Theorem paddasslem17 35122

Description: Lemma for paddass 35124. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.)

Hypotheses

Ref	Expression
paddass.a	|- A = ( Atoms ` K )
paddass.p	|- .+ = ( +P ` K )

Assertion

Ref	Expression
paddasslem17	|- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ -. ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) )

Proof of Theorem paddasslem17

Step	Hyp	Ref	Expression
1		ianor 509	. . . 4 |- ( -. ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) <-> ( -. ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) \/ -. ( Y =/= (/) /\ Z =/= (/) ) ) )
2		ianor 509	. . . . . 6 |- ( -. ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) <-> ( -. X =/= (/) \/ -. ( Y .+ Z ) =/= (/) ) )
3		nne 2798	. . . . . . 7 |- ( -. X =/= (/) <-> X = (/) )
4		nne 2798	. . . . . . 7 |- ( -. ( Y .+ Z ) =/= (/) <-> ( Y .+ Z ) = (/) )
5	3, 4	orbi12i 543	. . . . . 6 |- ( ( -. X =/= (/) \/ -. ( Y .+ Z ) =/= (/) ) <-> ( X = (/) \/ ( Y .+ Z ) = (/) ) )
6	2, 5	bitri 264	. . . . 5 |- ( -. ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) <-> ( X = (/) \/ ( Y .+ Z ) = (/) ) )
7		ianor 509	. . . . . 6 |- ( -. ( Y =/= (/) /\ Z =/= (/) ) <-> ( -. Y =/= (/) \/ -. Z =/= (/) ) )
8		nne 2798	. . . . . . 7 |- ( -. Y =/= (/) <-> Y = (/) )
9		nne 2798	. . . . . . 7 |- ( -. Z =/= (/) <-> Z = (/) )
10	8, 9	orbi12i 543	. . . . . 6 |- ( ( -. Y =/= (/) \/ -. Z =/= (/) ) <-> ( Y = (/) \/ Z = (/) ) )
11	7, 10	bitri 264	. . . . 5 |- ( -. ( Y =/= (/) /\ Z =/= (/) ) <-> ( Y = (/) \/ Z = (/) ) )
12	6, 11	orbi12i 543	. . . 4 |- ( ( -. ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) \/ -. ( Y =/= (/) /\ Z =/= (/) ) ) <-> ( ( X = (/) \/ ( Y .+ Z ) = (/) ) \/ ( Y = (/) \/ Z = (/) ) ) )
13	1, 12	bitri 264	. . 3 |- ( -. ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) <-> ( ( X = (/) \/ ( Y .+ Z ) = (/) ) \/ ( Y = (/) \/ Z = (/) ) ) )
14		paddass.a	. . . . . . . . . . 11 |- A = ( Atoms ` K )
15		paddass.p	. . . . . . . . . . 11 |- .+ = ( +P ` K )
16	14, 15	paddssat 35100	. . . . . . . . . 10 |- ( ( K e. HL /\ Y C_ A /\ Z C_ A ) -> ( Y .+ Z ) C_ A )
17	16	3adant3r1 1274	. . . . . . . . 9 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( Y .+ Z ) C_ A )
18	14, 15	padd02 35098	. . . . . . . . 9 |- ( ( K e. HL /\ ( Y .+ Z ) C_ A ) -> ( (/) .+ ( Y .+ Z ) ) = ( Y .+ Z ) )
19	17, 18	syldan 487	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( (/) .+ ( Y .+ Z ) ) = ( Y .+ Z ) )
20	14, 15	padd02 35098	. . . . . . . . . 10 |- ( ( K e. HL /\ Y C_ A ) -> ( (/) .+ Y ) = Y )
21	20	3ad2antr2 1227	. . . . . . . . 9 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( (/) .+ Y ) = Y )
22	21	oveq1d 6665	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( (/) .+ Y ) .+ Z ) = ( Y .+ Z ) )
23	19, 22	eqtr4d 2659	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( (/) .+ ( Y .+ Z ) ) = ( ( (/) .+ Y ) .+ Z ) )
24		oveq1 6657	. . . . . . . 8 |- ( X = (/) -> ( X .+ ( Y .+ Z ) ) = ( (/) .+ ( Y .+ Z ) ) )
25		oveq1 6657	. . . . . . . . 9 |- ( X = (/) -> ( X .+ Y ) = ( (/) .+ Y ) )
26	25	oveq1d 6665	. . . . . . . 8 |- ( X = (/) -> ( ( X .+ Y ) .+ Z ) = ( ( (/) .+ Y ) .+ Z ) )
27	24, 26	eqeq12d 2637	. . . . . . 7 |- ( X = (/) -> ( ( X .+ ( Y .+ Z ) ) = ( ( X .+ Y ) .+ Z ) <-> ( (/) .+ ( Y .+ Z ) ) = ( ( (/) .+ Y ) .+ Z ) ) )
28	23, 27	syl5ibrcom 237	. . . . . 6 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X = (/) -> ( X .+ ( Y .+ Z ) ) = ( ( X .+ Y ) .+ Z ) ) )
29		eqimss 3657	. . . . . 6 |- ( ( X .+ ( Y .+ Z ) ) = ( ( X .+ Y ) .+ Z ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) )
30	28, 29	syl6 35	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X = (/) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) ) )
31	14, 15	padd01 35097	. . . . . . . 8 |- ( ( K e. HL /\ X C_ A ) -> ( X .+ (/) ) = X )
32	31	3ad2antr1 1226	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ (/) ) = X )
33	14, 15	sspadd1 35101	. . . . . . . . 9 |- ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> X C_ ( X .+ Y ) )
34	33	3adant3r3 1276	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> X C_ ( X .+ Y ) )
35		simpl 473	. . . . . . . . 9 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> K e. HL )
36	14, 15	paddssat 35100	. . . . . . . . . 10 |- ( ( K e. HL /\ X C_ A /\ Y C_ A ) -> ( X .+ Y ) C_ A )
37	36	3adant3r3 1276	. . . . . . . . 9 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ Y ) C_ A )
38		simpr3 1069	. . . . . . . . 9 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> Z C_ A )
39	14, 15	sspadd1 35101	. . . . . . . . 9 |- ( ( K e. HL /\ ( X .+ Y ) C_ A /\ Z C_ A ) -> ( X .+ Y ) C_ ( ( X .+ Y ) .+ Z ) )
40	35, 37, 38, 39	syl3anc 1326	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ Y ) C_ ( ( X .+ Y ) .+ Z ) )
41	34, 40	sstrd 3613	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> X C_ ( ( X .+ Y ) .+ Z ) )
42	32, 41	eqsstrd 3639	. . . . . 6 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ (/) ) C_ ( ( X .+ Y ) .+ Z ) )
43		oveq2 6658	. . . . . . 7 |- ( ( Y .+ Z ) = (/) -> ( X .+ ( Y .+ Z ) ) = ( X .+ (/) ) )
44	43	sseq1d 3632	. . . . . 6 |- ( ( Y .+ Z ) = (/) -> ( ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) <-> ( X .+ (/) ) C_ ( ( X .+ Y ) .+ Z ) ) )
45	42, 44	syl5ibrcom 237	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( Y .+ Z ) = (/) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) ) )
46	30, 45	jaod 395	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( X = (/) \/ ( Y .+ Z ) = (/) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) ) )
47	14, 15	padd02 35098	. . . . . . . . . 10 |- ( ( K e. HL /\ Z C_ A ) -> ( (/) .+ Z ) = Z )
48	47	3ad2antr3 1228	. . . . . . . . 9 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( (/) .+ Z ) = Z )
49	48	oveq2d 6666	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ ( (/) .+ Z ) ) = ( X .+ Z ) )
50	32	oveq1d 6665	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( X .+ (/) ) .+ Z ) = ( X .+ Z ) )
51	49, 50	eqtr4d 2659	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ ( (/) .+ Z ) ) = ( ( X .+ (/) ) .+ Z ) )
52		oveq1 6657	. . . . . . . . 9 |- ( Y = (/) -> ( Y .+ Z ) = ( (/) .+ Z ) )
53	52	oveq2d 6666	. . . . . . . 8 |- ( Y = (/) -> ( X .+ ( Y .+ Z ) ) = ( X .+ ( (/) .+ Z ) ) )
54		oveq2 6658	. . . . . . . . 9 |- ( Y = (/) -> ( X .+ Y ) = ( X .+ (/) ) )
55	54	oveq1d 6665	. . . . . . . 8 |- ( Y = (/) -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ (/) ) .+ Z ) )
56	53, 55	eqeq12d 2637	. . . . . . 7 |- ( Y = (/) -> ( ( X .+ ( Y .+ Z ) ) = ( ( X .+ Y ) .+ Z ) <-> ( X .+ ( (/) .+ Z ) ) = ( ( X .+ (/) ) .+ Z ) ) )
57	51, 56	syl5ibrcom 237	. . . . . 6 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( Y = (/) -> ( X .+ ( Y .+ Z ) ) = ( ( X .+ Y ) .+ Z ) ) )
58	14, 15	padd01 35097	. . . . . . . . . 10 |- ( ( K e. HL /\ Y C_ A ) -> ( Y .+ (/) ) = Y )
59	58	3ad2antr2 1227	. . . . . . . . 9 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( Y .+ (/) ) = Y )
60	59	oveq2d 6666	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ ( Y .+ (/) ) ) = ( X .+ Y ) )
61	14, 15	padd01 35097	. . . . . . . . 9 |- ( ( K e. HL /\ ( X .+ Y ) C_ A ) -> ( ( X .+ Y ) .+ (/) ) = ( X .+ Y ) )
62	37, 61	syldan 487	. . . . . . . 8 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( X .+ Y ) .+ (/) ) = ( X .+ Y ) )
63	60, 62	eqtr4d 2659	. . . . . . 7 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( X .+ ( Y .+ (/) ) ) = ( ( X .+ Y ) .+ (/) ) )
64		oveq2 6658	. . . . . . . . 9 |- ( Z = (/) -> ( Y .+ Z ) = ( Y .+ (/) ) )
65	64	oveq2d 6666	. . . . . . . 8 |- ( Z = (/) -> ( X .+ ( Y .+ Z ) ) = ( X .+ ( Y .+ (/) ) ) )
66		oveq2 6658	. . . . . . . 8 |- ( Z = (/) -> ( ( X .+ Y ) .+ Z ) = ( ( X .+ Y ) .+ (/) ) )
67	65, 66	eqeq12d 2637	. . . . . . 7 |- ( Z = (/) -> ( ( X .+ ( Y .+ Z ) ) = ( ( X .+ Y ) .+ Z ) <-> ( X .+ ( Y .+ (/) ) ) = ( ( X .+ Y ) .+ (/) ) ) )
68	63, 67	syl5ibrcom 237	. . . . . 6 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( Z = (/) -> ( X .+ ( Y .+ Z ) ) = ( ( X .+ Y ) .+ Z ) ) )
69	57, 68	jaod 395	. . . . 5 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( Y = (/) \/ Z = (/) ) -> ( X .+ ( Y .+ Z ) ) = ( ( X .+ Y ) .+ Z ) ) )
70	69, 29	syl6 35	. . . 4 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( Y = (/) \/ Z = (/) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) ) )
71	46, 70	jaod 395	. . 3 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( ( ( X = (/) \/ ( Y .+ Z ) = (/) ) \/ ( Y = (/) \/ Z = (/) ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) ) )
72	13, 71	syl5bi 232	. 2 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) ) -> ( -. ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) ) )
73	72	3impia 1261	1 |- ( ( K e. HL /\ ( X C_ A /\ Y C_ A /\ Z C_ A ) /\ -. ( ( X =/= (/) /\ ( Y .+ Z ) =/= (/) ) /\ ( Y =/= (/) /\ Z =/= (/) ) ) ) -> ( X .+ ( Y .+ Z ) ) C_ ( ( X .+ Y ) .+ Z ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Atomscatm 34550   HLchlt 34637   +Pcpadd 35081

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-padd 35082

This theorem is referenced by:  paddasslem18  35123