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Theorem mndifsplit 18945
Description: Lemma for maducoeval2 18949. (Contributed by SO, 16-Jul-2018.)
Hypotheses
Ref Expression
mndifsplit.b  |-  B  =  ( Base `  M
)
mndifsplit.0g  |-  .0.  =  ( 0g `  M )
mndifsplit.pg  |-  .+  =  ( +g  `  M )
Assertion
Ref Expression
mndifsplit  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) )

Proof of Theorem mndifsplit
StepHypRef Expression
1 pm2.21 108 . . . 4  |-  ( -.  ( ph  /\  ps )  ->  ( ( ph  /\ 
ps )  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) ) )
21imp 429 . . 3  |-  ( ( -.  ( ph  /\  ps )  /\  ( ph  /\  ps ) )  ->  if ( (
ph  \/  ps ) ,  A ,  .0.  )  =  ( if (
ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) )
323ad2antl3 1160 . 2  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( ph  /\  ps ) )  ->  if ( (
ph  \/  ps ) ,  A ,  .0.  )  =  ( if (
ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) )
4 mndifsplit.b . . . . . 6  |-  B  =  ( Base `  M
)
5 mndifsplit.pg . . . . . 6  |-  .+  =  ( +g  `  M )
6 mndifsplit.0g . . . . . 6  |-  .0.  =  ( 0g `  M )
74, 5, 6mndrid 15762 . . . . 5  |-  ( ( M  e.  Mnd  /\  A  e.  B )  ->  ( A  .+  .0.  )  =  A )
873adant3 1016 . . . 4  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  ( A  .+  .0.  )  =  A )
98adantr 465 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( ph  /\  -.  ps )
)  ->  ( A  .+  .0.  )  =  A )
10 iftrue 3945 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  .0.  )  =  A )
11 iffalse 3948 . . . . 5  |-  ( -. 
ps  ->  if ( ps ,  A ,  .0.  )  =  .0.  )
1210, 11oveqan12d 6304 . . . 4  |-  ( (
ph  /\  -.  ps )  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  )
)  =  ( A 
.+  .0.  ) )
1312adantl 466 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( ph  /\  -.  ps )
)  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) )  =  ( A  .+  .0.  )
)
14 iftrue 3945 . . . . 5  |-  ( (
ph  \/  ps )  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  A )
1514orcs 394 . . . 4  |-  ( ph  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  A )
1615ad2antrl 727 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( ph  /\  -.  ps )
)  ->  if (
( ph  \/  ps ) ,  A ,  .0.  )  =  A
)
179, 13, 163eqtr4rd 2519 . 2  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( ph  /\  -.  ps )
)  ->  if (
( ph  \/  ps ) ,  A ,  .0.  )  =  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) )
184, 5, 6mndlid 15761 . . . . 5  |-  ( ( M  e.  Mnd  /\  A  e.  B )  ->  (  .0.  .+  A
)  =  A )
19183adant3 1016 . . . 4  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  (  .0.  .+  A )  =  A )
2019adantr 465 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  ps )
)  ->  (  .0.  .+  A )  =  A )
21 iffalse 3948 . . . . 5  |-  ( -. 
ph  ->  if ( ph ,  A ,  .0.  )  =  .0.  )
22 iftrue 3945 . . . . 5  |-  ( ps 
->  if ( ps ,  A ,  .0.  )  =  A )
2321, 22oveqan12d 6304 . . . 4  |-  ( ( -.  ph  /\  ps )  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  )
)  =  (  .0.  .+  A ) )
2423adantl 466 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  ps )
)  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) )  =  (  .0.  .+  A )
)
2514olcs 395 . . . 4  |-  ( ps 
->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  A )
2625ad2antll 728 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  ps )
)  ->  if (
( ph  \/  ps ) ,  A ,  .0.  )  =  A
)
2720, 24, 263eqtr4rd 2519 . 2  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  ps )
)  ->  if (
( ph  \/  ps ) ,  A ,  .0.  )  =  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) )
28 simp1 996 . . . . 5  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  M  e.  Mnd )
294, 6mndidcl 15759 . . . . . 6  |-  ( M  e.  Mnd  ->  .0.  e.  B )
3028, 29syl 16 . . . . 5  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  .0.  e.  B )
314, 5, 6mndlid 15761 . . . . 5  |-  ( ( M  e.  Mnd  /\  .0.  e.  B )  -> 
(  .0.  .+  .0.  )  =  .0.  )
3228, 30, 31syl2anc 661 . . . 4  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  (  .0.  .+  .0.  )  =  .0.  )
3332adantr 465 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  -.  ps ) )  ->  (  .0.  .+  .0.  )  =  .0.  )
3421, 11oveqan12d 6304 . . . 4  |-  ( ( -.  ph  /\  -.  ps )  ->  ( if (
ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) )  =  (  .0.  .+  .0.  )
)
3534adantl 466 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  -.  ps ) )  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) )  =  (  .0.  .+  .0.  )
)
36 ioran 490 . . . . 5  |-  ( -.  ( ph  \/  ps ) 
<->  ( -.  ph  /\  -.  ps ) )
37 iffalse 3948 . . . . 5  |-  ( -.  ( ph  \/  ps )  ->  if ( (
ph  \/  ps ) ,  A ,  .0.  )  =  .0.  )
3836, 37sylbir 213 . . . 4  |-  ( ( -.  ph  /\  -.  ps )  ->  if ( (
ph  \/  ps ) ,  A ,  .0.  )  =  .0.  )
3938adantl 466 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  -.  ps ) )  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  .0.  )
4033, 35, 393eqtr4rd 2519 . 2  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  -.  ps ) )  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) )
413, 17, 27, 404casesdan 948 1  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ifcif 3939   ` cfv 5588  (class class class)co 6285   Basecbs 14493   +g cplusg 14558   0gc0g 14698   Mndcmnd 15729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5551  df-fun 5590  df-fv 5596  df-riota 6246  df-ov 6288  df-0g 14700  df-mnd 15735
This theorem is referenced by:  maducoeval2  18949  madugsum  18952
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