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Theorem axlowdimlem14 24969
Description: Lemma for axlowdim 24975. Take two possible  Q from axlowdimlem10 24965. They are the same iff their distunguished values are the same. (Contributed by Scott Fenton, 21-Apr-2013.)
Hypotheses
Ref Expression
axlowdimlem14.1  |-  Q  =  ( { <. (
I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) )
axlowdimlem14.2  |-  R  =  ( { <. ( J  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( J  + 
1 ) } )  X.  { 0 } ) )
Assertion
Ref Expression
axlowdimlem14  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( Q  =  R  ->  I  =  J ) )

Proof of Theorem axlowdimlem14
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 axlowdimlem14.1 . . . . . . 7  |-  Q  =  ( { <. (
I  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( I  + 
1 ) } )  X.  { 0 } ) )
21axlowdimlem10 24965 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  Q  e.  ( EE `  N ) )
3 elee 24908 . . . . . . 7  |-  ( N  e.  NN  ->  ( Q  e.  ( EE `  N )  <->  Q :
( 1 ... N
) --> RR ) )
43adantr 451 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( Q  e.  ( EE `  N
)  <->  Q : ( 1 ... N ) --> RR ) )
52, 4mpbid 201 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  Q : ( 1 ... N ) --> RR )
6 ffn 5427 . . . . 5  |-  ( Q : ( 1 ... N ) --> RR  ->  Q  Fn  ( 1 ... N ) )
75, 6syl 15 . . . 4  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  Q  Fn  (
1 ... N ) )
8 axlowdimlem14.2 . . . . . . 7  |-  R  =  ( { <. ( J  +  1 ) ,  1 >. }  u.  ( ( ( 1 ... N )  \  { ( J  + 
1 ) } )  X.  { 0 } ) )
98axlowdimlem10 24965 . . . . . 6  |-  ( ( N  e.  NN  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  R  e.  ( EE `  N ) )
10 elee 24908 . . . . . . 7  |-  ( N  e.  NN  ->  ( R  e.  ( EE `  N )  <->  R :
( 1 ... N
) --> RR ) )
1110adantr 451 . . . . . 6  |-  ( ( N  e.  NN  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( R  e.  ( EE `  N
)  <->  R : ( 1 ... N ) --> RR ) )
129, 11mpbid 201 . . . . 5  |-  ( ( N  e.  NN  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  R : ( 1 ... N ) --> RR )
13 ffn 5427 . . . . 5  |-  ( R : ( 1 ... N ) --> RR  ->  R  Fn  ( 1 ... N ) )
1412, 13syl 15 . . . 4  |-  ( ( N  e.  NN  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  R  Fn  (
1 ... N ) )
15 eqfnfv 5660 . . . 4  |-  ( ( Q  Fn  ( 1 ... N )  /\  R  Fn  ( 1 ... N ) )  ->  ( Q  =  R  <->  A. i  e.  ( 1 ... N ) ( Q `  i
)  =  ( R `
 i ) ) )
167, 14, 15syl2an 463 . . 3  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  /\  ( N  e.  NN  /\  J  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( Q  =  R  <->  A. i  e.  (
1 ... N ) ( Q `  i )  =  ( R `  i ) ) )
17163impdi 1237 . 2  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( Q  =  R  <->  A. i  e.  ( 1 ... N ) ( Q `  i
)  =  ( R `
 i ) ) )
18 fznatpl1 24379 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( I  + 
1 )  e.  ( 1 ... N ) )
19183adant3 975 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( I  + 
1 )  e.  ( 1 ... N ) )
2019adantr 451 . . . . . 6  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  ( I  +  1 )  e.  ( 1 ... N
) )
21 ax-1ne0 8851 . . . . . . . 8  |-  1  =/=  0
2221a1i 10 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  1  =/=  0 )
231axlowdimlem11 24966 . . . . . . . 8  |-  ( Q `
 ( I  + 
1 ) )  =  1
2423a1i 10 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  ( Q `  ( I  +  1 ) )  =  1 )
25 elfzelz 10845 . . . . . . . . . . . . 13  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
2625zcnd 10165 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  CC )
27 elfzelz 10845 . . . . . . . . . . . . 13  |-  ( J  e.  ( 1 ... ( N  -  1 ) )  ->  J  e.  ZZ )
2827zcnd 10165 . . . . . . . . . . . 12  |-  ( J  e.  ( 1 ... ( N  -  1 ) )  ->  J  e.  CC )
29 ax-1cn 8840 . . . . . . . . . . . . 13  |-  1  e.  CC
30 addcan2 9042 . . . . . . . . . . . . 13  |-  ( ( I  e.  CC  /\  J  e.  CC  /\  1  e.  CC )  ->  (
( I  +  1 )  =  ( J  +  1 )  <->  I  =  J ) )
3129, 30mp3an3 1266 . . . . . . . . . . . 12  |-  ( ( I  e.  CC  /\  J  e.  CC )  ->  ( ( I  + 
1 )  =  ( J  +  1 )  <-> 
I  =  J ) )
3226, 28, 31syl2an 463 . . . . . . . . . . 11  |-  ( ( I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ( I  +  1 )  =  ( J  +  1 )  <->  I  =  J
) )
33323adant1 973 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ( I  +  1 )  =  ( J  +  1 )  <->  I  =  J
) )
3433necon3bid 2514 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ( I  +  1 )  =/=  ( J  +  1 )  <->  I  =/=  J
) )
3534biimpar 471 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  ( I  +  1 )  =/=  ( J  +  1 ) )
368axlowdimlem12 24967 . . . . . . . 8  |-  ( ( ( I  +  1 )  e.  ( 1 ... N )  /\  ( I  +  1
)  =/=  ( J  +  1 ) )  ->  ( R `  ( I  +  1
) )  =  0 )
3720, 35, 36syl2anc 642 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  ( R `  ( I  +  1 ) )  =  0 )
3822, 24, 373netr4d 2506 . . . . . 6  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  ( Q `  ( I  +  1 ) )  =/=  ( R `  ( I  +  1 ) ) )
39 df-ne 2481 . . . . . . . 8  |-  ( ( Q `  i )  =/=  ( R `  i )  <->  -.  ( Q `  i )  =  ( R `  i ) )
40 fveq2 5563 . . . . . . . . 9  |-  ( i  =  ( I  + 
1 )  ->  ( Q `  i )  =  ( Q `  ( I  +  1
) ) )
41 fveq2 5563 . . . . . . . . 9  |-  ( i  =  ( I  + 
1 )  ->  ( R `  i )  =  ( R `  ( I  +  1
) ) )
4240, 41neeq12d 2494 . . . . . . . 8  |-  ( i  =  ( I  + 
1 )  ->  (
( Q `  i
)  =/=  ( R `
 i )  <->  ( Q `  ( I  +  1 ) )  =/=  ( R `  ( I  +  1 ) ) ) )
4339, 42syl5bbr 250 . . . . . . 7  |-  ( i  =  ( I  + 
1 )  ->  ( -.  ( Q `  i
)  =  ( R `
 i )  <->  ( Q `  ( I  +  1 ) )  =/=  ( R `  ( I  +  1 ) ) ) )
4443rspcev 2918 . . . . . 6  |-  ( ( ( I  +  1 )  e.  ( 1 ... N )  /\  ( Q `  ( I  +  1 ) )  =/=  ( R `  ( I  +  1
) ) )  ->  E. i  e.  (
1 ... N )  -.  ( Q `  i
)  =  ( R `
 i ) )
4520, 38, 44syl2anc 642 . . . . 5  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  E. i  e.  ( 1 ... N
)  -.  ( Q `
 i )  =  ( R `  i
) )
4645ex 423 . . . 4  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( I  =/= 
J  ->  E. i  e.  ( 1 ... N
)  -.  ( Q `
 i )  =  ( R `  i
) ) )
47 df-ne 2481 . . . 4  |-  ( I  =/=  J  <->  -.  I  =  J )
48 rexnal 2588 . . . 4  |-  ( E. i  e.  ( 1 ... N )  -.  ( Q `  i
)  =  ( R `
 i )  <->  -.  A. i  e.  ( 1 ... N
) ( Q `  i )  =  ( R `  i ) )
4946, 47, 483imtr3g 260 . . 3  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( -.  I  =  J  ->  -.  A. i  e.  ( 1 ... N ) ( Q `  i )  =  ( R `  i ) ) )
5049con4d 97 . 2  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( A. i  e.  ( 1 ... N
) ( Q `  i )  =  ( R `  i )  ->  I  =  J ) )
5117, 50sylbid 206 1  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( Q  =  R  ->  I  =  J ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   A.wral 2577   E.wrex 2578    \ cdif 3183    u. cun 3184   {csn 3674   <.cop 3677    X. cxp 4724    Fn wfn 5287   -->wf 5288   ` cfv 5292  (class class class)co 5900   CCcc 8780   RRcr 8781   0cc0 8782   1c1 8783    + caddc 8785    - cmin 9082   NNcn 9791   ...cfz 10829   EEcee 24902
This theorem is referenced by:  axlowdimlem15  24970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-n0 10013  df-z 10072  df-uz 10278  df-fz 10830  df-ee 24905
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