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Theorem constr3pthlem3 23688
Description: Lemma for constr3pth 23691. (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 23677 . 2  |-  ( # `  F )  =  3
4 preq2 4056 . . . . 5  |-  ( (
# `  F )  =  3  ->  { 0 ,  ( # `  F
) }  =  {
0 ,  3 } )
54imaeq2d 5270 . . . 4  |-  ( (
# `  F )  =  3  ->  ( P " { 0 ,  ( # `  F
) } )  =  ( P " {
0 ,  3 } ) )
6 oveq2 6201 . . . . 5  |-  ( (
# `  F )  =  3  ->  (
1..^ ( # `  F
) )  =  ( 1..^ 3 ) )
76imaeq2d 5270 . . . 4  |-  ( (
# `  F )  =  3  ->  ( P " ( 1..^ (
# `  F )
) )  =  ( P " ( 1..^ 3 ) ) )
85, 7ineq12d 3654 . . 3  |-  ( (
# `  F )  =  3  ->  (
( P " {
0 ,  ( # `  F ) } )  i^i  ( P "
( 1..^ ( # `  F ) ) ) )  =  ( ( P " { 0 ,  3 } )  i^i  ( P "
( 1..^ 3 ) ) ) )
91, 2constr3lem6 23680 . . . 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 23684 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P : ( 0 ... 3 ) --> V )
11 ffn 5660 . . . . . . . . 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 10701 . . . . . . . . . 10  |-  3  e.  NN0
14 elnn0uz 11002 . . . . . . . . . 10  |-  ( 3  e.  NN0  <->  3  e.  (
ZZ>= `  0 ) )
1513, 14mpbi 208 . . . . . . . . 9  |-  3  e.  ( ZZ>= `  0 )
16 eluzfz1 11568 . . . . . . . . 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 11569 . . . . . . . . 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 5857 . . . . . . . 8  |-  ( ( P  Fn  ( 0 ... 3 )  /\  0  e.  ( 0 ... 3 )  /\  3  e.  ( 0 ... 3 ) )  ->  ( P " { 0 ,  3 } )  =  {
( P `  0
) ,  ( P `
 3 ) } )
2112, 17, 19, 20syl3anc 1219 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P " {
0 ,  3 } )  =  { ( P `  0 ) ,  ( P ` 
3 ) } )
22 3z 10783 . . . . . . . . . . 11  |-  3  e.  ZZ
23 fzoval 11664 . . . . . . . . . . 11  |-  ( 3  e.  ZZ  ->  (
1..^ 3 )  =  ( 1 ... (
3  -  1 ) ) )
2422, 23ax-mp 5 . . . . . . . . . 10  |-  ( 1..^ 3 )  =  ( 1 ... ( 3  -  1 ) )
25 3m1e2 10542 . . . . . . . . . . 11  |-  ( 3  -  1 )  =  2
2625oveq2i 6204 . . . . . . . . . 10  |-  ( 1 ... ( 3  -  1 ) )  =  ( 1 ... 2
)
2724, 26eqtri 2480 . . . . . . . . 9  |-  ( 1..^ 3 )  =  ( 1 ... 2 )
2827imaeq2i 5268 . . . . . . . 8  |-  ( P
" ( 1..^ 3 ) )  =  ( P " ( 1 ... 2 ) )
29 df-2 10484 . . . . . . . . . . . 12  |-  2  =  ( 1  +  1 )
3029oveq2i 6204 . . . . . . . . . . 11  |-  ( 1 ... 2 )  =  ( 1 ... (
1  +  1 ) )
31 1z 10780 . . . . . . . . . . . 12  |-  1  e.  ZZ
32 fzpr 11621 . . . . . . . . . . . 12  |-  ( 1  e.  ZZ  ->  (
1 ... ( 1  +  1 ) )  =  { 1 ,  ( 1  +  1 ) } )
3331, 32ax-mp 5 . . . . . . . . . . 11  |-  ( 1 ... ( 1  +  1 ) )  =  { 1 ,  ( 1  +  1 ) }
34 1p1e2 10539 . . . . . . . . . . . 12  |-  ( 1  +  1 )  =  2
3534preq2i 4059 . . . . . . . . . . 11  |-  { 1 ,  ( 1  +  1 ) }  =  { 1 ,  2 }
3630, 33, 353eqtri 2484 . . . . . . . . . 10  |-  ( 1 ... 2 )  =  { 1 ,  2 }
3736imaeq2i 5268 . . . . . . . . 9  |-  ( P
" ( 1 ... 2 ) )  =  ( P " {
1 ,  2 } )
38 1nn0 10699 . . . . . . . . . . . . 13  |-  1  e.  NN0
39 elnn0uz 11002 . . . . . . . . . . . . 13  |-  ( 1  e.  NN0  <->  1  e.  (
ZZ>= `  0 ) )
4038, 39mpbi 208 . . . . . . . . . . . 12  |-  1  e.  ( ZZ>= `  0 )
41 1le3 10642 . . . . . . . . . . . . 13  |-  1  <_  3
42 elfz5 11555 . . . . . . . . . . . . 13  |-  ( ( 1  e.  ( ZZ>= ` 
0 )  /\  3  e.  ZZ )  ->  (
1  e.  ( 0 ... 3 )  <->  1  <_  3 ) )
4341, 42mpbiri 233 . . . . . . . . . . . 12  |-  ( ( 1  e.  ( ZZ>= ` 
0 )  /\  3  e.  ZZ )  ->  1  e.  ( 0 ... 3
) )
4440, 22, 43mp2an 672 . . . . . . . . . . 11  |-  1  e.  ( 0 ... 3
)
4544a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  1  e.  ( 0 ... 3 ) )
46 2nn0 10700 . . . . . . . . . . . . 13  |-  2  e.  NN0
47 elnn0uz 11002 . . . . . . . . . . . . 13  |-  ( 2  e.  NN0  <->  2  e.  (
ZZ>= `  0 ) )
4846, 47mpbi 208 . . . . . . . . . . . 12  |-  2  e.  ( ZZ>= `  0 )
49 2re 10495 . . . . . . . . . . . . . 14  |-  2  e.  RR
50 3re 10499 . . . . . . . . . . . . . 14  |-  3  e.  RR
51 2lt3 10593 . . . . . . . . . . . . . 14  |-  2  <  3
5249, 50, 51ltleii 9601 . . . . . . . . . . . . 13  |-  2  <_  3
53 elfz5 11555 . . . . . . . . . . . . 13  |-  ( ( 2  e.  ( ZZ>= ` 
0 )  /\  3  e.  ZZ )  ->  (
2  e.  ( 0 ... 3 )  <->  2  <_  3 ) )
5452, 53mpbiri 233 . . . . . . . . . . . 12  |-  ( ( 2  e.  ( ZZ>= ` 
0 )  /\  3  e.  ZZ )  ->  2  e.  ( 0 ... 3
) )
5548, 22, 54mp2an 672 . . . . . . . . . . 11  |-  2  e.  ( 0 ... 3
)
5655a1i 11 . . . . . . . . . 10  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  2  e.  ( 0 ... 3 ) )
57 fnimapr 5857 . . . . . . . . . 10  |-  ( ( P  Fn  ( 0 ... 3 )  /\  1  e.  ( 0 ... 3 )  /\  2  e.  ( 0 ... 3 ) )  ->  ( P " { 1 ,  2 } )  =  {
( P `  1
) ,  ( P `
 2 ) } )
5812, 45, 56, 57syl3anc 1219 . . . . . . . . 9  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P " {
1 ,  2 } )  =  { ( P `  1 ) ,  ( P ` 
2 ) } )
5937, 58syl5eq 2504 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P " (
1 ... 2 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )
6028, 59syl5eq 2504 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P " (
1..^ 3 ) )  =  { ( P `
 1 ) ,  ( P `  2
) } )
6121, 60ineq12d 3654 . . . . . 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 ) } ) )
6261eqeq1d 2453 . . . . 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
) } )  =  (/) ) )
6362adantr 465 . . . 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 ) } )  =  (/) ) )
649, 63mpbird 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 ) ) )  =  (/) )
658, 64sylan9eq 2512 . 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
) ) ) )  =  (/) )
663, 65mpan 670 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2644    u. cun 3427    i^i cin 3428   (/)c0 3738   {cpr 3980   {ctp 3982   <.cop 3984   class class class wbr 4393   `'ccnv 4940   "cima 4944    Fn wfn 5514   -->wf 5515   ` cfv 5519  (class class class)co 6193   0cc0 9386   1c1 9387    + caddc 9389    <_ cle 9523    - cmin 9699   2c2 10475   3c3 10476   NN0cn0 10683   ZZcz 10750   ZZ>=cuz 10965   ...cfz 11547  ..^cfzo 11658   #chash 12213
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-rmo 2803  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-cda 8441  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
This theorem is referenced by:  constr3pth  23691
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