Theorem dalem25 38161

Description: Lemma for dath 38199. 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 38113	. . 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 38151	. . . . . . . . . 10 ⊢ (𝜓 → 𝐶 ≤ (𝑐 ∨ 𝑑))
9	8	3ad2ant3 1135	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ≤ (𝑐 ∨ 𝑑))
10	9	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 ≤ (𝑐 ∨ 𝑑))
11		simpr 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝑐 = 𝐺)
12		dalem23.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))
13	1	dalemkelat 38087	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Lat)
14	13	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
15	1	dalemkehl 38086	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ HL)
16	15	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
17	7	dalemccea 38146	. . . . . . . . . . . . . . . 16 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
18	17	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
19	1	dalempea 38089	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
20	19	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴)
21		eqid 2736	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 3, 4	hlatjcl 37829	. . . . . . . . . . . . . . 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 38147	. . . . . . . . . . . . . . . 16 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
25	24	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
26	1	dalemsea 38092	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
27	26	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴)
28	21, 3, 4	hlatjcl 37829	. . . . . . . . . . . . . . 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 18354	. . . . . . . . . . . . . 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 5140	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≤ (𝑑 ∨ 𝑆))
34	3, 4	hlatjcom 37830	. . . . . . . . . . . . 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 5131	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≤ (𝑆 ∨ 𝑑))
37	36	adantr 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐺 ≤ (𝑆 ∨ 𝑑))
38	11, 37	eqbrtrd 5127	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝑐 ≤ (𝑆 ∨ 𝑑))
39	2, 3, 4	hlatlej2 37838	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝑑 ≤ (𝑆 ∨ 𝑑))
42	7, 4	dalemcceb 38152	. . . . . . . . . . . 12 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
43	42	3ad2ant3 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
44	21, 4	atbase 37751	. . . . . . . . . . . . 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 37829	. . . . . . . . . . . 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 18339	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → ((𝑐 ≤ (𝑆 ∨ 𝑑) ∧ 𝑑 ≤ (𝑆 ∨ 𝑑)) ↔ (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑)))
52	38, 41, 51	mpbi2and 710	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑))
53	1, 4	dalemceb 38101	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
54	53	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ∈ (Base‘𝐾))
55	21, 3, 4	hlatjcl 37829	. . . . . . . . . . 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 18333	. . . . . . . . . 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 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → ((𝐶 ≤ (𝑐 ∨ 𝑑) ∧ (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑)) → 𝐶 ≤ (𝑆 ∨ 𝑑)))
60	10, 52, 59	mp2and 697	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 ≤ (𝑆 ∨ 𝑑))
61		dalem23.o	. . . . . . . . . . 11 ⊢ 𝑂 = (LPlanes‘𝐾)
62	1, 61	dalemyeb 38112	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
63	62	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
64	21, 2, 30	latmlem1 18358	. . . . . . . . 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 481	. . . . . . 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 38143	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ 𝑌)
71	70	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ≤ 𝑌)
72	21, 2, 30	latleeqm1 18356	. . . . . . . . 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 231	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐶 ∧ 𝑌) = 𝐶)
75	74	adantr 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → (𝐶 ∧ 𝑌) = 𝐶)
76	1, 2, 3, 4, 69	dalemsly 38118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌)
77	76	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ≤ 𝑌)
78	7	dalem-ddly 38149	. . . . . . . . 9 ⊢ (𝜓 → ¬ 𝑑 ≤ 𝑌)
79	78	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑑 ≤ 𝑌)
80	21, 2, 3, 30, 4	2atjm 37908	. . . . . . . 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 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → ((𝑆 ∨ 𝑑) ∧ 𝑌) = 𝑆)
83	67, 75, 82	3brtr3d 5136	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 ≤ 𝑆)
84		hlatl 37822	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
85	15, 84	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ AtLat)
86	1, 2, 3, 4, 61, 68	dalemcea 38123	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
87	2, 4	atcmp 37773	. . . . . . . 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 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → (𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆))
91	83, 90	mpbid 231	. . . 4 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 = 𝑆)
92	91	ex 413	. . 3 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 = 𝐺 → 𝐶 = 𝑆))
93	92	necon3d 2964	. 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 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943   class class class wbr 5105  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  lecple 17140  joincjn 18200  meetcmee 18201  Latclat 18320  Atomscatm 37725  AtLatcal 37726  HLchlt 37812  LPlanesclpl 37955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-proset 18184  df-poset 18202  df-plt 18219  df-lub 18235  df-glb 18236  df-join 18237  df-meet 18238  df-p0 18314  df-lat 18321  df-clat 18388  df-oposet 37638  df-ol 37640  df-oml 37641  df-covers 37728  df-ats 37729  df-atl 37760  df-cvlat 37784  df-hlat 37813  df-llines 37961  df-lplanes 37962

This theorem is referenced by:  dalem28  38163  dalem31N  38166