paddasslem17 - Mathbox for Norm Megill

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Theorem paddasslem17 40711

Description: Lemma for paddass 40713. 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 997	. . . 4 ⊢ (¬ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) ↔ (¬ (𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∨ ¬ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)))
2		ianor 997	. . . . . 6 ⊢ (¬ (𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ↔ (¬ 𝑋 ≠ ∅ ∨ ¬ (𝑌 + 𝑍) ≠ ∅))
3		nne 2961	. . . . . . 7 ⊢ (¬ 𝑋 ≠ ∅ ↔ 𝑋 = ∅)
4		nne 2961	. . . . . . 7 ⊢ (¬ (𝑌 + 𝑍) ≠ ∅ ↔ (𝑌 + 𝑍) = ∅)
5	3, 4	orbi12i 928	. . . . . 6 ⊢ ((¬ 𝑋 ≠ ∅ ∨ ¬ (𝑌 + 𝑍) ≠ ∅) ↔ (𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅))
6	2, 5	bitri 278	. . . . 5 ⊢ (¬ (𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ↔ (𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅))
7		ianor 997	. . . . . 6 ⊢ (¬ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅) ↔ (¬ 𝑌 ≠ ∅ ∨ ¬ 𝑍 ≠ ∅))
8		nne 2961	. . . . . . 7 ⊢ (¬ 𝑌 ≠ ∅ ↔ 𝑌 = ∅)
9		nne 2961	. . . . . . 7 ⊢ (¬ 𝑍 ≠ ∅ ↔ 𝑍 = ∅)
10	8, 9	orbi12i 928	. . . . . 6 ⊢ ((¬ 𝑌 ≠ ∅ ∨ ¬ 𝑍 ≠ ∅) ↔ (𝑌 = ∅ ∨ 𝑍 = ∅))
11	7, 10	bitri 278	. . . . 5 ⊢ (¬ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅) ↔ (𝑌 = ∅ ∨ 𝑍 = ∅))
12	6, 11	orbi12i 928	. . . 4 ⊢ ((¬ (𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∨ ¬ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) ↔ ((𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅) ∨ (𝑌 = ∅ ∨ 𝑍 = ∅)))
13	1, 12	bitri 278	. . 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 40689	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑌 + 𝑍) ⊆ 𝐴)
17	16	3adant3r1 1201	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑌 + 𝑍) ⊆ 𝐴)
18	14, 15	padd02 40687	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑌 + 𝑍) ⊆ 𝐴) → (∅ + (𝑌 + 𝑍)) = (𝑌 + 𝑍))
19	17, 18	syldan 603	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (∅ + (𝑌 + 𝑍)) = (𝑌 + 𝑍))
20	14, 15	padd02 40687	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → (∅ + 𝑌) = 𝑌)
21	20	3ad2antr2 1208	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (∅ + 𝑌) = 𝑌)
22	21	oveq1d 7431	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((∅ + 𝑌) + 𝑍) = (𝑌 + 𝑍))
23	19, 22	eqtr4d 2800	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (∅ + (𝑌 + 𝑍)) = ((∅ + 𝑌) + 𝑍))
24		oveq1 7423	. . . . . . . 8 ⊢ (𝑋 = ∅ → (𝑋 + (𝑌 + 𝑍)) = (∅ + (𝑌 + 𝑍)))
25		oveq1 7423	. . . . . . . . 9 ⊢ (𝑋 = ∅ → (𝑋 + 𝑌) = (∅ + 𝑌))
26	25	oveq1d 7431	. . . . . . . 8 ⊢ (𝑋 = ∅ → ((𝑋 + 𝑌) + 𝑍) = ((∅ + 𝑌) + 𝑍))
27	24, 26	eqeq12d 2778	. . . . . . 7 ⊢ (𝑋 = ∅ → ((𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍) ↔ (∅ + (𝑌 + 𝑍)) = ((∅ + 𝑌) + 𝑍)))
28	23, 27	syl5ibrcom 250	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 = ∅ → (𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍)))
29		eqimss 3992	. . . . . 6 ⊢ ((𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))
30	28, 29	syl6 36	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 = ∅ → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
31	14, 15	padd01 40686	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑋 + ∅) = 𝑋)
32	31	3ad2antr1 1207	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + ∅) = 𝑋)
33	14, 15	sspadd1 40690	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))
34	33	3adant3r3 1203	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑋 ⊆ (𝑋 + 𝑌))
35		simpl 488	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝐾 ∈ HL)
36	14, 15	paddssat 40689	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
37	36	3adant3r3 1203	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + 𝑌) ⊆ 𝐴)
38		simpr3 1215	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑍 ⊆ 𝐴)
39	14, 15	sspadd1 40690	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍))
40	35, 37, 38, 39	syl3anc 1398	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + 𝑌) ⊆ ((𝑋 + 𝑌) + 𝑍))
41	34, 40	sstrd 3944	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → 𝑋 ⊆ ((𝑋 + 𝑌) + 𝑍))
42	32, 41	eqsstrd 3968	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + ∅) ⊆ ((𝑋 + 𝑌) + 𝑍))
43		oveq2 7424	. . . . . . 7 ⊢ ((𝑌 + 𝑍) = ∅ → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + ∅))
44	43	sseq1d 3965	. . . . . 6 ⊢ ((𝑌 + 𝑍) = ∅ → ((𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍) ↔ (𝑋 + ∅) ⊆ ((𝑋 + 𝑌) + 𝑍)))
45	42, 44	syl5ibrcom 250	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑌 + 𝑍) = ∅ → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
46	30, 45	jaod 873	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
47	14, 15	padd02 40687	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑍 ⊆ 𝐴) → (∅ + 𝑍) = 𝑍)
48	47	3ad2antr3 1209	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (∅ + 𝑍) = 𝑍)
49	48	oveq2d 7432	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (∅ + 𝑍)) = (𝑋 + 𝑍))
50	32	oveq1d 7431	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + ∅) + 𝑍) = (𝑋 + 𝑍))
51	49, 50	eqtr4d 2800	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (∅ + 𝑍)) = ((𝑋 + ∅) + 𝑍))
52		oveq1 7423	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝑌 + 𝑍) = (∅ + 𝑍))
53	52	oveq2d 7432	. . . . . . . 8 ⊢ (𝑌 = ∅ → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + (∅ + 𝑍)))
54		oveq2 7424	. . . . . . . . 9 ⊢ (𝑌 = ∅ → (𝑋 + 𝑌) = (𝑋 + ∅))
55	54	oveq1d 7431	. . . . . . . 8 ⊢ (𝑌 = ∅ → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + ∅) + 𝑍))
56	53, 55	eqeq12d 2778	. . . . . . 7 ⊢ (𝑌 = ∅ → ((𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍) ↔ (𝑋 + (∅ + 𝑍)) = ((𝑋 + ∅) + 𝑍)))
57	51, 56	syl5ibrcom 250	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑌 = ∅ → (𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍)))
58	14, 15	padd01 40686	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴) → (𝑌 + ∅) = 𝑌)
59	58	3ad2antr2 1208	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑌 + ∅) = 𝑌)
60	59	oveq2d 7432	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (𝑌 + ∅)) = (𝑋 + 𝑌))
61	14, 15	padd01 40686	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ (𝑋 + 𝑌) ⊆ 𝐴) → ((𝑋 + 𝑌) + ∅) = (𝑋 + 𝑌))
62	37, 61	syldan 603	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) + ∅) = (𝑋 + 𝑌))
63	60, 62	eqtr4d 2800	. . . . . . 7 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (𝑌 + ∅)) = ((𝑋 + 𝑌) + ∅))
64		oveq2 7424	. . . . . . . . 9 ⊢ (𝑍 = ∅ → (𝑌 + 𝑍) = (𝑌 + ∅))
65	64	oveq2d 7432	. . . . . . . 8 ⊢ (𝑍 = ∅ → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + (𝑌 + ∅)))
66		oveq2 7424	. . . . . . . 8 ⊢ (𝑍 = ∅ → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑌) + ∅))
67	65, 66	eqeq12d 2778	. . . . . . 7 ⊢ (𝑍 = ∅ → ((𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍) ↔ (𝑋 + (𝑌 + ∅)) = ((𝑋 + 𝑌) + ∅)))
68	63, 67	syl5ibrcom 250	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑍 = ∅ → (𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍)))
69	57, 68	jaod 873	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑌 = ∅ ∨ 𝑍 = ∅) → (𝑋 + (𝑌 + 𝑍)) = ((𝑋 + 𝑌) + 𝑍)))
70	69, 29	syl6 36	. . . 4 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑌 = ∅ ∨ 𝑍 = ∅) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
71	46, 70	jaod 873	. . 3 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (((𝑋 = ∅ ∨ (𝑌 + 𝑍) = ∅) ∨ (𝑌 = ∅ ∨ 𝑍 = ∅)) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
72	13, 71	biimtrid 245	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (¬ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)))
73	72	3impia 1135	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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2957 ⊆ wss 3902 ∅c0 4282 ‘cfv 6537 (class class class)co 7416 Atomscatm 40138 HLchlt 40225 +_𝑃cpadd 40670

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7989 df-2nd 7990 df-padd 40671

This theorem is used by: paddasslem18 40712

W3C validator