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Theorem mndifsplit 19592
Description: Lemma for maducoeval2 19596. (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 pm2.21 111 . . . 4  |-  ( -.  ( ph  /\  ps )  ->  ( ( ph  /\ 
ps )  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) ) )
21imp 430 . . 3  |-  ( ( -.  ( ph  /\  ps )  /\  ( ph  /\  ps ) )  ->  if ( (
ph  \/  ps ) ,  A ,  .0.  )  =  ( if (
ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) ) )
323ad2antl3 1169 . 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 16509 . . . . 5  |-  ( ( M  e.  Mnd  /\  A  e.  B )  ->  ( A  .+  .0.  )  =  A )
873adant3 1025 . . . 4  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  ( A  .+  .0.  )  =  A )
98adantr 466 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( ph  /\  -.  ps )
)  ->  ( A  .+  .0.  )  =  A )
10 iftrue 3921 . . . . 5  |-  ( ph  ->  if ( ph ,  A ,  .0.  )  =  A )
11 iffalse 3924 . . . . 5  |-  ( -. 
ps  ->  if ( ps ,  A ,  .0.  )  =  .0.  )
1210, 11oveqan12d 6324 . . . 4  |-  ( (
ph  /\  -.  ps )  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  )
)  =  ( A 
.+  .0.  ) )
1312adantl 467 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( ph  /\  -.  ps )
)  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) )  =  ( A  .+  .0.  )
)
14 iftrue 3921 . . . . 5  |-  ( (
ph  \/  ps )  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  A )
1514orcs 395 . . . 4  |-  ( ph  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  A )
1615ad2antrl 732 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( ph  /\  -.  ps )
)  ->  if (
( ph  \/  ps ) ,  A ,  .0.  )  =  A
)
179, 13, 163eqtr4rd 2481 . 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 16508 . . . . 5  |-  ( ( M  e.  Mnd  /\  A  e.  B )  ->  (  .0.  .+  A
)  =  A )
19183adant3 1025 . . . 4  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  (  .0.  .+  A )  =  A )
2019adantr 466 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  ps )
)  ->  (  .0.  .+  A )  =  A )
21 iffalse 3924 . . . . 5  |-  ( -. 
ph  ->  if ( ph ,  A ,  .0.  )  =  .0.  )
22 iftrue 3921 . . . . 5  |-  ( ps 
->  if ( ps ,  A ,  .0.  )  =  A )
2321, 22oveqan12d 6324 . . . 4  |-  ( ( -.  ph  /\  ps )  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  )
)  =  (  .0.  .+  A ) )
2423adantl 467 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  ps )
)  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) )  =  (  .0.  .+  A )
)
2514olcs 396 . . . 4  |-  ( ps 
->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  A )
2625ad2antll 733 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  ps )
)  ->  if (
( ph  \/  ps ) ,  A ,  .0.  )  =  A
)
2720, 24, 263eqtr4rd 2481 . 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 1005 . . . . 5  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  M  e.  Mnd )
294, 6mndidcl 16505 . . . . . 6  |-  ( M  e.  Mnd  ->  .0.  e.  B )
3028, 29syl 17 . . . . 5  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  .0.  e.  B )
314, 5, 6mndlid 16508 . . . . 5  |-  ( ( M  e.  Mnd  /\  .0.  e.  B )  -> 
(  .0.  .+  .0.  )  =  .0.  )
3228, 30, 31syl2anc 665 . . . 4  |-  ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  ->  (  .0.  .+  .0.  )  =  .0.  )
3332adantr 466 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  -.  ps ) )  ->  (  .0.  .+  .0.  )  =  .0.  )
3421, 11oveqan12d 6324 . . . 4  |-  ( ( -.  ph  /\  -.  ps )  ->  ( if (
ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) )  =  (  .0.  .+  .0.  )
)
3534adantl 467 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  -.  ps ) )  ->  ( if ( ph ,  A ,  .0.  )  .+  if ( ps ,  A ,  .0.  ) )  =  (  .0.  .+  .0.  )
)
36 ioran 492 . . . . 5  |-  ( -.  ( ph  \/  ps ) 
<->  ( -.  ph  /\  -.  ps ) )
37 iffalse 3924 . . . . 5  |-  ( -.  ( ph  \/  ps )  ->  if ( (
ph  \/  ps ) ,  A ,  .0.  )  =  .0.  )
3836, 37sylbir 216 . . . 4  |-  ( ( -.  ph  /\  -.  ps )  ->  if ( (
ph  \/  ps ) ,  A ,  .0.  )  =  .0.  )
3938adantl 467 . . 3  |-  ( ( ( M  e.  Mnd  /\  A  e.  B  /\  -.  ( ph  /\  ps ) )  /\  ( -.  ph  /\  -.  ps ) )  ->  if ( ( ph  \/  ps ) ,  A ,  .0.  )  =  .0.  )
4033, 35, 393eqtr4rd 2481 . 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 958 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 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   ifcif 3915   ` cfv 5601  (class class class)co 6305   Basecbs 15084   +g cplusg 15152   0gc0g 15297   Mndcmnd 16486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-riota 6267  df-ov 6308  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488
This theorem is referenced by:  maducoeval2  19596  madugsum  19599
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