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Theorem axlowdimlem14 25896
Description: Lemma for axlowdim 25902. Take two possible  Q from axlowdimlem10 25892. 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 25892 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  Q  e.  ( EE `  N ) )
3 elee 25835 . . . . . . 7  |-  ( N  e.  NN  ->  ( Q  e.  ( EE `  N )  <->  Q :
( 1 ... N
) --> RR ) )
43adantr 453 . . . . . 6  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( Q  e.  ( EE `  N
)  <->  Q : ( 1 ... N ) --> RR ) )
52, 4mpbid 203 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  Q : ( 1 ... N ) --> RR )
6 ffn 5593 . . . . 5  |-  ( Q : ( 1 ... N ) --> RR  ->  Q  Fn  ( 1 ... N ) )
75, 6syl 16 . . . 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 25892 . . . . . 6  |-  ( ( N  e.  NN  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  R  e.  ( EE `  N ) )
10 elee 25835 . . . . . . 7  |-  ( N  e.  NN  ->  ( R  e.  ( EE `  N )  <->  R :
( 1 ... N
) --> RR ) )
1110adantr 453 . . . . . 6  |-  ( ( N  e.  NN  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( R  e.  ( EE `  N
)  <->  R : ( 1 ... N ) --> RR ) )
129, 11mpbid 203 . . . . 5  |-  ( ( N  e.  NN  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  R : ( 1 ... N ) --> RR )
13 ffn 5593 . . . . 5  |-  ( R : ( 1 ... N ) --> RR  ->  R  Fn  ( 1 ... N ) )
1412, 13syl 16 . . . 4  |-  ( ( N  e.  NN  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  R  Fn  (
1 ... N ) )
15 eqfnfv 5829 . . . 4  |-  ( ( Q  Fn  ( 1 ... N )  /\  R  Fn  ( 1 ... N ) )  ->  ( Q  =  R  <->  A. i  e.  ( 1 ... N ) ( Q `  i
)  =  ( R `
 i ) ) )
167, 14, 15syl2an 465 . . 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 1240 . 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 25200 . . . . . . . 8  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( I  + 
1 )  e.  ( 1 ... N ) )
19183adant3 978 . . . . . . 7  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( I  + 
1 )  e.  ( 1 ... N ) )
2019adantr 453 . . . . . 6  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  ( I  +  1 )  e.  ( 1 ... N
) )
21 ax-1ne0 9061 . . . . . . . 8  |-  1  =/=  0
2221a1i 11 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  1  =/=  0 )
231axlowdimlem11 25893 . . . . . . . 8  |-  ( Q `
 ( I  + 
1 ) )  =  1
2423a1i 11 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  ( Q `  ( I  +  1 ) )  =  1 )
25 elfzelz 11061 . . . . . . . . . . . . 13  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  ZZ )
2625zcnd 10378 . . . . . . . . . . . 12  |-  ( I  e.  ( 1 ... ( N  -  1 ) )  ->  I  e.  CC )
27 elfzelz 11061 . . . . . . . . . . . . 13  |-  ( J  e.  ( 1 ... ( N  -  1 ) )  ->  J  e.  ZZ )
2827zcnd 10378 . . . . . . . . . . . 12  |-  ( J  e.  ( 1 ... ( N  -  1 ) )  ->  J  e.  CC )
29 ax-1cn 9050 . . . . . . . . . . . . 13  |-  1  e.  CC
30 addcan2 9253 . . . . . . . . . . . . 13  |-  ( ( I  e.  CC  /\  J  e.  CC  /\  1  e.  CC )  ->  (
( I  +  1 )  =  ( J  +  1 )  <->  I  =  J ) )
3129, 30mp3an3 1269 . . . . . . . . . . . 12  |-  ( ( I  e.  CC  /\  J  e.  CC )  ->  ( ( I  + 
1 )  =  ( J  +  1 )  <-> 
I  =  J ) )
3226, 28, 31syl2an 465 . . . . . . . . . . 11  |-  ( ( I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ( I  +  1 )  =  ( J  +  1 )  <->  I  =  J
) )
33323adant1 976 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ( I  +  1 )  =  ( J  +  1 )  <->  I  =  J
) )
3433necon3bid 2638 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( ( I  +  1 )  =/=  ( J  +  1 )  <->  I  =/=  J
) )
3534biimpar 473 . . . . . . . 8  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  ( I  +  1 )  =/=  ( J  +  1 ) )
368axlowdimlem12 25894 . . . . . . . 8  |-  ( ( ( I  +  1 )  e.  ( 1 ... N )  /\  ( I  +  1
)  =/=  ( J  +  1 ) )  ->  ( R `  ( I  +  1
) )  =  0 )
3720, 35, 36syl2anc 644 . . . . . . 7  |-  ( ( ( N  e.  NN  /\  I  e.  ( 1 ... ( N  - 
1 ) )  /\  J  e.  ( 1 ... ( N  - 
1 ) ) )  /\  I  =/=  J
)  ->  ( R `  ( I  +  1 ) )  =  0 )
3822, 24, 373netr4d 2630 . . . . . 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 2603 . . . . . . . 8  |-  ( ( Q `  i )  =/=  ( R `  i )  <->  -.  ( Q `  i )  =  ( R `  i ) )
40 fveq2 5730 . . . . . . . . 9  |-  ( i  =  ( I  + 
1 )  ->  ( Q `  i )  =  ( Q `  ( I  +  1
) ) )
41 fveq2 5730 . . . . . . . . 9  |-  ( i  =  ( I  + 
1 )  ->  ( R `  i )  =  ( R `  ( I  +  1
) ) )
4240, 41neeq12d 2618 . . . . . . . 8  |-  ( i  =  ( I  + 
1 )  ->  (
( Q `  i
)  =/=  ( R `
 i )  <->  ( Q `  ( I  +  1 ) )  =/=  ( R `  ( I  +  1 ) ) ) )
4339, 42syl5bbr 252 . . . . . . 7  |-  ( i  =  ( I  + 
1 )  ->  ( -.  ( Q `  i
)  =  ( R `
 i )  <->  ( Q `  ( I  +  1 ) )  =/=  ( R `  ( I  +  1 ) ) ) )
4443rspcev 3054 . . . . . 6  |-  ( ( ( I  +  1 )  e.  ( 1 ... N )  /\  ( Q `  ( I  +  1 ) )  =/=  ( R `  ( I  +  1
) ) )  ->  E. i  e.  (
1 ... N )  -.  ( Q `  i
)  =  ( R `
 i ) )
4520, 38, 44syl2anc 644 . . . . 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 425 . . . 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 2603 . . . 4  |-  ( I  =/=  J  <->  -.  I  =  J )
48 rexnal 2718 . . . 4  |-  ( E. i  e.  ( 1 ... N )  -.  ( Q `  i
)  =  ( R `
 i )  <->  -.  A. i  e.  ( 1 ... N
) ( Q `  i )  =  ( R `  i ) )
4946, 47, 483imtr3g 262 . . 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 100 . 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 208 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   A.wral 2707   E.wrex 2708    \ cdif 3319    u. cun 3320   {csn 3816   <.cop 3819    X. cxp 4878    Fn wfn 5451   -->wf 5452   ` cfv 5456  (class class class)co 6083   CCcc 8990   RRcr 8991   0cc0 8992   1c1 8993    + caddc 8995    - cmin 9293   NNcn 10002   ...cfz 11045   EEcee 25829
This theorem is referenced by:  axlowdimlem15  25897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-n0 10224  df-z 10285  df-uz 10491  df-fz 11046  df-ee 25832
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