Theorem mndifsplit 21693

Description: Lemma for maducoeval2 21697. (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 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))

Proof of Theorem mndifsplit

Step	Hyp	Ref	Expression
1		pm2.21 123	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 ))))
2	1	imp 406	. . 3 ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
3	2	3ad2antl3 1185	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
4		mndifsplit.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
5		mndifsplit.pg	. . . . . 6 ⊢ + = (+g‘𝑀)
6		mndifsplit.0g	. . . . . 6 ⊢ 0 = (0g‘𝑀)
7	4, 5, 6	mndrid 18321	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐴 + 0 ) = 𝐴)
8	7	3adant3 1130	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → (𝐴 + 0 ) = 𝐴)
9	8	adantr 480	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝐴 + 0 ) = 𝐴)
10		iftrue 4462	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 0 ) = 𝐴)
11		iffalse 4465	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 0 ) = 0 )
12	10, 11	oveqan12d 7274	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
13	12	adantl 481	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
14		iftrue 4462	. . . . 5 ⊢ ((𝜑 ∨ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
15	14	orcs 871	. . . 4 ⊢ (𝜑 → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
16	15	ad2antrl 724	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
17	9, 13, 16	3eqtr4rd 2789	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
18	4, 5, 6	mndlid 18320	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
19	18	3adant3 1130	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → ( 0 + 𝐴) = 𝐴)
20	19	adantr 480	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → ( 0 + 𝐴) = 𝐴)
21		iffalse 4465	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 0 ) = 0 )
22		iftrue 4462	. . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 0 ) = 𝐴)
23	21, 22	oveqan12d 7274	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
24	23	adantl 481	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
25	14	olcs 872	. . . 4 ⊢ (𝜓 → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
26	25	ad2antll 725	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
27	20, 24, 26	3eqtr4rd 2789	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
28		simp1 1134	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → 𝑀 ∈ Mnd)
29	4, 6	mndidcl 18315	. . . . 5 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵)
30	4, 5, 6	mndlid 18320	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 )
31	28, 29, 30	syl2anc2 584	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → ( 0 + 0 ) = 0 )
32	31	adantr 480	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ( 0 + 0 ) = 0 )
33	21, 11	oveqan12d 7274	. . . 4 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
34	33	adantl 481	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
35		ioran 980	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
36		iffalse 4465	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 0 )
37	35, 36	sylbir 234	. . . 4 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 0 )
38	37	adantl 481	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 0 )
39	32, 34, 38	3eqtr4rd 2789	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
40	3, 17, 27, 39	4casesdan 1038	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ifcif 4456  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  Mndcmnd 18300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-riota 7212  df-ov 7258  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301

This theorem is referenced by:  maducoeval2  21697  madugsum  21700