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 e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) )

Proof of Theorem mndifsplit

Step	Hyp	Ref	Expression
1		pm2.21 120	. . . 4 |- ( -. ( ph /\ ps ) -> ( ( ph /\ ps ) -> if ( ( ph \/ ps ) , A , .0. ) = ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) ) )
2	1	imp 445	. . 3 |- ( ( -. ( ph /\ ps ) /\ ( ph /\ ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) )
3	2	3ad2antl3 1225	. 2 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( ph /\ ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = ( if ( ph , A , .0. ) .+ if ( ps , 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 e. Mnd /\ A e. B ) -> ( A .+ .0. ) = A )
8	7	3adant3 1081	. . . 4 |- ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) -> ( A .+ .0. ) = A )
9	8	adantr 481	. . 3 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( ph /\ -. ps ) ) -> ( A .+ .0. ) = A )
10		iftrue 4092	. . . . 5 |- ( ph -> if ( ph , A , .0. ) = A )
11		iffalse 4095	. . . . 5 |- ( -. ps -> if ( ps , A , .0. ) = .0. )
12	10, 11	oveqan12d 6669	. . . 4 |- ( ( ph /\ -. ps ) -> ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) = ( A .+ .0. ) )
13	12	adantl 482	. . 3 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( ph /\ -. ps ) ) -> ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) = ( A .+ .0. ) )
14		iftrue 4092	. . . . 5 |- ( ( ph \/ ps ) -> if ( ( ph \/ ps ) , A , .0. ) = A )
15	14	orcs 409	. . . 4 |- ( ph -> if ( ( ph \/ ps ) , A , .0. ) = A )
16	15	ad2antrl 764	. . 3 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( ph /\ -. ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = A )
17	9, 13, 16	3eqtr4rd 2667	. 2 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( ph /\ -. ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) )
18	4, 5, 6	mndlid 17311	. . . . 5 |- ( ( M e. Mnd /\ A e. B ) -> ( .0. .+ A ) = A )
19	18	3adant3 1081	. . . 4 |- ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) -> ( .0. .+ A ) = A )
20	19	adantr 481	. . 3 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( -. ph /\ ps ) ) -> ( .0. .+ A ) = A )
21		iffalse 4095	. . . . 5 |- ( -. ph -> if ( ph , A , .0. ) = .0. )
22		iftrue 4092	. . . . 5 |- ( ps -> if ( ps , A , .0. ) = A )
23	21, 22	oveqan12d 6669	. . . 4 |- ( ( -. ph /\ ps ) -> ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) = ( .0. .+ A ) )
24	23	adantl 482	. . 3 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( -. ph /\ ps ) ) -> ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) = ( .0. .+ A ) )
25	14	olcs 410	. . . 4 |- ( ps -> if ( ( ph \/ ps ) , A , .0. ) = A )
26	25	ad2antll 765	. . 3 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( -. ph /\ ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = A )
27	20, 24, 26	3eqtr4rd 2667	. 2 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( -. ph /\ ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) )
28		simp1 1061	. . . . 5 |- ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) -> M e. Mnd )
29	4, 6	mndidcl 17308	. . . . . 6 |- ( M e. Mnd -> .0. e. B )
30	28, 29	syl 17	. . . . 5 |- ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) -> .0. e. B )
31	4, 5, 6	mndlid 17311	. . . . 5 |- ( ( M e. Mnd /\ .0. e. B ) -> ( .0. .+ .0. ) = .0. )
32	28, 30, 31	syl2anc 693	. . . 4 |- ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) -> ( .0. .+ .0. ) = .0. )
33	32	adantr 481	. . 3 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( -. ph /\ -. ps ) ) -> ( .0. .+ .0. ) = .0. )
34	21, 11	oveqan12d 6669	. . . 4 |- ( ( -. ph /\ -. ps ) -> ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) = ( .0. .+ .0. ) )
35	34	adantl 482	. . 3 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( -. ph /\ -. ps ) ) -> ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) = ( .0. .+ .0. ) )
36		ioran 511	. . . . 5 |- ( -. ( ph \/ ps ) <-> ( -. ph /\ -. ps ) )
37		iffalse 4095	. . . . 5 |- ( -. ( ph \/ ps ) -> if ( ( ph \/ ps ) , A , .0. ) = .0. )
38	36, 37	sylbir 225	. . . 4 |- ( ( -. ph /\ -. ps ) -> if ( ( ph \/ ps ) , A , .0. ) = .0. )
39	38	adantl 482	. . 3 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( -. ph /\ -. ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = .0. )
40	33, 35, 39	3eqtr4rd 2667	. 2 |- ( ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) /\ ( -. ph /\ -. ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) )
41	3, 17, 27, 40	4casesdan 991	1 |- ( ( M e. Mnd /\ A e. B /\ -. ( ph /\ ps ) ) -> if ( ( ph \/ ps ) , A , .0. ) = ( if ( ph , A , .0. ) .+ if ( ps , A , .0. ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ifcif 4086   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   0gc0g 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