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Theorem constr3pthlem3 24859
Description: Lemma for constr3pth 24862. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
Hypotheses
Ref Expression
constr3cycl.f  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
constr3cycl.p  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
Assertion
Ref Expression
constr3pthlem3  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C  /\  C  =/=  A ) )  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  (/) )

Proof of Theorem constr3pthlem3
StepHypRef Expression
1 constr3cycl.f . . 3  |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. ,  <. 1 ,  ( `' E `  { B ,  C } ) >. ,  <. 2 ,  ( `' E `  { C ,  A } ) >. }
2 constr3cycl.p . . 3  |-  P  =  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  A >. } )
31, 2constr3lem2 24848 . 2  |-  ( # `  F )  =  3
4 preq2 4096 . . . . 5  |-  ( (
# `  F )  =  3  ->  { 0 ,  ( # `  F
) }  =  {
0 ,  3 } )
54imaeq2d 5325 . . . 4  |-  ( (
# `  F )  =  3  ->  ( P " { 0 ,  ( # `  F
) } )  =  ( P " {
0 ,  3 } ) )
6 oveq2 6278 . . . . 5  |-  ( (
# `  F )  =  3  ->  (
1..^ ( # `  F
) )  =  ( 1..^ 3 ) )
76imaeq2d 5325 . . . 4  |-  ( (
# `  F )  =  3  ->  ( P " ( 1..^ (
# `  F )
) )  =  ( P " ( 1..^ 3 ) ) )
85, 7ineq12d 3687 . . 3  |-  ( (
# `  F )  =  3  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  ( ( P " { 0 ,  3 } )  i^i  ( P "
( 1..^ 3 ) ) ) )
91, 2constr3lem6 24851 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C  /\  C  =/=  A ) )  ->  ( { ( P ` 
0 ) ,  ( P `  3 ) }  i^i  { ( P `  1 ) ,  ( P ` 
2 ) } )  =  (/) )
101, 2constr3trllem4 24855 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P : ( 0 ... 3 ) --> V )
11 ffn 5713 . . . . . . . . 9  |-  ( P : ( 0 ... 3 ) --> V  ->  P  Fn  ( 0 ... 3 ) )
1210, 11syl 16 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P  Fn  ( 0 ... 3 ) )
13 3nn0 10809 . . . . . . . . . 10  |-  3  e.  NN0
14 elnn0uz 11119 . . . . . . . . . 10  |-  ( 3  e.  NN0  <->  3  e.  (
ZZ>= `  0 ) )
1513, 14mpbi 208 . . . . . . . . 9  |-  3  e.  ( ZZ>= `  0 )
16 eluzfz1 11696 . . . . . . . . 9  |-  ( 3  e.  ( ZZ>= `  0
)  ->  0  e.  ( 0 ... 3
) )
1715, 16mp1i 12 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  0  e.  ( 0 ... 3 ) )
18 eluzfz2 11697 . . . . . . . . 9  |-  ( 3  e.  ( ZZ>= `  0
)  ->  3  e.  ( 0 ... 3
) )
1915, 18mp1i 12 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  3  e.  ( 0 ... 3 ) )
20 fnimapr 5912 . . . . . . . 8  |-  ( ( P  Fn  ( 0 ... 3 )  /\  0  e.  ( 0 ... 3 )  /\  3  e.  ( 0 ... 3 ) )  ->  ( P " { 0 ,  3 } )  =  {
( P `  0
) ,  ( P `
 3 ) } )
2112, 17, 19, 20syl3anc 1226 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P " {
0 ,  3 } )  =  { ( P `  0 ) ,  ( P ` 
3 ) } )
22 3z 10893 . . . . . . . . . . 11  |-  3  e.  ZZ
23 fzoval 11805 . . . . . . . . . . 11  |-  ( 3  e.  ZZ  ->  (
1..^ 3 )  =  ( 1 ... (
3  -  1 ) ) )
2422, 23ax-mp 5 . . . . . . . . . 10  |-  ( 1..^ 3 )  =  ( 1 ... ( 3  -  1 ) )
25 3m1e2 10648 . . . . . . . . . . 11  |-  ( 3  -  1 )  =  2
2625oveq2i 6281 . . . . . . . . . 10  |-  ( 1 ... ( 3  -  1 ) )  =  ( 1 ... 2
)
2724, 26eqtri 2483 . . . . . . . . 9  |-  ( 1..^ 3 )  =  ( 1 ... 2 )
2827imaeq2i 5323 . . . . . . . 8  |-  ( P
" ( 1..^ 3 ) )  =  ( P " ( 1 ... 2 ) )
29 df-2 10590 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
3029oveq2i 6281 . . . . . . . . . . 11  |-  ( 1 ... 2 )  =  ( 1 ... (
1  +  1 ) )
31 1z 10890 . . . . . . . . . . . 12  |-  1  e.  ZZ
32 fzpr 11739 . . . . . . . . . . . 12  |-  ( 1  e.  ZZ  ->  (
1 ... ( 1  +  1 ) )  =  { 1 ,  ( 1  +  1 ) } )
3331, 32ax-mp 5 . . . . . . . . . . 11  |-  ( 1 ... ( 1  +  1 ) )  =  { 1 ,  ( 1  +  1 ) }
34 1p1e2 10645 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
3534preq2i 4099 . . . . . . . . . . 11  |-  { 1 ,  ( 1  +  1 ) }  =  { 1 ,  2 }
3630, 33, 353eqtri 2487 . . . . . . . . . 10  |-  ( 1 ... 2 )  =  { 1 ,  2 }
3736imaeq2i 5323 . . . . . . . . 9  |-  ( P
" ( 1 ... 2 ) )  =  ( P " {
1 ,  2 } )
38 1eluzge0 11125 . . . . . . . . . . . 12  |-  1  e.  ( ZZ>= `  0 )
39 1le3 10748 . . . . . . . . . . . . 13  |-  1  <_  3
40 elfz5 11683 . . . . . . . . . . . . 13  |-  ( ( 1  e.  ( ZZ>= ` 
0 )  /\  3  e.  ZZ )  ->  (
1  e.  ( 0 ... 3 )  <->  1  <_  3 ) )
4139, 40mpbiri 233 . . . . . . . . . . . 12  |-  ( ( 1  e.  ( ZZ>= ` 
0 )  /\  3  e.  ZZ )  ->  1  e.  ( 0 ... 3
) )
4238, 22, 41mp2an 670 . . . . . . . . . . 11  |-  1  e.  ( 0 ... 3
)
4342a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  1  e.  ( 0 ... 3 ) )
44 2eluzge0 11126 . . . . . . . . . . . 12  |-  2  e.  ( ZZ>= `  0 )
45 2re 10601 . . . . . . . . . . . . . 14  |-  2  e.  RR
46 3re 10605 . . . . . . . . . . . . . 14  |-  3  e.  RR
47 2lt3 10699 . . . . . . . . . . . . . 14  |-  2  <  3
4845, 46, 47ltleii 9696 . . . . . . . . . . . . 13  |-  2  <_  3
49 elfz5 11683 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ( ZZ>= ` 
0 )  /\  3  e.  ZZ )  ->  (
2  e.  ( 0 ... 3 )  <->  2  <_  3 ) )
5048, 49mpbiri 233 . . . . . . . . . . . 12  |-  ( ( 2  e.  ( ZZ>= ` 
0 )  /\  3  e.  ZZ )  ->  2  e.  ( 0 ... 3
) )
5144, 22, 50mp2an 670 . . . . . . . . . . 11  |-  2  e.  ( 0 ... 3
)
5251a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  2  e.  ( 0 ... 3 ) )
53 fnimapr 5912 . . . . . . . . . 10  |-  ( ( P  Fn  ( 0 ... 3 )  /\  1  e.  ( 0 ... 3 )  /\  2  e.  ( 0 ... 3 ) )  ->  ( P " { 1 ,  2 } )  =  {
( P `  1
) ,  ( P `
 2 ) } )
5412, 43, 52, 53syl3anc 1226 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P " {
1 ,  2 } )  =  { ( P `  1 ) ,  ( P ` 
2 ) } )
5537, 54syl5eq 2507 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P " (
1 ... 2 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )
5628, 55syl5eq 2507 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P " (
1..^ 3 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )
5721, 56ineq12d 3687 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( P " { 0 ,  3 } )  i^i  ( P " ( 1..^ 3 ) ) )  =  ( { ( P `
 0 ) ,  ( P `  3
) }  i^i  {
( P `  1
) ,  ( P `
 2 ) } ) )
5857eqeq1d 2456 . . . . 5  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( ( ( P
" { 0 ,  3 } )  i^i  ( P " (
1..^ 3 ) ) )  =  (/)  <->  ( {
( P `  0
) ,  ( P `
 3 ) }  i^i  { ( P `
 1 ) ,  ( P `  2
) } )  =  (/) ) )
5958adantr 463 . . . 4  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C  /\  C  =/=  A ) )  ->  (
( ( P " { 0 ,  3 } )  i^i  ( P " ( 1..^ 3 ) ) )  =  (/) 
<->  ( { ( P `
 0 ) ,  ( P `  3
) }  i^i  {
( P `  1
) ,  ( P `
 2 ) } )  =  (/) ) )
609, 59mpbird 232 . . 3  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C  /\  C  =/=  A ) )  ->  (
( P " {
0 ,  3 } )  i^i  ( P
" ( 1..^ 3 ) ) )  =  (/) )
618, 60sylan9eq 2515 . 2  |-  ( ( ( # `  F
)  =  3  /\  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/= 
A ) ) )  ->  ( ( P
" { 0 ,  ( # `  F
) } )  i^i  ( P " (
1..^ ( # `  F
) ) ) )  =  (/) )
623, 61mpan 668 1  |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V
)  /\  ( A  =/=  B  /\  B  =/= 
C  /\  C  =/=  A ) )  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649    u. cun 3459    i^i cin 3460   (/)c0 3783   {cpr 4018   {ctp 4020   <.cop 4022   class class class wbr 4439   `'ccnv 4987   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   0cc0 9481   1c1 9482    + caddc 9484    <_ cle 9618    - cmin 9796   2c2 10581   3c3 10582   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11082   ...cfz 11675  ..^cfzo 11799   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-fzo 11800  df-hash 12388
This theorem is referenced by:  constr3pth  24862
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