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Theorem mndifsplit 21173
Description: Lemma for maducoeval2 21177. (Contributed by SO, 16-Jul-2018.)
Hypotheses
Ref Expression
mndifsplit.b 𝐵 = (Base‘𝑀)
mndifsplit.0g 0 = (0g𝑀)
mndifsplit.pg + = (+g𝑀)
Assertion
Ref Expression
mndifsplit ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))

Proof of Theorem mndifsplit
StepHypRef Expression
1 pm2.21 123 . . . 4 (¬ (𝜑𝜓) → ((𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 ))))
21imp 407 . . 3 ((¬ (𝜑𝜓) ∧ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
323ad2antl3 1179 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
4 mndifsplit.b . . . . . 6 𝐵 = (Base‘𝑀)
5 mndifsplit.pg . . . . . 6 + = (+g𝑀)
6 mndifsplit.0g . . . . . 6 0 = (0g𝑀)
74, 5, 6mndrid 17920 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → (𝐴 + 0 ) = 𝐴)
873adant3 1124 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → (𝐴 + 0 ) = 𝐴)
98adantr 481 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝐴 + 0 ) = 𝐴)
10 iftrue 4469 . . . . 5 (𝜑 → if(𝜑, 𝐴, 0 ) = 𝐴)
11 iffalse 4472 . . . . 5 𝜓 → if(𝜓, 𝐴, 0 ) = 0 )
1210, 11oveqan12d 7164 . . . 4 ((𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
1312adantl 482 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
14 iftrue 4469 . . . . 5 ((𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1514orcs 871 . . . 4 (𝜑 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1615ad2antrl 724 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
179, 13, 163eqtr4rd 2864 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
184, 5, 6mndlid 17919 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → ( 0 + 𝐴) = 𝐴)
19183adant3 1124 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
2019adantr 481 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
21 iffalse 4472 . . . . 5 𝜑 → if(𝜑, 𝐴, 0 ) = 0 )
22 iftrue 4469 . . . . 5 (𝜓 → if(𝜓, 𝐴, 0 ) = 𝐴)
2321, 22oveqan12d 7164 . . . 4 ((¬ 𝜑𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
2423adantl 482 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
2514olcs 872 . . . 4 (𝜓 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2625ad2antll 725 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2720, 24, 263eqtr4rd 2864 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
28 simp1 1128 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → 𝑀 ∈ Mnd)
294, 6mndidcl 17914 . . . . 5 (𝑀 ∈ Mnd → 0𝐵)
304, 5, 6mndlid 17919 . . . . 5 ((𝑀 ∈ Mnd ∧ 0𝐵) → ( 0 + 0 ) = 0 )
3128, 29, 30syl2anc2 585 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 0 ) = 0 )
3231adantr 481 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ( 0 + 0 ) = 0 )
3321, 11oveqan12d 7164 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
3433adantl 482 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
35 ioran 977 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
36 iffalse 4472 . . . . 5 (¬ (𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3735, 36sylbir 236 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3837adantl 482 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3932, 34, 383eqtr4rd 2864 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
403, 17, 27, 394casesdan 1033 1 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  ifcif 4463  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Mndcmnd 17899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7103  df-ov 7148  df-0g 16703  df-mgm 17840  df-sgrp 17889  df-mnd 17900
This theorem is referenced by:  maducoeval2  21177  madugsum  21180
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