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Theorem axlowdimlem14 24583
Description: Lemma for axlowdim 24589. Take two possible  Q from axlowdimlem10 24579. 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 24579 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  Q  e.  ( EE `  N ) )
3 elee 24522 . . . . . . 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 5389 . . . . 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 24579 . . . . . 6  |-  ( ( N  e.  NN  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  R  e.  ( EE `  N ) )
10 elee 24522 . . . . . . 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 5389 . . . . 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 5622 . . . 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 24093 . . . . . . . 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 8806 . . . . . . . 8  |-  1  =/=  0
2221a1i 10 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  1  =/=  0 )
231axlowdimlem11 24580 . . . . . . . 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 10798 . . . . . . . . . . . . 13  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
2625zcnd 10118 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  CC )
27 elfzelz 10798 . . . . . . . . . . . . 13  |-  ( J  e.  ( 1 ... ( N  -  1 ) )  ->  J  e.  ZZ )
2827zcnd 10118 . . . . . . . . . . . 12  |-  ( J  e.  ( 1 ... ( N  -  1 ) )  ->  J  e.  CC )
29 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
30 addcan2 8997 . . . . . . . . . . . . 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 2481 . . . . . . . . 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 24581 . . . . . . . 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 2473 . . . . . 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 2448 . . . . . . . 8  |-  ( ( Q `  i )  =/=  ( R `  i )  <->  -.  ( Q `  i )  =  ( R `  i ) )
40 fveq2 5525 . . . . . . . . 9  |-  ( i  =  ( I  + 
1 )  ->  ( Q `  i )  =  ( Q `  ( I  +  1
) ) )
41 fveq2 5525 . . . . . . . . 9  |-  ( i  =  ( I  + 
1 )  ->  ( R `  i )  =  ( R `  ( I  +  1
) ) )
4240, 41neeq12d 2461 . . . . . . . 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 2884 . . . . . 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 2448 . . . 4  |-  ( I  =/=  J  <->  -.  I  =  J )
48 rexnal 2554 . . . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    \ cdif 3149    u. cun 3150   {csn 3640   <.cop 3643    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    - cmin 9037   NNcn 9746   ...cfz 10782   EEcee 24516
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-map 6774  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-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-ee 24519
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