paddasslem17 - Mathbox for Norm Megill

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Theorem paddasslem17 39793

Description: Lemma for paddass 39795. 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 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ ¬ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅))) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))

**Proof of Theorem paddasslem17**
Step	Hyp	Ref	Expression
1		ianor 982	. . . 4 ⊢ (¬ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) ↔ (¬ (𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∨ ¬ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)))
2		ianor 982	. . . . . 6 ⊢ (¬ (𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ↔ (¬ 𝑋 ≠ ∅ ∨ ¬ (𝑌 + 𝑍) ≠ ∅))
3		nne 2950	. . . . . . 7 ⊢ (¬ 𝑋 ≠ ∅ ↔ 𝑋 = ∅)
4		nne 2950	. . . . . . 7 ⊢ (¬ (𝑌 + 𝑍) ≠ ∅ ↔ (𝑌 + 𝑍) = ∅)
5	3, 4	orbi12i 913	. . . . . 6 ⊢ ((¬ 𝑋 ≠ ∅ ∨ ¬ (𝑌 + 𝑍) ≠ ∅) ↔ (𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅))
6	2, 5	bitri 275	. . . . 5 ⊢ (¬ (𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ↔ (𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅))
7		ianor 982	. . . . . 6 ⊢ (¬ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅) ↔ (¬ 𝑌 ≠ ∅ ∨ ¬ 𝑍 ≠ ∅))
8		nne 2950	. . . . . . 7 ⊢ (¬ 𝑌 ≠ ∅ ↔ 𝑌 = ∅)
9		nne 2950	. . . . . . 7 ⊢ (¬ 𝑍 ≠ ∅ ↔ 𝑍 = ∅)
10	8, 9	orbi12i 913	. . . . . 6 ⊢ ((¬ 𝑌 ≠ ∅ ∨ ¬ 𝑍 ≠ ∅) ↔ (𝑌 = ∅ ∨ 𝑍 = ∅))
11	7, 10	bitri 275	. . . . 5 ⊢ (¬ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅) ↔ (𝑌 = ∅ ∨ 𝑍 = ∅))
12	6, 11	orbi12i 913	. . . 4 ⊢ ((¬ (𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∨ ¬ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) ↔ ((𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅) ∨ (𝑌 = ∅ ∨ 𝑍 = ∅)))
13	1, 12	bitri 275	. . 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 39771	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑌 + 𝑍) ⊆ 𝐴)
17	16	3adant3r1 1182	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑌 + 𝑍) ⊆ 𝐴)
18	14, 15	padd02 39769	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑌 + 𝑍) ⊆ 𝐴) → (∅ + (𝑌 + 𝑍)) = (𝑌 + 𝑍))
19	17, 18	syldan 590	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (∅ + (𝑌 + 𝑍)) = (𝑌 + 𝑍))
20	14, 15	padd02 39769	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → (∅ + 𝑌) = 𝑌)
21	20	3ad2antr2 1189	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (∅ + 𝑌) = 𝑌)
22	21	oveq1d 7463	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((∅ + 𝑌) + 𝑍) = (𝑌 + 𝑍))
23	19, 22	eqtr4d 2783	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (∅ + (𝑌 + 𝑍)) = ((∅ + 𝑌) + 𝑍))
24		oveq1 7455	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝑋 + (𝑌 + 𝑍)) = (∅ + (𝑌 + 𝑍)))
25		oveq1 7455	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (𝑋 + 𝑌) = (∅ + 𝑌))
26	25	oveq1d 7463	. . . . . . . 8 ⊢ (𝑋 = ∅ → ((𝑋 + 𝑌) + 𝑍) = ((∅ + 𝑌) + 𝑍))
27	24, 26	eqeq12d 2756	. . . . . . 7 ⊢ (𝑋 = ∅ → ((𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍) ↔ (∅ + (𝑌 + 𝑍)) = ((∅ + 𝑌) + 𝑍)))
28	23, 27	syl5ibrcom 247	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 = ∅ → (𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍)))
29		eqimss 4067	. . . . . 6 ⊢ ((𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))
30	28, 29	syl6 35	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 = ∅ → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
31	14, 15	padd01 39768	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑋 + ∅) = 𝑋)
32	31	3ad2antr1 1188	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + ∅) = 𝑋)
33	14, 15	sspadd1 39772	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))
34	33	3adant3r3 1184	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑋 ⊆ (𝑋 + 𝑌))
35		simpl 482	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝐾 ∈ HL)
36	14, 15	paddssat 39771	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
37	36	3adant3r3 1184	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + 𝑌) ⊆ 𝐴)
38		simpr3 1196	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑍 ⊆ 𝐴)
39	14, 15	sspadd1 39772	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍))
40	35, 37, 38, 39	syl3anc 1371	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍))
41	34, 40	sstrd 4019	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑋 ⊆ ((𝑋 + 𝑌) + 𝑍))
42	32, 41	eqsstrd 4047	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + ∅) ⊆ ((𝑋 + 𝑌) + 𝑍))
43		oveq2 7456	. . . . . . 7 ⊢ ((𝑌 + 𝑍) = ∅ → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + ∅))
44	43	sseq1d 4040	. . . . . 6 ⊢ ((𝑌 + 𝑍) = ∅ → ((𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍) ↔ (𝑋 + ∅) ⊆ ((𝑋 + 𝑌) + 𝑍)))
45	42, 44	syl5ibrcom 247	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑌 + 𝑍) = ∅ → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
46	30, 45	jaod 858	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
47	14, 15	padd02 39769	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ⊆ 𝐴) → (∅ + 𝑍) = 𝑍)
48	47	3ad2antr3 1190	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (∅ + 𝑍) = 𝑍)
49	48	oveq2d 7464	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (∅ + 𝑍)) = (𝑋 + 𝑍))
50	32	oveq1d 7463	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + ∅) + 𝑍) = (𝑋 + 𝑍))
51	49, 50	eqtr4d 2783	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (∅ + 𝑍)) = ((𝑋 + ∅) + 𝑍))
52		oveq1 7455	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝑌 + 𝑍) = (∅ + 𝑍))
53	52	oveq2d 7464	. . . . . . . 8 ⊢ (𝑌 = ∅ → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + (∅ + 𝑍)))
54		oveq2 7456	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝑋 + 𝑌) = (𝑋 + ∅))
55	54	oveq1d 7463	. . . . . . . 8 ⊢ (𝑌 = ∅ → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + ∅) + 𝑍))
56	53, 55	eqeq12d 2756	. . . . . . 7 ⊢ (𝑌 = ∅ → ((𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍) ↔ (𝑋 + (∅ + 𝑍)) = ((𝑋 + ∅) + 𝑍)))
57	51, 56	syl5ibrcom 247	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑌 = ∅ → (𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍)))
58	14, 15	padd01 39768	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → (𝑌 + ∅) = 𝑌)
59	58	3ad2antr2 1189	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑌 + ∅) = 𝑌)
60	59	oveq2d 7464	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (𝑌 + ∅)) = (𝑋 + 𝑌))
61	14, 15	padd01 39768	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴) → ((𝑋 + 𝑌) + ∅) = (𝑋 + 𝑌))
62	37, 61	syldan 590	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) + ∅) = (𝑋 + 𝑌))
63	60, 62	eqtr4d 2783	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (𝑌 + ∅)) = ((𝑋 + 𝑌) + ∅))
64		oveq2 7456	. . . . . . . . 9 ⊢ (𝑍 = ∅ → (𝑌 + 𝑍) = (𝑌 + ∅))
65	64	oveq2d 7464	. . . . . . . 8 ⊢ (𝑍 = ∅ → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + (𝑌 + ∅)))
66		oveq2 7456	. . . . . . . 8 ⊢ (𝑍 = ∅ → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑌) + ∅))
67	65, 66	eqeq12d 2756	. . . . . . 7 ⊢ (𝑍 = ∅ → ((𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍) ↔ (𝑋 + (𝑌 + ∅)) = ((𝑋 + 𝑌) + ∅)))
68	63, 67	syl5ibrcom 247	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑍 = ∅ → (𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍)))
69	57, 68	jaod 858	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑌 = ∅ ∨ 𝑍 = ∅) → (𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍)))
70	69, 29	syl6 35	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑌 = ∅ ∨ 𝑍 = ∅) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
71	46, 70	jaod 858	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (((𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅) ∨ (𝑌 = ∅ ∨ 𝑍 = ∅)) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
72	13, 71	biimtrid 242	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (¬ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
73	72	3impia 1117	1 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ ¬ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅))) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ⊆ wss 3976 ∅c0 4352 ‘cfv 6573 (class class class)co 7448 Atomscatm 39219 HLchlt 39306 +_𝑃cpadd 39752

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-1st 8030 df-2nd 8031 df-padd 39753

This theorem is referenced by: paddasslem18 39794

W3C validator