . . . . . 7  |-  <" A B C ">  =  ( <" A B "> concat  <" C "> )
14 s2cl 12614 . . . . . . . . 9  |-  ( ( A  e.  S  /\  B  e.  S )  ->  <" A B ">  e. Word  S
)
1514adantr 465 . . . . . . . 8  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <" A B ">  e. Word  S )
16 cats1un 12481 . . . . . . . 8  |-  ( (
<" A B ">  e. Word  S  /\  C  e.  S )  ->  ( <" A B "> concat 
<" C "> )  =  ( <" A B ">  u. 
{ <. ( # `  <" A B "> ) ,  C >. } ) )
1715, 7, 16syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <" A B "> concat  <" C "> )  =  (
<" A B ">  u.  { <. ( # `
 <" A B "> ) ,  C >. } ) )
1813, 17syl5eq 2504 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <" A B C ">  =  (
<" A B ">  u.  { <. ( # `
 <" A B "> ) ,  C >. } ) )
19 s2prop 12635 . . . . . . . 8  |-  ( ( A  e.  S  /\  B  e.  S )  ->  <" A B ">  =  { <. 0 ,  A >. , 
<. 1 ,  B >. } )
2019adantr 465 . . . . . . 7  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <" A B ">  =  { <. 0 ,  A >. ,  <. 1 ,  B >. } )
2120uneq1d 3610 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <" A B ">  u.  { <. ( # `  <" A B "> ) ,  C >. } )  =  ( {
<. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. (
# `  <" A B "> ) ,  C >. } ) )
2218, 21eqtrd 2492 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <" A B C ">  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. (
# `  <" A B "> ) ,  C >. } ) )
2322uneq1d 3610 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <" A B C ">  u.  {
<. ( # `  <" A B C "> ) ,  D >. } )  =  ( ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. (
# `  <" A B "> ) ,  C >. } )  u. 
{ <. ( # `  <" A B C "> ) ,  D >. } ) )
2412, 23eqtrd 2492 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <" A B C "> concat  <" D "> )  =  ( ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. ( # `  <" A B "> ) ,  C >. } )  u.  { <. (
# `  <" A B C "> ) ,  D >. } ) )
25 unass 3614 . . . 4  |-  ( ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. (
# `  <" A B "> ) ,  C >. } )  u. 
{ <. ( # `  <" A B C "> ) ,  D >. } )  =  ( {
<. 0 ,  A >. ,  <. 1 ,  B >. }  u.  ( {
<. ( # `  <" A B "> ) ,  C >. }  u.  { <. ( # `
 <" A B C "> ) ,  D >. } ) )
2625a1i 11 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( ( { <. 0 ,  A >. , 
<. 1 ,  B >. }  u.  { <. (
# `  <" A B "> ) ,  C >. } )  u. 
{ <. ( # `  <" A B C "> ) ,  D >. } )  =  ( {
<. 0 ,  A >. ,  <. 1 ,  B >. }  u.  ( {
<. ( # `  <" A B "> ) ,  C >. }  u.  { <. ( # `
 <" A B C "> ) ,  D >. } ) ) )
27 df-pr 3981 . . . . 5  |-  { <. (
# `  <" A B "> ) ,  C >. ,  <. ( # `
 <" A B C "> ) ,  D >. }  =  ( { <. ( # `  <" A B "> ) ,  C >. }  u.  { <. ( # `
 <" A B C "> ) ,  D >. } )
28 s2len 12625 . . . . . . . 8  |-  ( # `  <" A B "> )  =  2
2928a1i 11 . . . . . . 7  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( # `  <" A B "> )  =  2 )
3029opeq1d 4166 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <. ( # `  <" A B "> ) ,  C >.  = 
<. 2 ,  C >. )
31 s3len 12629 . . . . . . . 8  |-  ( # `  <" A B C "> )  =  3
3231a1i 11 . . . . . . 7  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( # `  <" A B C "> )  =  3 )
3332opeq1d 4166 . . . . . 6  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <. ( # `  <" A B C "> ) ,  D >.  = 
<. 3 ,  D >. )
3430, 33preq12d 4063 . . . . 5  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  { <. ( # `  <" A B "> ) ,  C >. , 
<. ( # `  <" A B C "> ) ,  D >. }  =  { <. 2 ,  C >. ,  <. 3 ,  D >. } )
3527, 34syl5eqr 2506 . . . 4  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( { <. ( # `
 <" A B "> ) ,  C >. }  u.  { <. ( # `  <" A B C "> ) ,  D >. } )  =  { <. 2 ,  C >. , 
<. 3 ,  D >. } )
3635uneq2d 3611 . . 3  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  ( { <. ( # `  <" A B "> ) ,  C >. }  u.  { <. ( # `
 <" A B C "> ) ,  D >. } ) )  =  ( { <. 0 ,  A >. , 
<. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  D >. } ) )
3724, 26, 363eqtrd 2496 . 2  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  -> 
( <" A B C "> concat  <" D "> )  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  D >. } ) )
381, 37syl5eq 2504 1  |-  ( ( ( A  e.  S  /\  B  e.  S
)  /\  ( C  e.  S  /\  D  e.  S ) )  ->  <" A B C D ">  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  D >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3427   {csn 3978   {cpr 3980   <.cop 3984   ` cfv 5519  (class class class)co 6193   0cc0 9386   1c1 9387   2c2 10475   3c3 10476   #chash 12213  Word cword 12332   concat cconcat 12334   <"cs1 12335   <"cs2 12579   <"cs3 12580   <"cs4 12581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-1o 7023  df-oadd 7027  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-card 8213  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-3 10485  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-fzo 11659  df-hash 12214  df-word 12340  df-concat 12342  df-s1 12343  df-s2 12586  df-s3 12587  df-s4 12588
This theorem is referenced by:  s4f1o  12639  s4dom  12640
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