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Theorem axlowdimlem13 24378
Description: Lemma for axlowdim 24385. 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 10545 . . . . . . . . 9  |-  2  =/=  0
21neii 2581 . . . . . . . 8  |-  -.  2  =  0
3 eqcom 2391 . . . . . . . . 9  |-  ( 2  =  0  <->  0  = 
2 )
4 1pneg1e0 10561 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
54eqcomi 2395 . . . . . . . . . 10  |-  0  =  ( 1  + 
-u 1 )
6 df-2 10511 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
75, 6eqeq12i 2402 . . . . . . . . 9  |-  ( 0  =  2  <->  ( 1  +  -u 1 )  =  ( 1  +  1 ) )
8 ax-1cn 9461 . . . . . . . . . 10  |-  1  e.  CC
9 neg1cn 10556 . . . . . . . . . 10  |-  -u 1  e.  CC
108, 9, 8addcani 9684 . . . . . . . . 9  |-  ( ( 1  +  -u 1
)  =  ( 1  +  1 )  <->  -u 1  =  1 )
113, 7, 103bitri 271 . . . . . . . 8  |-  ( 2  =  0  <->  -u 1  =  1 )
122, 11mtbi 296 . . . . . . 7  |-  -.  -u 1  =  1
1312intnanr 913 . . . . . 6  |-  -.  ( -u 1  =  1  /\  0  =  0 )
14 ax-1ne0 9472 . . . . . . . . 9  |-  1  =/=  0
1514neii 2581 . . . . . . . 8  |-  -.  1  =  0
16 negeq0 9786 . . . . . . . . 9  |-  ( 1  e.  CC  ->  (
1  =  0  <->  -u 1  =  0 ) )
178, 16ax-mp 5 . . . . . . . 8  |-  ( 1  =  0  <->  -u 1  =  0 )
1815, 17mtbi 296 . . . . . . 7  |-  -.  -u 1  =  0
1918intnanr 913 . . . . . 6  |-  -.  ( -u 1  =  0  /\  0  =  1 )
2013, 19pm3.2ni 852 . . . . 5  |-  -.  (
( -u 1  =  1  /\  0  =  0 )  \/  ( -u
1  =  0  /\  0  =  1 ) )
21 negex 9731 . . . . . 6  |-  -u 1  e.  _V
22 c0ex 9501 . . . . . 6  |-  0  e.  _V
23 1ex 9502 . . . . . 6  |-  1  e.  _V
2421, 22, 23, 22preq12b 4120 . . . . 5  |-  ( {
-u 1 ,  0 }  =  { 1 ,  0 }  <->  ( ( -u 1  =  1  /\  0  =  0 )  \/  ( -u 1  =  0  /\  0  =  1 ) ) )
2520, 24mtbir 297 . . . 4  |-  -.  { -u 1 ,  0 }  =  { 1 ,  0 }
26 3ex 10528 . . . . . . . . 9  |-  3  e.  _V
2726rnsnop 5397 . . . . . . . 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 11037 . . . . . . . . . . . 12  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
30 eluzfz1 11614 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
3129, 30sylbi 195 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
32 df-3 10512 . . . . . . . . . . . . . . . 16  |-  3  =  ( 2  +  1 )
33 1e0p1 10923 . . . . . . . . . . . . . . . 16  |-  1  =  ( 0  +  1 )
3432, 33eqeq12i 2402 . . . . . . . . . . . . . . 15  |-  ( 3  =  1  <->  ( 2  +  1 )  =  ( 0  +  1 ) )
35 2cn 10523 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
36 0cn 9499 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
3735, 36, 8addcan2i 9685 . . . . . . . . . . . . . . 15  |-  ( ( 2  +  1 )  =  ( 0  +  1 )  <->  2  = 
0 )
3834, 37bitri 249 . . . . . . . . . . . . . 14  |-  ( 3  =  1  <->  2  = 
0 )
3938necon3bii 2650 . . . . . . . . . . . . 13  |-  ( 3  =/=  1  <->  2  =/=  0 )
401, 39mpbir 209 . . . . . . . . . . . 12  |-  3  =/=  1
4140necomi 2652 . . . . . . . . . . 11  |-  1  =/=  3
4231, 41jctir 536 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1  e.  ( 1 ... N )  /\  1  =/=  3 ) )
43 eldifsn 4069 . . . . . . . . . 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 463 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  1  e.  ( ( 1 ... N
)  \  { 3 } ) )
46 ne0i 3717 . . . . . . . 8  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( 1 ... N )  \  { 3 } )  =/=  (/) )
47 rnxp 5347 . . . . . . . 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 3573 . . . . . 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 5324 . . . . . 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 3947 . . . . . 6  |-  { -u
1 ,  0 }  =  ( { -u
1 }  u.  {
0 } )
5249, 50, 513eqtr4g 2448 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  { -u 1 ,  0 } )
53 ovex 6224 . . . . . . . . 9  |-  ( I  +  1 )  e. 
_V
5453rnsnop 5397 . . . . . . . 8  |-  ran  { <. ( I  +  1 ) ,  1 >. }  =  { 1 }
5554a1i 11 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. (
I  +  1 ) ,  1 >. }  =  { 1 } )
56 nnz 10803 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  ZZ )
57 fzssp1 11648 . . . . . . . . . . . 12  |-  ( 1 ... ( N  - 
1 ) )  C_  ( 1 ... (
( N  -  1 )  +  1 ) )
58 zcn 10786 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
59 npcan1 9902 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
6059oveq2d 6212 . . . . . . . . . . . . 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 3465 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6356, 62syl 16 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6463sselda 3417 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( 1 ... N ) )
65 elfzelz 11609 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
6665zred 10884 . . . . . . . . . . 11  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  RR )
67 id 22 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  e.  RR )
68 ltp1 10297 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  <  ( I  +  1 ) )
6967, 68ltned 9632 . . . . . . . . . . 11  |-  ( I  e.  RR  ->  I  =/=  ( I  +  1 ) )
7066, 69syl 16 . . . . . . . . . 10  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  =/=  ( I  +  1 ) )
7170adantl 464 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  =/=  (
I  +  1 ) )
72 eldifsn 4069 . . . . . . . . 9  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  <-> 
( I  e.  ( 1 ... N )  /\  I  =/=  (
I  +  1 ) ) )
7364, 71, 72sylanbrc 662 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( ( 1 ... N
)  \  { (
I  +  1 ) } ) )
74 ne0i 3717 . . . . . . . 8  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  =/=  (/) )
75 rnxp 5347 . . . . . . . 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 3573 . . . . . 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 5324 . . . . . 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 3947 . . . . . 6  |-  { 1 ,  0 }  =  ( { 1 }  u.  { 0 } )
8077, 78, 793eqtr4g 2448 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  { 1 ,  0 } )
8152, 80eqeq12d 2404 . . . 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 301 . . 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 5141 . . 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 2402 . . 3  |-  ( P  =  Q  <->  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) )
8887necon3abii 2642 . 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 366    /\ wa 367    = wceq 1399    e. wcel 1826    =/= wne 2577    \ cdif 3386    u. cun 3387    C_ wss 3389   (/)c0 3711   {csn 3944   {cpr 3946   <.cop 3950    X. cxp 4911   ran crn 4914   ` cfv 5496  (class class class)co 6196   CCcc 9401   RRcr 9402   0cc0 9403   1c1 9404    + caddc 9406    - cmin 9718   -ucneg 9719   NNcn 10452   2c2 10502   3c3 10503   ZZcz 10781   ZZ>=cuz 11001   ...cfz 11593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-nn 10453  df-2 10511  df-3 10512  df-n0 10713  df-z 10782  df-uz 11002  df-fz 11594
This theorem is referenced by:  axlowdimlem15  24380
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