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Theorem axlowdimlem13 23926
Description: Lemma for axlowdim 23933. Establish that  P and 
Q are different points. (Contributed by Scott Fenton, 21-Apr-2013.)
Hypotheses
Ref Expression
axlowdimlem13.1  |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } ) )
axlowdimlem13.2  |-  Q  =  ( { <. (
I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) )
Assertion
Ref Expression
axlowdimlem13  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  P  =/=  Q
)

Proof of Theorem axlowdimlem13
StepHypRef Expression
1 2ne0 10617 . . . . . . . . 9  |-  2  =/=  0
2 df-ne 2657 . . . . . . . . 9  |-  ( 2  =/=  0  <->  -.  2  =  0 )
31, 2mpbi 208 . . . . . . . 8  |-  -.  2  =  0
4 eqcom 2469 . . . . . . . . 9  |-  ( 2  =  0  <->  0  = 
2 )
5 1pneg1e0 10633 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
65eqcomi 2473 . . . . . . . . . 10  |-  0  =  ( 1  + 
-u 1 )
7 df-2 10583 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
86, 7eqeq12i 2480 . . . . . . . . 9  |-  ( 0  =  2  <->  ( 1  +  -u 1 )  =  ( 1  +  1 ) )
9 ax-1cn 9539 . . . . . . . . . 10  |-  1  e.  CC
10 neg1cn 10628 . . . . . . . . . 10  |-  -u 1  e.  CC
119, 10, 9addcani 9761 . . . . . . . . 9  |-  ( ( 1  +  -u 1
)  =  ( 1  +  1 )  <->  -u 1  =  1 )
124, 8, 113bitri 271 . . . . . . . 8  |-  ( 2  =  0  <->  -u 1  =  1 )
133, 12mtbi 298 . . . . . . 7  |-  -.  -u 1  =  1
1413intnanr 908 . . . . . 6  |-  -.  ( -u 1  =  1  /\  0  =  0 )
15 ax-1ne0 9550 . . . . . . . . 9  |-  1  =/=  0
16 df-ne 2657 . . . . . . . . 9  |-  ( 1  =/=  0  <->  -.  1  =  0 )
1715, 16mpbi 208 . . . . . . . 8  |-  -.  1  =  0
18 negeq0 9862 . . . . . . . . 9  |-  ( 1  e.  CC  ->  (
1  =  0  <->  -u 1  =  0 ) )
199, 18ax-mp 5 . . . . . . . 8  |-  ( 1  =  0  <->  -u 1  =  0 )
2017, 19mtbi 298 . . . . . . 7  |-  -.  -u 1  =  0
2120intnanr 908 . . . . . 6  |-  -.  ( -u 1  =  0  /\  0  =  1 )
2214, 21pm3.2ni 851 . . . . 5  |-  -.  (
( -u 1  =  1  /\  0  =  0 )  \/  ( -u
1  =  0  /\  0  =  1 ) )
23 negex 9807 . . . . . 6  |-  -u 1  e.  _V
24 c0ex 9579 . . . . . 6  |-  0  e.  _V
259elexi 3116 . . . . . 6  |-  1  e.  _V
2623, 24, 25, 24preq12b 4195 . . . . 5  |-  ( {
-u 1 ,  0 }  =  { 1 ,  0 }  <->  ( ( -u 1  =  1  /\  0  =  0 )  \/  ( -u 1  =  0  /\  0  =  1 ) ) )
2722, 26mtbir 299 . . . 4  |-  -.  { -u 1 ,  0 }  =  { 1 ,  0 }
28 3ex 10600 . . . . . . . . 9  |-  3  e.  _V
2928rnsnop 5480 . . . . . . . 8  |-  ran  { <. 3 ,  -u 1 >. }  =  { -u
1 }
3029a1i 11 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. 3 ,  -u 1 >. }  =  { -u 1 } )
31 elnnuz 11107 . . . . . . . . . . . 12  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
32 eluzfz1 11682 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
3331, 32sylbi 195 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
34 df-3 10584 . . . . . . . . . . . . . . . 16  |-  3  =  ( 2  +  1 )
35 1e0p1 10993 . . . . . . . . . . . . . . . 16  |-  1  =  ( 0  +  1 )
3634, 35eqeq12i 2480 . . . . . . . . . . . . . . 15  |-  ( 3  =  1  <->  ( 2  +  1 )  =  ( 0  +  1 ) )
37 2cn 10595 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
38 0cn 9577 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
3937, 38, 9addcan2i 9762 . . . . . . . . . . . . . . 15  |-  ( ( 2  +  1 )  =  ( 0  +  1 )  <->  2  = 
0 )
4036, 39bitri 249 . . . . . . . . . . . . . 14  |-  ( 3  =  1  <->  2  = 
0 )
4140necon3bii 2728 . . . . . . . . . . . . 13  |-  ( 3  =/=  1  <->  2  =/=  0 )
421, 41mpbir 209 . . . . . . . . . . . 12  |-  3  =/=  1
4342necomi 2730 . . . . . . . . . . 11  |-  1  =/=  3
4433, 43jctir 538 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1  e.  ( 1 ... N )  /\  1  =/=  3 ) )
45 eldifsn 4145 . . . . . . . . . 10  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  <-> 
( 1  e.  ( 1 ... N )  /\  1  =/=  3
) )
4644, 45sylibr 212 . . . . . . . . 9  |-  ( N  e.  NN  ->  1  e.  ( ( 1 ... N )  \  {
3 } ) )
4746adantr 465 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  1  e.  ( ( 1 ... N
)  \  { 3 } ) )
48 ne0i 3784 . . . . . . . 8  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( 1 ... N )  \  { 3 } )  =/=  (/) )
49 rnxp 5428 . . . . . . . 8  |-  ( ( ( 1 ... N
)  \  { 3 } )  =/=  (/)  ->  ran  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } )  =  { 0 } )
5047, 48, 493syl 20 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } )  =  { 0 } )
5130, 50uneq12d 3652 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ran  { <. 3 ,  -u 1 >. }  u.  ran  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( {
-u 1 }  u.  { 0 } ) )
52 rnun 5405 . . . . . 6  |-  ran  ( { <. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( ran 
{ <. 3 ,  -u
1 >. }  u.  ran  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } ) )
53 df-pr 4023 . . . . . 6  |-  { -u
1 ,  0 }  =  ( { -u
1 }  u.  {
0 } )
5451, 52, 533eqtr4g 2526 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  { -u 1 ,  0 } )
55 ovex 6300 . . . . . . . . 9  |-  ( I  +  1 )  e. 
_V
5655rnsnop 5480 . . . . . . . 8  |-  ran  { <. ( I  +  1 ) ,  1 >. }  =  { 1 }
5756a1i 11 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. (
I  +  1 ) ,  1 >. }  =  { 1 } )
58 nnz 10875 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  ZZ )
59 fzssp1 11715 . . . . . . . . . . . 12  |-  ( 1 ... ( N  - 
1 ) )  C_  ( 1 ... (
( N  -  1 )  +  1 ) )
60 zcn 10858 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
61 npcan 9818 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
629, 61mpan2 671 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
6362oveq2d 6291 . . . . . . . . . . . . 13  |-  ( N  e.  CC  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
6460, 63syl 16 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
6559, 64syl5sseq 3545 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6658, 65syl 16 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6766sselda 3497 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( 1 ... N ) )
68 elfzelz 11677 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
6968zred 10955 . . . . . . . . . . 11  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  RR )
70 id 22 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  e.  RR )
71 ltp1 10369 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  <  ( I  +  1 ) )
7270, 71ltned 9709 . . . . . . . . . . 11  |-  ( I  e.  RR  ->  I  =/=  ( I  +  1 ) )
7369, 72syl 16 . . . . . . . . . 10  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  =/=  ( I  +  1 ) )
7473adantl 466 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  =/=  (
I  +  1 ) )
75 eldifsn 4145 . . . . . . . . 9  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  <-> 
( I  e.  ( 1 ... N )  /\  I  =/=  (
I  +  1 ) ) )
7667, 74, 75sylanbrc 664 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( ( 1 ... N
)  \  { (
I  +  1 ) } ) )
77 ne0i 3784 . . . . . . . 8  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  =/=  (/) )
78 rnxp 5428 . . . . . . . 8  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  =/=  (/)  ->  ran  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } )  =  { 0 } )
7976, 77, 783syl 20 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } )  =  { 0 } )
8057, 79uneq12d 3652 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ran  { <. ( I  +  1 ) ,  1 >. }  u.  ran  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  ( { 1 }  u.  { 0 } ) )
81 rnun 5405 . . . . . 6  |-  ran  ( { <. ( I  + 
1 ) ,  1
>. }  u.  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  ( ran  { <. ( I  +  1 ) ,  1 >. }  u.  ran  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )
82 df-pr 4023 . . . . . 6  |-  { 1 ,  0 }  =  ( { 1 }  u.  { 0 } )
8380, 81, 823eqtr4g 2526 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  { 1 ,  0 } )
8454, 83eqeq12d 2482 . . . 4  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ran  ( { <. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ran  ( { <. ( I  + 
1 ) ,  1
>. }  u.  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  <->  { -u 1 ,  0 }  =  { 1 ,  0 } ) )
8527, 84mtbiri 303 . . 3  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  -.  ran  ( {
<. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ran  ( { <. ( I  + 
1 ) ,  1
>. }  u.  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) )
86 rneq 5219 . . 3  |-  ( ( { <. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  ->  ran  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  ran  ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) )
8785, 86nsyl 121 . 2  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  -.  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) )
88 axlowdimlem13.1 . . . 4  |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } ) )
89 axlowdimlem13.2 . . . 4  |-  Q  =  ( { <. (
I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) )
9088, 89eqeq12i 2480 . . 3  |-  ( P  =  Q  <->  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) )
9190necon3abii 2720 . 2  |-  ( P  =/=  Q  <->  -.  ( { <. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) )
9287, 91sylibr 212 1  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  P  =/=  Q
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762    =/= wne 2655    \ cdif 3466    u. cun 3467    C_ wss 3469   (/)c0 3778   {csn 4020   {cpr 4022   <.cop 4026    X. cxp 4990   ran crn 4993   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    - cmin 9794   -ucneg 9795   NNcn 10525   2c2 10574   3c3 10575   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662
This theorem is referenced by:  axlowdimlem15  23928
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