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Theorem subaddv 25671
Description: Relationship between subtraction and addition. (Contributed by FL, 30-May-2014.)
Hypotheses
Ref Expression
subaddv.1  |-  + w  =  (  + cv `  N )
subaddv.2  |-  - w  =  (  - cv  `  N
)
Assertion
Ref Expression
subaddv  |-  ( ( N  e.  NN  /\  ( A  e.  ( CC  ^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N
) )  /\  C  e.  ( CC  ^m  (
1 ... N ) ) ) )  ->  (
( A - w B )  =  C  <-> 
( B + w C )  =  A ) )

Proof of Theorem subaddv
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 subaddv.1 . . . . 5  |-  + w  =  (  + cv `  N )
2 subaddv.2 . . . . 5  |-  - w  =  (  - cv  `  N
)
31, 2issubcv 25670 . . . 4  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) )  /\  B  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( A - w B )  =  ( iota_ x  e.  ( CC  ^m  ( 1 ... N ) ) ( B + w
x )  =  A ) )
43eqeq1d 2291 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) )  /\  B  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( ( A - w B )  =  C  <->  ( iota_ x  e.  ( CC  ^m  ( 1 ... N
) ) ( B + w x )  =  A )  =  C ) )
543adant3r3 1162 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( CC  ^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N
) )  /\  C  e.  ( CC  ^m  (
1 ... N ) ) ) )  ->  (
( A - w B )  =  C  <-> 
( iota_ x  e.  ( CC  ^m  ( 1 ... N ) ) ( B + w
x )  =  A )  =  C ) )
6 simpll 730 . . . . . . 7  |-  ( ( ( C  e.  ( CC  ^m  ( 1 ... N ) )  /\  ( B  e.  ( CC  ^m  (
1 ... N ) )  /\  A  e.  ( CC  ^m  ( 1 ... N ) ) ) )  /\  N  e.  NN )  ->  C  e.  ( CC  ^m  (
1 ... N ) ) )
71negveud 25668 . . . . . . . . . . . 12  |-  ( ( N  e.  NN  /\  B  e.  ( CC  ^m  ( 1 ... N
) )  /\  A  e.  ( CC  ^m  (
1 ... N ) ) )  ->  E! x  e.  ( CC  ^m  (
1 ... N ) ) ( B + w
x )  =  A )
873exp 1150 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  ( B  e.  ( CC  ^m  ( 1 ... N
) )  ->  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  E! x  e.  ( CC  ^m  ( 1 ... N
) ) ( B + w x )  =  A ) ) )
98com3l 75 . . . . . . . . . 10  |-  ( B  e.  ( CC  ^m  ( 1 ... N
) )  ->  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  ( N  e.  NN  ->  E! x  e.  ( CC 
^m  ( 1 ... N ) ) ( B + w x
)  =  A ) ) )
109imp 418 . . . . . . . . 9  |-  ( ( B  e.  ( CC 
^m  ( 1 ... N ) )  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( N  e.  NN  ->  E! x  e.  ( CC  ^m  ( 1 ... N ) ) ( B + w
x )  =  A ) )
1110adantl 452 . . . . . . . 8  |-  ( ( C  e.  ( CC 
^m  ( 1 ... N ) )  /\  ( B  e.  ( CC  ^m  ( 1 ... N ) )  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) ) )  ->  ( N  e.  NN  ->  E! x  e.  ( CC  ^m  (
1 ... N ) ) ( B + w
x )  =  A ) )
1211imp 418 . . . . . . 7  |-  ( ( ( C  e.  ( CC  ^m  ( 1 ... N ) )  /\  ( B  e.  ( CC  ^m  (
1 ... N ) )  /\  A  e.  ( CC  ^m  ( 1 ... N ) ) ) )  /\  N  e.  NN )  ->  E! x  e.  ( CC  ^m  ( 1 ... N
) ) ( B + w x )  =  A )
13 oveq2 5866 . . . . . . . . 9  |-  ( x  =  C  ->  ( B + w x )  =  ( B + w C ) )
1413eqeq1d 2291 . . . . . . . 8  |-  ( x  =  C  ->  (
( B + w
x )  =  A  <-> 
( B + w C )  =  A ) )
1514riota2 6327 . . . . . . 7  |-  ( ( C  e.  ( CC 
^m  ( 1 ... N ) )  /\  E! x  e.  ( CC  ^m  ( 1 ... N ) ) ( B + w x
)  =  A )  ->  ( ( B + w C )  =  A  <->  ( iota_ x  e.  ( CC  ^m  ( 1 ... N
) ) ( B + w x )  =  A )  =  C ) )
166, 12, 15syl2anc 642 . . . . . 6  |-  ( ( ( C  e.  ( CC  ^m  ( 1 ... N ) )  /\  ( B  e.  ( CC  ^m  (
1 ... N ) )  /\  A  e.  ( CC  ^m  ( 1 ... N ) ) ) )  /\  N  e.  NN )  ->  (
( B + w C )  =  A  <-> 
( iota_ x  e.  ( CC  ^m  ( 1 ... N ) ) ( B + w
x )  =  A )  =  C ) )
1716ex 423 . . . . 5  |-  ( ( C  e.  ( CC 
^m  ( 1 ... N ) )  /\  ( B  e.  ( CC  ^m  ( 1 ... N ) )  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) ) )  ->  ( N  e.  NN  ->  ( ( B + w C )  =  A  <->  ( iota_ x  e.  ( CC  ^m  ( 1 ... N
) ) ( B + w x )  =  A )  =  C ) ) )
18173impb 1147 . . . 4  |-  ( ( C  e.  ( CC 
^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N
) )  /\  A  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( N  e.  NN  ->  ( ( B + w C )  =  A  <->  ( iota_ x  e.  ( CC  ^m  ( 1 ... N
) ) ( B + w x )  =  A )  =  C ) ) )
19183com13 1156 . . 3  |-  ( ( A  e.  ( CC 
^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N
) )  /\  C  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( N  e.  NN  ->  ( ( B + w C )  =  A  <->  ( iota_ x  e.  ( CC  ^m  ( 1 ... N
) ) ( B + w x )  =  A )  =  C ) ) )
2019impcom 419 . 2  |-  ( ( N  e.  NN  /\  ( A  e.  ( CC  ^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N
) )  /\  C  e.  ( CC  ^m  (
1 ... N ) ) ) )  ->  (
( B + w C )  =  A  <-> 
( iota_ x  e.  ( CC  ^m  ( 1 ... N ) ) ( B + w
x )  =  A )  =  C ) )
215, 20bitr4d 247 1  |-  ( ( N  e.  NN  /\  ( A  e.  ( CC  ^m  ( 1 ... N ) )  /\  B  e.  ( CC  ^m  ( 1 ... N
) )  /\  C  e.  ( CC  ^m  (
1 ... N ) ) ) )  ->  (
( A - w B )  =  C  <-> 
( B + w C )  =  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E!wreu 2545   ` cfv 5255  (class class class)co 5858   iota_crio 6297    ^m cmap 6772   CCcc 8735   1c1 8738   NNcn 9746   ...cfz 10782    + cvcplcv 25644    - cv cmcv 25664
This theorem is referenced by:  issubrv  25672
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-rep 4131  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
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-rmo 2551  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-ltxr 8872  df-nn 9747  df-addcv 25645  df-nullcv 25651  df-subcatv 25665
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