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Theorem mndifsplit 21244
 Description: Lemma for maducoeval2 21248. (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 410 . . 3 ((¬ (𝜑𝜓) ∧ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
323ad2antl3 1184 . 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 17927 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → (𝐴 + 0 ) = 𝐴)
873adant3 1129 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → (𝐴 + 0 ) = 𝐴)
98adantr 484 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝐴 + 0 ) = 𝐴)
10 iftrue 4434 . . . . 5 (𝜑 → if(𝜑, 𝐴, 0 ) = 𝐴)
11 iffalse 4437 . . . . 5 𝜓 → if(𝜓, 𝐴, 0 ) = 0 )
1210, 11oveqan12d 7158 . . . 4 ((𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
1312adantl 485 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
14 iftrue 4434 . . . . 5 ((𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1514orcs 872 . . . 4 (𝜑 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1615ad2antrl 727 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
179, 13, 163eqtr4rd 2847 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
184, 5, 6mndlid 17926 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → ( 0 + 𝐴) = 𝐴)
19183adant3 1129 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
2019adantr 484 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
21 iffalse 4437 . . . . 5 𝜑 → if(𝜑, 𝐴, 0 ) = 0 )
22 iftrue 4434 . . . . 5 (𝜓 → if(𝜓, 𝐴, 0 ) = 𝐴)
2321, 22oveqan12d 7158 . . . 4 ((¬ 𝜑𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
2423adantl 485 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
2514olcs 873 . . . 4 (𝜓 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2625ad2antll 728 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2720, 24, 263eqtr4rd 2847 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
28 simp1 1133 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → 𝑀 ∈ Mnd)
294, 6mndidcl 17921 . . . . 5 (𝑀 ∈ Mnd → 0𝐵)
304, 5, 6mndlid 17926 . . . . 5 ((𝑀 ∈ Mnd ∧ 0𝐵) → ( 0 + 0 ) = 0 )
3128, 29, 30syl2anc2 588 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 0 ) = 0 )
3231adantr 484 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ( 0 + 0 ) = 0 )
3321, 11oveqan12d 7158 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
3433adantl 485 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
35 ioran 981 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
36 iffalse 4437 . . . . 5 (¬ (𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3735, 36sylbir 238 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3837adantl 485 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3932, 34, 383eqtr4rd 2847 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
403, 17, 27, 394casesdan 1037 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 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  ifcif 4428  ‘cfv 6328  (class class class)co 7139  Basecbs 16478  +gcplusg 16560  0gc0g 16708  Mndcmnd 17906 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-riota 7097  df-ov 7142  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907 This theorem is referenced by:  maducoeval2  21248  madugsum  21251
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