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Theorem axlowdimlem13 23203
Description: Lemma for axlowdim 23210. 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 10417 . . . . . . . . 9  |-  2  =/=  0
2 df-ne 2611 . . . . . . . . 9  |-  ( 2  =/=  0  <->  -.  2  =  0 )
31, 2mpbi 208 . . . . . . . 8  |-  -.  2  =  0
4 eqcom 2445 . . . . . . . . 9  |-  ( 2  =  0  <->  0  = 
2 )
5 1pneg1e0 10433 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
65eqcomi 2447 . . . . . . . . . 10  |-  0  =  ( 1  + 
-u 1 )
7 df-2 10383 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
86, 7eqeq12i 2456 . . . . . . . . 9  |-  ( 0  =  2  <->  ( 1  +  -u 1 )  =  ( 1  +  1 ) )
9 ax-1cn 9343 . . . . . . . . . 10  |-  1  e.  CC
10 neg1cn 10428 . . . . . . . . . 10  |-  -u 1  e.  CC
119, 10, 9addcani 9565 . . . . . . . . 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 906 . . . . . 6  |-  -.  ( -u 1  =  1  /\  0  =  0 )
15 ax-1ne0 9354 . . . . . . . . 9  |-  1  =/=  0
16 df-ne 2611 . . . . . . . . 9  |-  ( 1  =/=  0  <->  -.  1  =  0 )
1715, 16mpbi 208 . . . . . . . 8  |-  -.  1  =  0
18 negeq0 9666 . . . . . . . . 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 906 . . . . . 6  |-  -.  ( -u 1  =  0  /\  0  =  1 )
2214, 21pm3.2ni 850 . . . . 5  |-  -.  (
( -u 1  =  1  /\  0  =  0 )  \/  ( -u
1  =  0  /\  0  =  1 ) )
23 negex 9611 . . . . . 6  |-  -u 1  e.  _V
24 c0ex 9383 . . . . . 6  |-  0  e.  _V
259elexi 2985 . . . . . 6  |-  1  e.  _V
2623, 24, 25, 24preq12b 4051 . . . . 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 10400 . . . . . . . . 9  |-  3  e.  _V
2928rnsnop 5323 . . . . . . . 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 10900 . . . . . . . . . . . 12  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
32 eluzfz1 11461 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
3331, 32sylbi 195 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
34 df-3 10384 . . . . . . . . . . . . . . . 16  |-  3  =  ( 2  +  1 )
35 1e0p1 10786 . . . . . . . . . . . . . . . 16  |-  1  =  ( 0  +  1 )
3634, 35eqeq12i 2456 . . . . . . . . . . . . . . 15  |-  ( 3  =  1  <->  ( 2  +  1 )  =  ( 0  +  1 ) )
37 2cn 10395 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
38 0cn 9381 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
3937, 38, 9addcan2i 9566 . . . . . . . . . . . . . . 15  |-  ( ( 2  +  1 )  =  ( 0  +  1 )  <->  2  = 
0 )
4036, 39bitri 249 . . . . . . . . . . . . . 14  |-  ( 3  =  1  <->  2  = 
0 )
4140necon3bii 2643 . . . . . . . . . . . . 13  |-  ( 3  =/=  1  <->  2  =/=  0 )
421, 41mpbir 209 . . . . . . . . . . . 12  |-  3  =/=  1
4342necomi 2697 . . . . . . . . . . 11  |-  1  =/=  3
4433, 43jctir 538 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1  e.  ( 1 ... N )  /\  1  =/=  3 ) )
45 eldifsn 4003 . . . . . . . . . 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 3646 . . . . . . . 8  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( 1 ... N )  \  { 3 } )  =/=  (/) )
49 rnxp 5271 . . . . . . . 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 3514 . . . . . 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 5248 . . . . . 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 3883 . . . . . 6  |-  { -u
1 ,  0 }  =  ( { -u
1 }  u.  {
0 } )
5451, 52, 533eqtr4g 2500 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  { -u 1 ,  0 } )
55 ovex 6119 . . . . . . . . 9  |-  ( I  +  1 )  e. 
_V
5655rnsnop 5323 . . . . . . . 8  |-  ran  { <. ( I  +  1 ) ,  1 >. }  =  { 1 }
5756a1i 11 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. (
I  +  1 ) ,  1 >. }  =  { 1 } )
58 nnz 10671 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  ZZ )
59 fzssp1 11504 . . . . . . . . . . . 12  |-  ( 1 ... ( N  - 
1 ) )  C_  ( 1 ... (
( N  -  1 )  +  1 ) )
60 zcn 10654 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
61 npcan 9622 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( ( N  - 
1 )  +  1 )  =  N )
629, 61mpan2 671 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
6362oveq2d 6110 . . . . . . . . . . . . 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 3407 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6658, 65syl 16 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6766sselda 3359 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( 1 ... N ) )
68 elfzelz 11456 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
6968zred 10750 . . . . . . . . . . 11  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  RR )
70 id 22 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  e.  RR )
71 ltp1 10170 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  <  ( I  +  1 ) )
7270, 71ltned 9513 . . . . . . . . . . 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 4003 . . . . . . . . 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 3646 . . . . . . . 8  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  =/=  (/) )
78 rnxp 5271 . . . . . . . 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 3514 . . . . . 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 5248 . . . . . 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 3883 . . . . . 6  |-  { 1 ,  0 }  =  ( { 1 }  u.  { 0 } )
8380, 81, 823eqtr4g 2500 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) )  =  { 1 ,  0 } )
8454, 83eqeq12d 2457 . . . 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 5068 . . 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 2456 . . 3  |-  ( P  =  Q  <->  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) )
9190necon3abii 2641 . 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 1369    e. wcel 1756    =/= wne 2609    \ cdif 3328    u. cun 3329    C_ wss 3331   (/)c0 3640   {csn 3880   {cpr 3882   <.cop 3886    X. cxp 4841   ran crn 4844   ` cfv 5421  (class class class)co 6094   CCcc 9283   RRcr 9284   0cc0 9285   1c1 9286    + caddc 9288    - cmin 9598   -ucneg 9599   NNcn 10325   2c2 10374   3c3 10375   ZZcz 10649   ZZ>=cuz 10864   ...cfz 11440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-1cn 9343  ax-icn 9344  ax-addcl 9345  ax-addrcl 9346  ax-mulcl 9347  ax-mulrcl 9348  ax-mulcom 9349  ax-addass 9350  ax-mulass 9351  ax-distr 9352  ax-i2m1 9353  ax-1ne0 9354  ax-1rid 9355  ax-rnegex 9356  ax-rrecex 9357  ax-cnre 9358  ax-pre-lttri 9359  ax-pre-lttrn 9360  ax-pre-ltadd 9361  ax-pre-mulgt0 9362
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-reu 2725  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-pss 3347  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-tp 3885  df-op 3887  df-uni 4095  df-iun 4176  df-br 4296  df-opab 4354  df-mpt 4355  df-tr 4389  df-eprel 4635  df-id 4639  df-po 4644  df-so 4645  df-fr 4682  df-we 4684  df-ord 4725  df-on 4726  df-lim 4727  df-suc 4728  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-riota 6055  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-om 6480  df-1st 6580  df-2nd 6581  df-recs 6835  df-rdg 6869  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-sub 9600  df-neg 9601  df-nn 10326  df-2 10383  df-3 10384  df-n0 10583  df-z 10650  df-uz 10865  df-fz 11441
This theorem is referenced by:  axlowdimlem15  23205
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