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Theorem axlowdimlem13 24582
Description: Lemma for axlowdim 24589. 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 9829 . . . . . . . . 9  |-  2  =/=  0
2 df-ne 2448 . . . . . . . . 9  |-  ( 2  =/=  0  <->  -.  2  =  0 )
31, 2mpbi 199 . . . . . . . 8  |-  -.  2  =  0
4 eqcom 2285 . . . . . . . . 9  |-  ( 2  =  0  <->  0  = 
2 )
5 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
65negidi 9115 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
76eqcomi 2287 . . . . . . . . . 10  |-  0  =  ( 1  + 
-u 1 )
8 df-2 9804 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
97, 8eqeq12i 2296 . . . . . . . . 9  |-  ( 0  =  2  <->  ( 1  +  -u 1 )  =  ( 1  +  1 ) )
10 neg1cn 9813 . . . . . . . . . 10  |-  -u 1  e.  CC
115, 10, 5addcani 9005 . . . . . . . . 9  |-  ( ( 1  +  -u 1
)  =  ( 1  +  1 )  <->  -u 1  =  1 )
124, 9, 113bitri 262 . . . . . . . 8  |-  ( 2  =  0  <->  -u 1  =  1 )
133, 12mtbi 289 . . . . . . 7  |-  -.  -u 1  =  1
1413intnanr 881 . . . . . 6  |-  -.  ( -u 1  =  1  /\  0  =  0 )
15 ax-1ne0 8806 . . . . . . . . 9  |-  1  =/=  0
16 df-ne 2448 . . . . . . . . 9  |-  ( 1  =/=  0  <->  -.  1  =  0 )
1715, 16mpbi 199 . . . . . . . 8  |-  -.  1  =  0
18 negeq0 9101 . . . . . . . . 9  |-  ( 1  e.  CC  ->  (
1  =  0  <->  -u 1  =  0 ) )
195, 18ax-mp 8 . . . . . . . 8  |-  ( 1  =  0  <->  -u 1  =  0 )
2017, 19mtbi 289 . . . . . . 7  |-  -.  -u 1  =  0
2120intnanr 881 . . . . . 6  |-  -.  ( -u 1  =  0  /\  0  =  1 )
2214, 21pm3.2ni 827 . . . . 5  |-  -.  (
( -u 1  =  1  /\  0  =  0 )  \/  ( -u
1  =  0  /\  0  =  1 ) )
23 negex 9050 . . . . . 6  |-  -u 1  e.  _V
24 0cn 8831 . . . . . . 7  |-  0  e.  CC
2524elexi 2797 . . . . . 6  |-  0  e.  _V
265elexi 2797 . . . . . 6  |-  1  e.  _V
2723, 25, 26, 25preq12b 3788 . . . . 5  |-  ( {
-u 1 ,  0 }  =  { 1 ,  0 }  <->  ( ( -u 1  =  1  /\  0  =  0 )  \/  ( -u 1  =  0  /\  0  =  1 ) ) )
2822, 27mtbir 290 . . . 4  |-  -.  { -u 1 ,  0 }  =  { 1 ,  0 }
29 3re 9817 . . . . . . . . . 10  |-  3  e.  RR
3029elexi 2797 . . . . . . . . 9  |-  3  e.  _V
3130rnsnop 5153 . . . . . . . 8  |-  ran  { <. 3 ,  -u 1 >. }  =  { -u
1 }
3231a1i 10 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. 3 ,  -u 1 >. }  =  { -u 1 } )
33 elnnuz 10264 . . . . . . . . . . . . 13  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
34 eluzfz1 10803 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
3533, 34sylbi 187 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
36 df-3 9805 . . . . . . . . . . . . . . . . 17  |-  3  =  ( 2  +  1 )
37 1e0p1 10152 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 0  +  1 )
3836, 37eqeq12i 2296 . . . . . . . . . . . . . . . 16  |-  ( 3  =  1  <->  ( 2  +  1 )  =  ( 0  +  1 ) )
39 2cn 9816 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
4039, 24, 5addcan2i 9006 . . . . . . . . . . . . . . . 16  |-  ( ( 2  +  1 )  =  ( 0  +  1 )  <->  2  = 
0 )
4138, 40bitri 240 . . . . . . . . . . . . . . 15  |-  ( 3  =  1  <->  2  = 
0 )
4241necon3bii 2478 . . . . . . . . . . . . . 14  |-  ( 3  =/=  1  <->  2  =/=  0 )
431, 42mpbir 200 . . . . . . . . . . . . 13  |-  3  =/=  1
4443necomi 2528 . . . . . . . . . . . 12  |-  1  =/=  3
4535, 44jctir 524 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
1  e.  ( 1 ... N )  /\  1  =/=  3 ) )
46 eldifsn 3749 . . . . . . . . . . 11  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  <-> 
( 1  e.  ( 1 ... N )  /\  1  =/=  3
) )
4745, 46sylibr 203 . . . . . . . . . 10  |-  ( N  e.  NN  ->  1  e.  ( ( 1 ... N )  \  {
3 } ) )
4847adantr 451 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  1  e.  ( ( 1 ... N
)  \  { 3 } ) )
49 ne0i 3461 . . . . . . . . 9  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( 1 ... N )  \  { 3 } )  =/=  (/) )
5048, 49syl 15 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ( 1 ... N )  \  { 3 } )  =/=  (/) )
51 rnxp 5106 . . . . . . . 8  |-  ( ( ( 1 ... N
)  \  { 3 } )  =/=  (/)  ->  ran  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } )  =  { 0 } )
5250, 51syl 15 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } )  =  { 0 } )
5332, 52uneq12d 3330 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ran  { <. 3 ,  -u 1 >. }  u.  ran  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( {
-u 1 }  u.  { 0 } ) )
54 rnun 5089 . . . . . 6  |-  ran  ( { <. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( ran 
{ <. 3 ,  -u
1 >. }  u.  ran  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } ) )
55 df-pr 3647 . . . . . 6  |-  { -u
1 ,  0 }  =  ( { -u
1 }  u.  {
0 } )
5653, 54, 553eqtr4g 2340 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  { -u 1 ,  0 } )
57 ovex 5883 . . . . . . . . 9  |-  ( I  +  1 )  e. 
_V
5857rnsnop 5153 . . . . . . . 8  |-  ran  { <. ( I  +  1 ) ,  1 >. }  =  { 1 }
5958a1i 10 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. (
I  +  1 ) ,  1 >. }  =  { 1 } )
60 nnz 10045 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  ZZ )
61 fzssp1 10834 . . . . . . . . . . . . 13  |-  ( 1 ... ( N  - 
1 ) )  C_  ( 1 ... (
( N  -  1 )  +  1 ) )
62 zcn 10029 . . . . . . . . . . . . . 14  |-  ( N  e.  ZZ  ->  N  e.  CC )
63 npcan 9060 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
645, 63mpan2 652 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
6564oveq2d 5874 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
6662, 65syl 15 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
6761, 66syl5sseq 3226 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6860, 67syl 15 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6968sselda 3180 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( 1 ... N ) )
70 elfzelz 10798 . . . . . . . . . . . . 13  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
7170zred 10117 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  RR )
72 id 19 . . . . . . . . . . . . . 14  |-  ( I  e.  RR  ->  I  e.  RR )
73 ltp1 9594 . . . . . . . . . . . . . 14  |-  ( I  e.  RR  ->  I  <  ( I  +  1 ) )
74 ltne 8917 . . . . . . . . . . . . . 14  |-  ( ( I  e.  RR  /\  I  <  ( I  + 
1 ) )  -> 
( I  +  1 )  =/=  I )
7572, 73, 74syl2anc 642 . . . . . . . . . . . . 13  |-  ( I  e.  RR  ->  (
I  +  1 )  =/=  I )
7675necomd 2529 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  =/=  ( I  +  1 ) )
7771, 76syl 15 . . . . . . . . . . 11  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  =/=  ( I  +  1 ) )
7877adantl 452 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  =/=  (
I  +  1 ) )
79 eldifsn 3749 . . . . . . . . . 10  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  <-> 
( I  e.  ( 1 ... N )  /\  I  =/=  (
I  +  1 ) ) )
8069, 78, 79sylanbrc 645 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( ( 1 ... N
)  \  { (
I  +  1 ) } ) )
81 ne0i 3461 . . . . . . . . 9  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  =/=  (/) )
8280, 81syl 15 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  =/=  (/) )
83 rnxp 5106 . . . . . . . 8  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  =/=  (/)  ->  ran  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } )  =  { 0 } )
8482, 83syl 15 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } )  =  { 0 } )
8559, 84uneq12d 3330 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ran  { <. ( I  +  1 ) ,  1 >. }  u.  ran  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  ( { 1 }  u.  { 0 } ) )
86 rnun 5089 . . . . . 6  |-  ran  ( { <. ( I  + 
1 ) ,  1
>. }  u.  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  ( ran  { <. ( I  +  1 ) ,  1 >. }  u.  ran  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )
87 df-pr 3647 . . . . . 6  |-  { 1 ,  0 }  =  ( { 1 }  u.  { 0 } )
8885, 86, 873eqtr4g 2340 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  { 1 ,  0 } )
8956, 88eqeq12d 2297 . . . 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 } ) )
9028, 89mtbiri 294 . . 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 } ) ) )
91 rneq 4904 . . 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 } ) ) )
9290, 91nsyl 113 . 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 } ) ) )
93 axlowdimlem13.1 . . . 4  |-  P  =  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } ) )
94 axlowdimlem13.2 . . . 4  |-  Q  =  ( { <. (
I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) )
9593, 94eqeq12i 2296 . . 3  |-  ( P  =  Q  <->  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) )
9695necon3abii 2476 . 2  |-  ( P  =/=  Q  <->  -.  ( { <. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) )
9792, 96sylibr 203 1  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  P  =/=  Q
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    u. cun 3150    C_ wss 3152   (/)c0 3455   {csn 3640   {cpr 3641   <.cop 3643   class class class wbr 4023    X. cxp 4687   ran crn 4690   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    - cmin 9037   -ucneg 9038   NNcn 9746   2c2 9795   3c3 9796   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782
This theorem is referenced by:  axlowdimlem15  24584
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783
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