Theorem dalem25 39655

Description: Lemma for dath 39693. Show that the dummy center of perspectivity 𝑐 is different from auxiliary atom 𝐺. (Contributed by NM, 3-Aug-2012.)

Hypotheses

Ref	Expression
dalem.ph	⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
dalem.l	⊢ ≤ = (le‘𝐾)
dalem.j	⊢ ∨ = (join‘𝐾)
dalem.a	⊢ 𝐴 = (Atoms‘𝐾)
dalem.ps	⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
dalem23.m	⊢ ∧ = (meet‘𝐾)
dalem23.o	⊢ 𝑂 = (LPlanes‘𝐾)
dalem23.y	⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
dalem23.z	⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
dalem23.g	⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))

Assertion

Ref	Expression
dalem25	⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≠ 𝐺)

Proof of Theorem dalem25

Step	Hyp	Ref	Expression
1		dalem.ph	. . . 4 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))
2		dalem.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dalem.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		dalem.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
5	1, 2, 3, 4	dalemcnes 39607	. . 3 ⊢ (𝜑 → 𝐶 ≠ 𝑆)
6	5	3ad2ant1 1133	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ≠ 𝑆)
7		dalem.ps	. . . . . . . . . . 11 ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))
8	7	dalemclccjdd 39645	. . . . . . . . . 10 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑))
9	8	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑))
10	9	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 ≤ (𝑐 ∨ 𝑑))
11		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝑐 = 𝐺)
12		dalem23.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
13	1	dalemkelat 39581	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Lat)
14	13	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
15	1	dalemkehl 39580	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ HL)
16	15	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
17	7	dalemccea 39640	. . . . . . . . . . . . . . . 16 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
18	17	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
19	1	dalempea 39583	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
20	19	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴)
21		eqid 2740	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 3, 4	hlatjcl 39323	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
23	16, 18, 20, 22	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
24	7	dalemddea 39641	. . . . . . . . . . . . . . . 16 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
25	24	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
26	1	dalemsea 39586	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
27	26	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴)
28	21, 3, 4	hlatjcl 39323	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
29	16, 25, 27, 28	syl3anc 1371	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
30		dalem23.m	. . . . . . . . . . . . . . 15 ⊢ ∧ = (meet‘𝐾)
31	21, 2, 30	latmle2 18535	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑑 ∨ 𝑆))
32	14, 23, 29, 31	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑑 ∨ 𝑆))
33	12, 32	eqbrtrid 5201	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≤ (𝑑 ∨ 𝑆))
34	3, 4	hlatjcom 39324	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) = (𝑆 ∨ 𝑑))
35	16, 25, 27, 34	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) = (𝑆 ∨ 𝑑))
36	33, 35	breqtrd 5192	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≤ (𝑆 ∨ 𝑑))
37	36	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐺 ≤ (𝑆 ∨ 𝑑))
38	11, 37	eqbrtrd 5188	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝑐 ≤ (𝑆 ∨ 𝑑))
39	2, 3, 4	hlatlej2 39332	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → 𝑑 ≤ (𝑆 ∨ 𝑑))
40	16, 27, 25, 39	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ≤ (𝑆 ∨ 𝑑))
41	40	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝑑 ≤ (𝑆 ∨ 𝑑))
42	7, 4	dalemcceb 39646	. . . . . . . . . . . 12 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
43	42	3ad2ant3 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
44	21, 4	atbase 39245	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝐴 → 𝑑 ∈ (Base‘𝐾))
45	24, 44	syl 17	. . . . . . . . . . . 12 ⊢ (𝜓 → 𝑑 ∈ (Base‘𝐾))
46	45	3ad2ant3 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ (Base‘𝐾))
47	21, 3, 4	hlatjcl 39323	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → (𝑆 ∨ 𝑑) ∈ (Base‘𝐾))
48	16, 27, 25, 47	syl3anc 1371	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑆 ∨ 𝑑) ∈ (Base‘𝐾))
49	21, 2, 3	latjle12 18520	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∈ (Base‘𝐾) ∧ 𝑑 ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑑) ∈ (Base‘𝐾))) → ((𝑐 ≤ (𝑆 ∨ 𝑑) ∧ 𝑑 ≤ (𝑆 ∨ 𝑑)) ↔ (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑)))
50	14, 43, 46, 48, 49	syl13anc 1372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ≤ (𝑆 ∨ 𝑑) ∧ 𝑑 ≤ (𝑆 ∨ 𝑑)) ↔ (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑)))
51	50	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → ((𝑐 ≤ (𝑆 ∨ 𝑑) ∧ 𝑑 ≤ (𝑆 ∨ 𝑑)) ↔ (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑)))
52	38, 41, 51	mpbi2and 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑))
53	1, 4	dalemceb 39595	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
54	53	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ∈ (Base‘𝐾))
55	21, 3, 4	hlatjcl 39323	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
56	16, 18, 25, 55	syl3anc 1371	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
57	21, 2	lattr 18514	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑐 ∨ 𝑑) ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑑) ∈ (Base‘𝐾))) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑)) → 𝐶 ≤ (𝑆 ∨ 𝑑)))
58	14, 54, 56, 48, 57	syl13anc 1372	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑)) → 𝐶 ≤ (𝑆 ∨ 𝑑)))
59	58	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑)) → 𝐶 ≤ (𝑆 ∨ 𝑑)))
60	10, 52, 59	mp2and 698	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 ≤ (𝑆 ∨ 𝑑))
61		dalem23.o	. . . . . . . . . . 11 ⊢ 𝑂 = (LPlanes‘𝐾)
62	1, 61	dalemyeb 39606	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
63	62	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
64	21, 2, 30	latmlem1 18539	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ (𝐶 ∈ (Base‘𝐾) ∧ (𝑆 ∨ 𝑑) ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾))) → (𝐶 ≤ (𝑆 ∨ 𝑑) → (𝐶 ∧ 𝑌) ≤ ((𝑆 ∨ 𝑑) ∧ 𝑌)))
65	14, 54, 48, 63, 64	syl13anc 1372	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐶 ≤ (𝑆 ∨ 𝑑) → (𝐶 ∧ 𝑌) ≤ ((𝑆 ∨ 𝑑) ∧ 𝑌)))
66	65	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → (𝐶 ≤ (𝑆 ∨ 𝑑) → (𝐶 ∧ 𝑌) ≤ ((𝑆 ∨ 𝑑) ∧ 𝑌)))
67	60, 66	mpd 15	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → (𝐶 ∧ 𝑌) ≤ ((𝑆 ∨ 𝑑) ∧ 𝑌))
68		dalem23.y	. . . . . . . . . 10 ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)
69		dalem23.z	. . . . . . . . . 10 ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)
70	1, 2, 3, 4, 61, 68, 69	dalem17 39637	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ 𝑌)
71	70	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ≤ 𝑌)
72	21, 2, 30	latleeqm1 18537	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 ≤ 𝑌 ↔ (𝐶 ∧ 𝑌) = 𝐶))
73	14, 54, 63, 72	syl3anc 1371	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐶 ≤ 𝑌 ↔ (𝐶 ∧ 𝑌) = 𝐶))
74	71, 73	mpbid 232	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐶 ∧ 𝑌) = 𝐶)
75	74	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → (𝐶 ∧ 𝑌) = 𝐶)
76	1, 2, 3, 4, 69	dalemsly 39612	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌)
77	76	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ≤ 𝑌)
78	7	dalem-ddly 39643	. . . . . . . . 9 ⊢ (𝜓 → ¬ 𝑑 ≤ 𝑌)
79	78	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑑 ≤ 𝑌)
80	21, 2, 3, 30, 4	2atjm 39402	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ (𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ∧ 𝑌 ∈ (Base‘𝐾)) ∧ (𝑆 ≤ 𝑌 ∧ ¬ 𝑑 ≤ 𝑌)) → ((𝑆 ∨ 𝑑) ∧ 𝑌) = 𝑆)
81	16, 27, 25, 63, 77, 79, 80	syl132anc 1388	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑆 ∨ 𝑑) ∧ 𝑌) = 𝑆)
82	81	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → ((𝑆 ∨ 𝑑) ∧ 𝑌) = 𝑆)
83	67, 75, 82	3brtr3d 5197	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 ≤ 𝑆)
84		hlatl 39316	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
85	15, 84	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ AtLat)
86	1, 2, 3, 4, 61, 68	dalemcea 39617	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
87	2, 4	atcmp 39267	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝐶 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆))
88	85, 86, 26, 87	syl3anc 1371	. . . . . . 7 ⊢ (𝜑 → (𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆))
89	88	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆))
90	89	adantr 480	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → (𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆))
91	83, 90	mpbid 232	. . . 4 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 = 𝑆)
92	91	ex 412	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 = 𝐺 → 𝐶 = 𝑆))
93	92	necon3d 2967	. 2 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐶 ≠ 𝑆 → 𝑐 ≠ 𝐺))
94	6, 93	mpd 15	1 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ≠ 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  joincjn 18381  meetcmee 18382  Latclat 18501  Atomscatm 39219  AtLatcal 39220  HLchlt 39306  LPlanesclpl 39449

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-rmo 3388  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-riota 7404  df-ov 7451  df-oprab 7452  df-proset 18365  df-poset 18383  df-plt 18400  df-lub 18416  df-glb 18417  df-join 18418  df-meet 18419  df-p0 18495  df-lat 18502  df-clat 18569  df-oposet 39132  df-ol 39134  df-oml 39135  df-covers 39222  df-ats 39223  df-atl 39254  df-cvlat 39278  df-hlat 39307  df-llines 39455  df-lplanes 39456

This theorem is referenced by:  dalem28  39657  dalem31N  39660