MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mndifsplit Structured version   Visualization version   GIF version

Theorem mndifsplit 22658
Description: Lemma for maducoeval2 22662. (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 406 . . 3 ((¬ (𝜑𝜓) ∧ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
323ad2antl3 1186 . 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 18781 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → (𝐴 + 0 ) = 𝐴)
873adant3 1131 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → (𝐴 + 0 ) = 𝐴)
98adantr 480 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝐴 + 0 ) = 𝐴)
10 iftrue 4537 . . . . 5 (𝜑 → if(𝜑, 𝐴, 0 ) = 𝐴)
11 iffalse 4540 . . . . 5 𝜓 → if(𝜓, 𝐴, 0 ) = 0 )
1210, 11oveqan12d 7450 . . . 4 ((𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
1312adantl 481 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
14 iftrue 4537 . . . . 5 ((𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1514orcs 875 . . . 4 (𝜑 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1615ad2antrl 728 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
179, 13, 163eqtr4rd 2786 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
184, 5, 6mndlid 18780 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → ( 0 + 𝐴) = 𝐴)
19183adant3 1131 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
2019adantr 480 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
21 iffalse 4540 . . . . 5 𝜑 → if(𝜑, 𝐴, 0 ) = 0 )
22 iftrue 4537 . . . . 5 (𝜓 → if(𝜓, 𝐴, 0 ) = 𝐴)
2321, 22oveqan12d 7450 . . . 4 ((¬ 𝜑𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
2423adantl 481 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
2514olcs 876 . . . 4 (𝜓 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2625ad2antll 729 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2720, 24, 263eqtr4rd 2786 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
28 simp1 1135 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → 𝑀 ∈ Mnd)
294, 6mndidcl 18775 . . . . 5 (𝑀 ∈ Mnd → 0𝐵)
304, 5, 6mndlid 18780 . . . . 5 ((𝑀 ∈ Mnd ∧ 0𝐵) → ( 0 + 0 ) = 0 )
3128, 29, 30syl2anc2 585 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 0 ) = 0 )
3231adantr 480 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ( 0 + 0 ) = 0 )
3321, 11oveqan12d 7450 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
3433adantl 481 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
35 ioran 985 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
36 iffalse 4540 . . . . 5 (¬ (𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3735, 36sylbir 235 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3837adantl 481 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3932, 34, 383eqtr4rd 2786 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
403, 17, 27, 394casesdan 1041 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 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  ifcif 4531  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Mndcmnd 18760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761
This theorem is referenced by:  maducoeval2  22662  madugsum  22665
  Copyright terms: Public domain W3C validator