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Theorem axlowdimlem13 24233
Description: Lemma for axlowdim 24240. 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 10635 . . . . . . . . 9  |-  2  =/=  0
21neii 2642 . . . . . . . 8  |-  -.  2  =  0
3 eqcom 2452 . . . . . . . . 9  |-  ( 2  =  0  <->  0  = 
2 )
4 1pneg1e0 10651 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
54eqcomi 2456 . . . . . . . . . 10  |-  0  =  ( 1  + 
-u 1 )
6 df-2 10601 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
75, 6eqeq12i 2463 . . . . . . . . 9  |-  ( 0  =  2  <->  ( 1  +  -u 1 )  =  ( 1  +  1 ) )
8 ax-1cn 9553 . . . . . . . . . 10  |-  1  e.  CC
9 neg1cn 10646 . . . . . . . . . 10  |-  -u 1  e.  CC
108, 9, 8addcani 9776 . . . . . . . . 9  |-  ( ( 1  +  -u 1
)  =  ( 1  +  1 )  <->  -u 1  =  1 )
113, 7, 103bitri 271 . . . . . . . 8  |-  ( 2  =  0  <->  -u 1  =  1 )
122, 11mtbi 298 . . . . . . 7  |-  -.  -u 1  =  1
1312intnanr 915 . . . . . 6  |-  -.  ( -u 1  =  1  /\  0  =  0 )
14 ax-1ne0 9564 . . . . . . . . 9  |-  1  =/=  0
1514neii 2642 . . . . . . . 8  |-  -.  1  =  0
16 negeq0 9878 . . . . . . . . 9  |-  ( 1  e.  CC  ->  (
1  =  0  <->  -u 1  =  0 ) )
178, 16ax-mp 5 . . . . . . . 8  |-  ( 1  =  0  <->  -u 1  =  0 )
1815, 17mtbi 298 . . . . . . 7  |-  -.  -u 1  =  0
1918intnanr 915 . . . . . 6  |-  -.  ( -u 1  =  0  /\  0  =  1 )
2013, 19pm3.2ni 854 . . . . 5  |-  -.  (
( -u 1  =  1  /\  0  =  0 )  \/  ( -u
1  =  0  /\  0  =  1 ) )
21 negex 9823 . . . . . 6  |-  -u 1  e.  _V
22 c0ex 9593 . . . . . 6  |-  0  e.  _V
23 1ex 9594 . . . . . 6  |-  1  e.  _V
2421, 22, 23, 22preq12b 4191 . . . . 5  |-  ( {
-u 1 ,  0 }  =  { 1 ,  0 }  <->  ( ( -u 1  =  1  /\  0  =  0 )  \/  ( -u 1  =  0  /\  0  =  1 ) ) )
2520, 24mtbir 299 . . . 4  |-  -.  { -u 1 ,  0 }  =  { 1 ,  0 }
26 3ex 10618 . . . . . . . . 9  |-  3  e.  _V
2726rnsnop 5479 . . . . . . . 8  |-  ran  { <. 3 ,  -u 1 >. }  =  { -u
1 }
2827a1i 11 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. 3 ,  -u 1 >. }  =  { -u 1 } )
29 elnnuz 11127 . . . . . . . . . . . 12  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
30 eluzfz1 11703 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
3129, 30sylbi 195 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
32 df-3 10602 . . . . . . . . . . . . . . . 16  |-  3  =  ( 2  +  1 )
33 1e0p1 11013 . . . . . . . . . . . . . . . 16  |-  1  =  ( 0  +  1 )
3432, 33eqeq12i 2463 . . . . . . . . . . . . . . 15  |-  ( 3  =  1  <->  ( 2  +  1 )  =  ( 0  +  1 ) )
35 2cn 10613 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
36 0cn 9591 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
3735, 36, 8addcan2i 9777 . . . . . . . . . . . . . . 15  |-  ( ( 2  +  1 )  =  ( 0  +  1 )  <->  2  = 
0 )
3834, 37bitri 249 . . . . . . . . . . . . . 14  |-  ( 3  =  1  <->  2  = 
0 )
3938necon3bii 2711 . . . . . . . . . . . . 13  |-  ( 3  =/=  1  <->  2  =/=  0 )
401, 39mpbir 209 . . . . . . . . . . . 12  |-  3  =/=  1
4140necomi 2713 . . . . . . . . . . 11  |-  1  =/=  3
4231, 41jctir 538 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1  e.  ( 1 ... N )  /\  1  =/=  3 ) )
43 eldifsn 4140 . . . . . . . . . 10  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  <-> 
( 1  e.  ( 1 ... N )  /\  1  =/=  3
) )
4442, 43sylibr 212 . . . . . . . . 9  |-  ( N  e.  NN  ->  1  e.  ( ( 1 ... N )  \  {
3 } ) )
4544adantr 465 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  1  e.  ( ( 1 ... N
)  \  { 3 } ) )
46 ne0i 3776 . . . . . . . 8  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( 1 ... N )  \  { 3 } )  =/=  (/) )
47 rnxp 5427 . . . . . . . 8  |-  ( ( ( 1 ... N
)  \  { 3 } )  =/=  (/)  ->  ran  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } )  =  { 0 } )
4845, 46, 473syl 20 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } )  =  { 0 } )
4928, 48uneq12d 3644 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ran  { <. 3 ,  -u 1 >. }  u.  ran  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( {
-u 1 }  u.  { 0 } ) )
50 rnun 5404 . . . . . 6  |-  ran  ( { <. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( ran 
{ <. 3 ,  -u
1 >. }  u.  ran  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } ) )
51 df-pr 4017 . . . . . 6  |-  { -u
1 ,  0 }  =  ( { -u
1 }  u.  {
0 } )
5249, 50, 513eqtr4g 2509 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  { -u 1 ,  0 } )
53 ovex 6309 . . . . . . . . 9  |-  ( I  +  1 )  e. 
_V
5453rnsnop 5479 . . . . . . . 8  |-  ran  { <. ( I  +  1 ) ,  1 >. }  =  { 1 }
5554a1i 11 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. (
I  +  1 ) ,  1 >. }  =  { 1 } )
56 nnz 10893 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  ZZ )
57 fzssp1 11736 . . . . . . . . . . . 12  |-  ( 1 ... ( N  - 
1 ) )  C_  ( 1 ... (
( N  -  1 )  +  1 ) )
58 zcn 10876 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
59 npcan1 9991 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
6059oveq2d 6297 . . . . . . . . . . . . 13  |-  ( N  e.  CC  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
6158, 60syl 16 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
6257, 61syl5sseq 3537 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6356, 62syl 16 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6463sselda 3489 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( 1 ... N ) )
65 elfzelz 11698 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
6665zred 10975 . . . . . . . . . . 11  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  RR )
67 id 22 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  e.  RR )
68 ltp1 10387 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  <  ( I  +  1 ) )
6967, 68ltned 9724 . . . . . . . . . . 11  |-  ( I  e.  RR  ->  I  =/=  ( I  +  1 ) )
7066, 69syl 16 . . . . . . . . . 10  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  =/=  ( I  +  1 ) )
7170adantl 466 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  =/=  (
I  +  1 ) )
72 eldifsn 4140 . . . . . . . . 9  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  <-> 
( I  e.  ( 1 ... N )  /\  I  =/=  (
I  +  1 ) ) )
7364, 71, 72sylanbrc 664 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( ( 1 ... N
)  \  { (
I  +  1 ) } ) )
74 ne0i 3776 . . . . . . . 8  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  =/=  (/) )
75 rnxp 5427 . . . . . . . 8  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  =/=  (/)  ->  ran  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } )  =  { 0 } )
7673, 74, 753syl 20 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } )  =  { 0 } )
7755, 76uneq12d 3644 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ran  { <. ( I  +  1 ) ,  1 >. }  u.  ran  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  ( { 1 }  u.  { 0 } ) )
78 rnun 5404 . . . . . 6  |-  ran  ( { <. ( I  + 
1 ) ,  1
>. }  u.  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  ( ran  { <. ( I  +  1 ) ,  1 >. }  u.  ran  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )
79 df-pr 4017 . . . . . 6  |-  { 1 ,  0 }  =  ( { 1 }  u.  { 0 } )
8077, 78, 793eqtr4g 2509 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  { 1 ,  0 } )
8152, 80eqeq12d 2465 . . . 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 } ) )
8225, 81mtbiri 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 } ) ) )
83 rneq 5218 . . 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 } ) ) )
8482, 83nsyl 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 } ) ) )
85 axlowdimlem13.1 . . . 4  |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } ) )
86 axlowdimlem13.2 . . . 4  |-  Q  =  ( { <. (
I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) )
8785, 86eqeq12i 2463 . . 3  |-  ( P  =  Q  <->  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) )
8887necon3abii 2703 . 2  |-  ( P  =/=  Q  <->  -.  ( { <. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) )
8984, 88sylibr 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 1383    e. wcel 1804    =/= wne 2638    \ cdif 3458    u. cun 3459    C_ wss 3461   (/)c0 3770   {csn 4014   {cpr 4016   <.cop 4020    X. cxp 4987   ran crn 4990   ` cfv 5578  (class class class)co 6281   CCcc 9493   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    - cmin 9810   -ucneg 9811   NNcn 10543   2c2 10592   3c3 10593   ZZcz 10871   ZZ>=cuz 11091   ...cfz 11682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11092  df-fz 11683
This theorem is referenced by:  axlowdimlem15  24235
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