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Theorem mndifsplit 20490
Description: Lemma for maducoeval2 20494. (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 pm2.21 120 . . . 4 (¬ (𝜑𝜓) → ((𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 ))))
21imp 444 . . 3 ((¬ (𝜑𝜓) ∧ (𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
323ad2antl3 1245 . 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 17359 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → (𝐴 + 0 ) = 𝐴)
873adant3 1101 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → (𝐴 + 0 ) = 𝐴)
98adantr 480 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝐴 + 0 ) = 𝐴)
10 iftrue 4125 . . . . 5 (𝜑 → if(𝜑, 𝐴, 0 ) = 𝐴)
11 iffalse 4128 . . . . 5 𝜓 → if(𝜓, 𝐴, 0 ) = 0 )
1210, 11oveqan12d 6709 . . . 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 4125 . . . . 5 ((𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1514orcs 408 . . . 4 (𝜑 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
1615ad2antrl 764 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
179, 13, 163eqtr4rd 2696 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
184, 5, 6mndlid 17358 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵) → ( 0 + 𝐴) = 𝐴)
19183adant3 1101 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
2019adantr 480 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → ( 0 + 𝐴) = 𝐴)
21 iffalse 4128 . . . . 5 𝜑 → if(𝜑, 𝐴, 0 ) = 0 )
22 iftrue 4125 . . . . 5 (𝜓 → if(𝜓, 𝐴, 0 ) = 𝐴)
2321, 22oveqan12d 6709 . . . 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 409 . . . 4 (𝜓 → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2625ad2antll 765 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 𝐴)
2720, 24, 263eqtr4rd 2696 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
28 simp1 1081 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → 𝑀 ∈ Mnd)
294, 6mndidcl 17355 . . . . . 6 (𝑀 ∈ Mnd → 0𝐵)
3028, 29syl 17 . . . . 5 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → 0𝐵)
314, 5, 6mndlid 17358 . . . . 5 ((𝑀 ∈ Mnd ∧ 0𝐵) → ( 0 + 0 ) = 0 )
3228, 30, 31syl2anc 694 . . . 4 ((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) → ( 0 + 0 ) = 0 )
3332adantr 480 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ( 0 + 0 ) = 0 )
3421, 11oveqan12d 6709 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
3534adantl 481 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
36 ioran 510 . . . . 5 (¬ (𝜑𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
37 iffalse 4128 . . . . 5 (¬ (𝜑𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3836, 37sylbir 225 . . . 4 ((¬ 𝜑 ∧ ¬ 𝜓) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
3938adantl 481 . . 3 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = 0 )
4033, 35, 393eqtr4rd 2696 . 2 (((𝑀 ∈ Mnd ∧ 𝐴𝐵 ∧ ¬ (𝜑𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
413, 17, 27, 404casesdan 1010 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  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  ifcif 4119  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Mndcmnd 17341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342
This theorem is referenced by:  maducoeval2  20494  madugsum  20497
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