Theorem mndifsplit 20442

Description: Lemma for maducoeval2 20446. (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		pm2.21 120	. . . 4 ⊢ (¬ (𝜑 ∧ 𝜓) → ((𝜑 ∧ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 ))))
2	1	imp 445	. . 3 ⊢ ((¬ (𝜑 ∧ 𝜓) ∧ (𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
3	2	3ad2antl3 1225	. 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 17312	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → (𝐴 + 0 ) = 𝐴)
8	7	3adant3 1081	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → (𝐴 + 0 ) = 𝐴)
9	8	adantr 481	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (𝐴 + 0 ) = 𝐴)
10		iftrue 4092	. . . . 5 ⊢ (𝜑 → if(𝜑, 𝐴, 0 ) = 𝐴)
11		iffalse 4095	. . . . 5 ⊢ (¬ 𝜓 → if(𝜓, 𝐴, 0 ) = 0 )
12	10, 11	oveqan12d 6669	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
13	12	adantl 482	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = (𝐴 + 0 ))
14		iftrue 4092	. . . . 5 ⊢ ((𝜑 ∨ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
15	14	orcs 409	. . . 4 ⊢ (𝜑 → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
16	15	ad2antrl 764	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (𝜑 ∧ ¬ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
17	9, 13, 16	3eqtr4rd 2667	. 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 17311	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
19	18	3adant3 1081	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → ( 0 + 𝐴) = 𝐴)
20	19	adantr 481	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → ( 0 + 𝐴) = 𝐴)
21		iffalse 4095	. . . . 5 ⊢ (¬ 𝜑 → if(𝜑, 𝐴, 0 ) = 0 )
22		iftrue 4092	. . . . 5 ⊢ (𝜓 → if(𝜓, 𝐴, 0 ) = 𝐴)
23	21, 22	oveqan12d 6669	. . . 4 ⊢ ((¬ 𝜑 ∧ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
24	23	adantl 482	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 𝐴))
25	14	olcs 410	. . . 4 ⊢ (𝜓 → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
26	25	ad2antll 765	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 𝐴)
27	20, 24, 26	3eqtr4rd 2667	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
28		simp1 1061	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → 𝑀 ∈ Mnd)
29	4, 6	mndidcl 17308	. . . . . 6 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵)
30	28, 29	syl 17	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → 0 ∈ 𝐵)
31	4, 5, 6	mndlid 17311	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 )
32	28, 30, 31	syl2anc 693	. . . 4 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) → ( 0 + 0 ) = 0 )
33	32	adantr 481	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → ( 0 + 0 ) = 0 )
34	21, 11	oveqan12d 6669	. . . 4 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
35	34	adantl 482	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )) = ( 0 + 0 ))
36		ioran 511	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) ↔ (¬ 𝜑 ∧ ¬ 𝜓))
37		iffalse 4095	. . . . 5 ⊢ (¬ (𝜑 ∨ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 0 )
38	36, 37	sylbir 225	. . . 4 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 0 )
39	38	adantl 482	. . 3 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = 0 )
40	33, 35, 39	3eqtr4rd 2667	. 2 ⊢ (((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵 ∧ ¬ (𝜑 ∧ 𝜓)) ∧ (¬ 𝜑 ∧ ¬ 𝜓)) → if((𝜑 ∨ 𝜓), 𝐴, 0 ) = (if(𝜑, 𝐴, 0 ) + if(𝜓, 𝐴, 0 )))
41	3, 17, 27, 40	4casesdan 991	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   ∨ wo 383   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ifcif 4086  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Mndcmnd 17294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295

This theorem is referenced by:  maducoeval2  20446  madugsum  20449