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Theorem axlowdimlem13 24977
Description: Lemma for axlowdim 24984. 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 10699 . . . . . . . . 9  |-  2  =/=  0
21neii 2625 . . . . . . . 8  |-  -.  2  =  0
3 eqcom 2457 . . . . . . . . 9  |-  ( 2  =  0  <->  0  = 
2 )
4 1pneg1e0 10715 . . . . . . . . . . 11  |-  ( 1  +  -u 1 )  =  0
54eqcomi 2459 . . . . . . . . . 10  |-  0  =  ( 1  + 
-u 1 )
6 df-2 10665 . . . . . . . . . 10  |-  2  =  ( 1  +  1 )
75, 6eqeq12i 2464 . . . . . . . . 9  |-  ( 0  =  2  <->  ( 1  +  -u 1 )  =  ( 1  +  1 ) )
8 ax-1cn 9594 . . . . . . . . . 10  |-  1  e.  CC
9 neg1cn 10710 . . . . . . . . . 10  |-  -u 1  e.  CC
108, 9, 8addcani 9823 . . . . . . . . 9  |-  ( ( 1  +  -u 1
)  =  ( 1  +  1 )  <->  -u 1  =  1 )
113, 7, 103bitri 275 . . . . . . . 8  |-  ( 2  =  0  <->  -u 1  =  1 )
122, 11mtbi 300 . . . . . . 7  |-  -.  -u 1  =  1
1312intnanr 925 . . . . . 6  |-  -.  ( -u 1  =  1  /\  0  =  0 )
14 ax-1ne0 9605 . . . . . . . . 9  |-  1  =/=  0
1514neii 2625 . . . . . . . 8  |-  -.  1  =  0
16 negeq0 9925 . . . . . . . . 9  |-  ( 1  e.  CC  ->  (
1  =  0  <->  -u 1  =  0 ) )
178, 16ax-mp 5 . . . . . . . 8  |-  ( 1  =  0  <->  -u 1  =  0 )
1815, 17mtbi 300 . . . . . . 7  |-  -.  -u 1  =  0
1918intnanr 925 . . . . . 6  |-  -.  ( -u 1  =  0  /\  0  =  1 )
2013, 19pm3.2ni 864 . . . . 5  |-  -.  (
( -u 1  =  1  /\  0  =  0 )  \/  ( -u
1  =  0  /\  0  =  1 ) )
21 negex 9870 . . . . . 6  |-  -u 1  e.  _V
22 c0ex 9634 . . . . . 6  |-  0  e.  _V
23 1ex 9635 . . . . . 6  |-  1  e.  _V
2421, 22, 23, 22preq12b 4150 . . . . 5  |-  ( {
-u 1 ,  0 }  =  { 1 ,  0 }  <->  ( ( -u 1  =  1  /\  0  =  0 )  \/  ( -u 1  =  0  /\  0  =  1 ) ) )
2520, 24mtbir 301 . . . 4  |-  -.  { -u 1 ,  0 }  =  { 1 ,  0 }
26 3ex 10682 . . . . . . . . 9  |-  3  e.  _V
2726rnsnop 5316 . . . . . . . 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 11192 . . . . . . . . . . . 12  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
30 eluzfz1 11803 . . . . . . . . . . . 12  |-  ( N  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... N
) )
3129, 30sylbi 199 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  1  e.  ( 1 ... N
) )
32 df-3 10666 . . . . . . . . . . . . . . . 16  |-  3  =  ( 2  +  1 )
33 1e0p1 11076 . . . . . . . . . . . . . . . 16  |-  1  =  ( 0  +  1 )
3432, 33eqeq12i 2464 . . . . . . . . . . . . . . 15  |-  ( 3  =  1  <->  ( 2  +  1 )  =  ( 0  +  1 ) )
35 2cn 10677 . . . . . . . . . . . . . . . 16  |-  2  e.  CC
36 0cn 9632 . . . . . . . . . . . . . . . 16  |-  0  e.  CC
3735, 36, 8addcan2i 9824 . . . . . . . . . . . . . . 15  |-  ( ( 2  +  1 )  =  ( 0  +  1 )  <->  2  = 
0 )
3834, 37bitri 253 . . . . . . . . . . . . . 14  |-  ( 3  =  1  <->  2  = 
0 )
3938necon3bii 2675 . . . . . . . . . . . . 13  |-  ( 3  =/=  1  <->  2  =/=  0 )
401, 39mpbir 213 . . . . . . . . . . . 12  |-  3  =/=  1
4140necomi 2677 . . . . . . . . . . 11  |-  1  =/=  3
4231, 41jctir 541 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1  e.  ( 1 ... N )  /\  1  =/=  3 ) )
43 eldifsn 4096 . . . . . . . . . 10  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  <-> 
( 1  e.  ( 1 ... N )  /\  1  =/=  3
) )
4442, 43sylibr 216 . . . . . . . . 9  |-  ( N  e.  NN  ->  1  e.  ( ( 1 ... N )  \  {
3 } ) )
4544adantr 467 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  1  e.  ( ( 1 ... N
)  \  { 3 } ) )
46 ne0i 3736 . . . . . . . 8  |-  ( 1  e.  ( ( 1 ... N )  \  { 3 } )  ->  ( ( 1 ... N )  \  { 3 } )  =/=  (/) )
47 rnxp 5266 . . . . . . . 8  |-  ( ( ( 1 ... N
)  \  { 3 } )  =/=  (/)  ->  ran  ( ( ( 1 ... N )  \  { 3 } )  X.  { 0 } )  =  { 0 } )
4845, 46, 473syl 18 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( (
( 1 ... N
)  \  { 3 } )  X.  {
0 } )  =  { 0 } )
4928, 48uneq12d 3588 . . . . . 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 5243 . . . . . 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 3970 . . . . . 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 6316 . . . . . . . . 9  |-  ( I  +  1 )  e. 
_V
5453rnsnop 5316 . . . . . . . 8  |-  ran  { <. ( I  +  1 ) ,  1 >. }  =  { 1 }
5554a1i 11 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  { <. (
I  +  1 ) ,  1 >. }  =  { 1 } )
56 nnz 10956 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  ZZ )
57 fzssp1 11838 . . . . . . . . . . . 12  |-  ( 1 ... ( N  - 
1 ) )  C_  ( 1 ... (
( N  -  1 )  +  1 ) )
58 zcn 10939 . . . . . . . . . . . . 13  |-  ( N  e.  ZZ  ->  N  e.  CC )
59 npcan1 10041 . . . . . . . . . . . . . 14  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
6059oveq2d 6304 . . . . . . . . . . . . 13  |-  ( N  e.  CC  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
6158, 60syl 17 . . . . . . . . . . . 12  |-  ( N  e.  ZZ  ->  (
1 ... ( ( N  -  1 )  +  1 ) )  =  ( 1 ... N
) )
6257, 61syl5sseq 3479 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6356, 62syl 17 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (
1 ... ( N  - 
1 ) )  C_  ( 1 ... N
) )
6463sselda 3431 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( 1 ... N ) )
65 elfzelz 11797 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
6665zred 11037 . . . . . . . . . . 11  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  RR )
67 id 22 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  e.  RR )
68 ltp1 10440 . . . . . . . . . . . 12  |-  ( I  e.  RR  ->  I  <  ( I  +  1 ) )
6967, 68ltned 9768 . . . . . . . . . . 11  |-  ( I  e.  RR  ->  I  =/=  ( I  +  1 ) )
7066, 69syl 17 . . . . . . . . . 10  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  =/=  ( I  +  1 ) )
7170adantl 468 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  =/=  (
I  +  1 ) )
72 eldifsn 4096 . . . . . . . . 9  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  <-> 
( I  e.  ( 1 ... N )  /\  I  =/=  (
I  +  1 ) ) )
7364, 71, 72sylanbrc 669 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  I  e.  ( ( 1 ... N
)  \  { (
I  +  1 ) } ) )
74 ne0i 3736 . . . . . . . 8  |-  ( I  e.  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  ->  ( ( 1 ... N )  \  { ( I  + 
1 ) } )  =/=  (/) )
75 rnxp 5266 . . . . . . . 8  |-  ( ( ( 1 ... N
)  \  { (
I  +  1 ) } )  =/=  (/)  ->  ran  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } )  =  { 0 } )
7673, 74, 753syl 18 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ran  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } )  =  { 0 } )
7755, 76uneq12d 3588 . . . . . 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 5243 . . . . . 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 3970 . . . . . 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 305 . . 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 5059 . . 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 125 . 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 2464 . . 3  |-  ( P  =  Q  <->  ( { <. 3 ,  -u 1 >. }  u.  ( ( ( 1 ... N
)  \  { 3 } )  X.  {
0 } ) )  =  ( { <. ( I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) ) )
8887necon3abii 2669 . 2  |-  ( P  =/=  Q  <->  -.  ( { <. 3 ,  -u
1 >. }  u.  (
( ( 1 ... N )  \  {
3 } )  X. 
{ 0 } ) )  =  ( {
<. ( I  +  1 ) ,  1 >. }  u.  ( (
( 1 ... N
)  \  { (
I  +  1 ) } )  X.  {
0 } ) ) )
8984, 88sylibr 216 1  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  P  =/=  Q
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1443    e. wcel 1886    =/= wne 2621    \ cdif 3400    u. cun 3401    C_ wss 3403   (/)c0 3730   {csn 3967   {cpr 3969   <.cop 3973    X. cxp 4831   ran crn 4834   ` cfv 5581  (class class class)co 6288   CCcc 9534   RRcr 9535   0cc0 9536   1c1 9537    + caddc 9539    - cmin 9857   -ucneg 9858   NNcn 10606   2c2 10656   3c3 10657   ZZcz 10934   ZZ>=cuz 11156   ...cfz 11781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-3 10666  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782
This theorem is referenced by:  axlowdimlem15  24979
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