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Theorem axlowdimlem13 24654
Description: Lemma for axlowdim 24661. 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 9845 . . . . . . . . 9  |-  2  =/=  0
2 df-ne 2461 . . . . . . . . 9  |-  ( 2  =/=  0  <->  -.  2  =  0 )
31, 2mpbi 199 . . . . . . . 8  |-  -.  2  =  0
4 eqcom 2298 . . . . . . . . 9  |-  ( 2  =  0  <->  0  = 
2 )
5 ax-1cn 8811 . . . . . . . . . . . 12  |-  1  e.  CC
65negidi 9131 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
76eqcomi 2300 . . . . . . . . . 10  |-  0  =  ( 1  + 
-u 1 )
8 df-2 9820 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
97, 8eqeq12i 2309 . . . . . . . . 9  |-  ( 0  =  2  <->  ( 1  +  -u 1 )  =  ( 1  +  1 ) )
10 neg1cn 9829 . . . . . . . . . 10  |-  -u 1  e.  CC
115, 10, 5addcani 9021 . . . . . . . . 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 8822 . . . . . . . . 9  |-  1  =/=  0
16 df-ne 2461 . . . . . . . . 9  |-  ( 1  =/=  0  <->  -.  1  =  0 )
1715, 16mpbi 199 . . . . . . . 8  |-  -.  1  =  0
18 negeq0 9117 . . . . . . . . 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 9066 . . . . . 6  |-  -u 1  e.  _V
24 0cn 8847 . . . . . . 7  |-  0  e.  CC
2524elexi 2810 . . . . . 6  |-  0  e.  _V
265elexi 2810 . . . . . 6  |-  1  e.  _V
2723, 25, 26, 25preq12b 3804 . . . . 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 9833 . . . . . . . . . 10  |-  3  e.  RR
3029elexi 2810 . . . . . . . . 9  |-  3  e.  _V
3130rnsnop 5169 . . . . . . . 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 10280 . . . . . . . . . . . . 13  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
34 eluzfz1 10819 . . . . . . . . . . . . 13  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
3533, 34sylbi 187 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
36 df-3 9821 . . . . . . . . . . . . . . . . 17  |-  3  =  ( 2  +  1 )
37 1e0p1 10168 . . . . . . . . . . . . . . . . 17  |-  1  =  ( 0  +  1 )
3836, 37eqeq12i 2309 . . . . . . . . . . . . . . . 16  |-  ( 3  =  1  <->  ( 2  +  1 )  =  ( 0  +  1 ) )
39 2cn 9832 . . . . . . . . . . . . . . . . 17  |-  2  e.  CC
4039, 24, 5addcan2i 9022 . . . . . . . . . . . . . . . 16  |-  ( ( 2  +  1 )  =  ( 0  +  1 )  <->  2  = 
0 )
4138, 40bitri 240 . . . . . . . . . . . . . . 15  |-  ( 3  =  1  <->  2  = 
0 )
4241necon3bii 2491 . . . . . . . . . . . . . 14  |-  ( 3  =/=  1  <->  2  =/=  0 )
431, 42mpbir 200 . . . . . . . . . . . . 13  |-  3  =/=  1
4443necomi 2541 . . . . . . . . . . . 12  |-  1  =/=  3
4535, 44jctir 524 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
1  e.  ( 1 ... N )  /\  1  =/=  3 ) )
46 eldifsn 3762 . . . . . . . . . . 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 3474 . . . . . . . . 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 5122 . . . . . . . 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 3343 . . . . . 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 5105 . . . . . 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 3660 . . . . . 6  |-  { -u
1 ,  0 }  =  ( { -u
1 }  u.  {
0 } )
5653, 54, 553eqtr4g 2353 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  { -u 1 ,  0 } )
57 ovex 5899 . . . . . . . . 9  |-  ( I  +  1 )  e. 
_V
5857rnsnop 5169 . . . . . . . 8  |-  ran  { <. ( I  +  1 ) ,  1 >. }  =  { 1 }
5958a1i 10 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. (
I  +  1 ) ,  1 >. }  =  { 1 } )
60 nnz 10061 . . . . . . . . . . . 12  |-  ( N  e.  NN  ->  N  e.  ZZ )
61 fzssp1 10850 . . . . . . . . . . . . 13  |-  ( 1 ... ( N  - 
1 ) )  C_  ( 1 ... (
( N  -  1 )  +  1 ) )
62 zcn 10045 . . . . . . . . . . . . . 14  |-  ( N  e.  ZZ  ->  N  e.  CC )
63 npcan 9076 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
645, 63mpan2 652 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
6564oveq2d 5890 . . . . . . . . . . . . . 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 3239 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6860, 67syl 15 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6968sselda 3193 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( 1 ... N ) )
70 elfzelz 10814 . . . . . . . . . . . . 13  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
7170zred 10133 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  RR )
72 id 19 . . . . . . . . . . . . . 14  |-  ( I  e.  RR  ->  I  e.  RR )
73 ltp1 9610 . . . . . . . . . . . . . 14  |-  ( I  e.  RR  ->  I  <  ( I  +  1 ) )
74 ltne 8933 . . . . . . . . . . . . . 14  |-  ( ( I  e.  RR  /\  I  <  ( I  + 
1 ) )  -> 
( I  +  1 )  =/=  I )
7572, 73, 74syl2anc 642 . . . . . . . . . . . . 13  |-  ( I  e.  RR  ->  (
I  +  1 )  =/=  I )
7675necomd 2542 . . . . . . . . . . . 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 3762 . . . . . . . . . 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 3474 . . . . . . . . 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 5122 . . . . . . . 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 3343 . . . . . 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 5105 . . . . . 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 3660 . . . . . 6  |-  { 1 ,  0 }  =  ( { 1 }  u.  { 0 } )
8885, 86, 873eqtr4g 2353 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  { 1 ,  0 } )
8956, 88eqeq12d 2310 . . . 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 4920 . . 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 2309 . . 3  |-  ( P  =  Q  <->  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) )
9695necon3abii 2489 . 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 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   <.cop 3656   class class class wbr 4039    X. cxp 4703   ran crn 4706   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    - cmin 9053   -ucneg 9054   NNcn 9762   2c2 9811   3c3 9812   ZZcz 10040   ZZ>=cuz 10246   ...cfz 10798
This theorem is referenced by:  axlowdimlem15  24656
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799
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