Theorem mndifsplit 22830

Description: Lemma for maducoeval2 22834. (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		pm2.21 124	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 ))))
2	1	imp 412	. . 3 ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
3	2	3ad2antl3 1206	. 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 18842	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐴 + 0 ) = 𝐴)
8	7	3adant3 1150	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → (𝐴 + 0 ) = 𝐴)
9	8	adantr 486	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝐴 + 0 ) = 𝐴)
10		iftrue 4498	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 0 ) = 𝐴)
11		iffalse 4501	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 0 ) = 0 )
12	10, 11	oveqan12d 7442	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
13	12	adantl 487	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
14		iftrue 4498	. . . . 5 ⊢ ((𝜑 ∨ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
15	14	orcs 889	. . . 4 ⊢ (𝜑 → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
16	15	ad2antrl 741	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
17	9, 13, 16	3eqtr4rd 2812	. 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 18841	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
19	18	3adant3 1150	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → ( 0 + 𝐴) = 𝐴)
20	19	adantr 486	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → ( 0 + 𝐴) = 𝐴)
21		iffalse 4501	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 0 ) = 0 )
22		iftrue 4498	. . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 0 ) = 𝐴)
23	21, 22	oveqan12d 7442	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
24	23	adantl 487	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
25	14	olcs 890	. . . 4 ⊢ (𝜓 → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
26	25	ad2antll 742	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
27	20, 24, 26	3eqtr4rd 2812	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
28		simp1 1154	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → 𝑀 ∈ Mnd)
29	4, 6	mndidcl 18836	. . . . 5 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵)
30	4, 5, 6	mndlid 18841	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 )
31	28, 29, 30	syl2anc2 597	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → ( 0 + 0 ) = 0 )
32	31	adantr 486	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ( 0 + 0 ) = 0 )
33	21, 11	oveqan12d 7442	. . . 4 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
34	33	adantl 487	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
35		ioran 999	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
36		iffalse 4501	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 0 )
37	35, 36	sylbir 238	. . . 4 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 0 )
38	37	adantl 487	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 0 )
39	32, 34, 38	3eqtr4rd 2812	. 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 1057	1 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ifcif 4492  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  0_gc0g 17517  Mndcmnd 18821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-riota 7380  df-ov 7426  df-0g 17519  df-mgm 18723  df-sgrp 18806  df-mnd 18822

This theorem is used by:  maducoeval2  22834  madugsum  22837