Theorem dalem25 39687

Description: Lemma for dath 39725. 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 39639	. . 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 39677	. . . . . . . . . 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 39613	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐾 ∈ Lat)
14	13	3ad2ant1 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ Lat)
15	1	dalemkehl 39612	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐾 ∈ HL)
16	15	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐾 ∈ HL)
17	7	dalemccea 39672	. . . . . . . . . . . . . . . 16 ⊢ (𝜓 → 𝑐 ∈ 𝐴)
18	17	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ 𝐴)
19	1	dalempea 39615	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
20	19	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑃 ∈ 𝐴)
21		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (Base‘𝐾) = (Base‘𝐾)
22	21, 3, 4	hlatjcl 39355	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
23	16, 18, 20, 22	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑃) ∈ (Base‘𝐾))
24	7	dalemddea 39673	. . . . . . . . . . . . . . . 16 ⊢ (𝜓 → 𝑑 ∈ 𝐴)
25	24	3ad2ant3 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑑 ∈ 𝐴)
26	1	dalemsea 39618	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
27	26	3ad2ant1 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ∈ 𝐴)
28	21, 3, 4	hlatjcl 39355	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
29	16, 25, 27, 28	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) ∈ (Base‘𝐾))
30		dalem23.m	. . . . . . . . . . . . . . 15 ⊢ ∧ = (meet‘𝐾)
31	21, 2, 30	latmle2 18430	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Lat ∧ (𝑐 ∨ 𝑃) ∈ (Base‘𝐾) ∧ (𝑑 ∨ 𝑆) ∈ (Base‘𝐾)) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑑 ∨ 𝑆))
32	14, 23, 29, 31	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆)) ≤ (𝑑 ∨ 𝑆))
33	12, 32	eqbrtrid 5144	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐺 ≤ (𝑑 ∨ 𝑆))
34	3, 4	hlatjcom 39356	. . . . . . . . . . . . 13 ⊢ ((𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝑑 ∨ 𝑆) = (𝑆 ∨ 𝑑))
35	16, 25, 27, 34	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑑 ∨ 𝑆) = (𝑆 ∨ 𝑑))
36	33, 35	breqtrd 5135	. . . . . . . . . . 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 5131	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝑐 ≤ (𝑆 ∨ 𝑑))
39	2, 3, 4	hlatlej2 39364	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → 𝑑 ≤ (𝑆 ∨ 𝑑))
40	16, 27, 25, 39	syl3anc 1373	. . . . . . . . . 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 39678	. . . . . . . . . . . 12 ⊢ (𝜓 → 𝑐 ∈ (Base‘𝐾))
43	42	3ad2ant3 1135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑐 ∈ (Base‘𝐾))
44	21, 4	atbase 39277	. . . . . . . . . . . . 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 39355	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → (𝑆 ∨ 𝑑) ∈ (Base‘𝐾))
48	16, 27, 25, 47	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑆 ∨ 𝑑) ∈ (Base‘𝐾))
49	21, 2, 3	latjle12 18415	. . . . . . . . . . 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 1374	. . . . . . . . . 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 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → (𝑐 ∨ 𝑑) ≤ (𝑆 ∨ 𝑑))
53	1, 4	dalemceb 39627	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ (Base‘𝐾))
54	53	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ∈ (Base‘𝐾))
55	21, 3, 4	hlatjcl 39355	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
56	16, 18, 25, 55	syl3anc 1373	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝑐 ∨ 𝑑) ∈ (Base‘𝐾))
57	21, 2	lattr 18409	. . . . . . . . . 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 1374	. . . . . . . . 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 699	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 ≤ (𝑆 ∨ 𝑑))
61		dalem23.o	. . . . . . . . . . 11 ⊢ 𝑂 = (LPlanes‘𝐾)
62	1, 61	dalemyeb 39638	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ (Base‘𝐾))
63	62	3ad2ant1 1133	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑌 ∈ (Base‘𝐾))
64	21, 2, 30	latmlem1 18434	. . . . . . . . 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 1374	. . . . . . . 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 39669	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐶 ≤ 𝑌)
71	70	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐶 ≤ 𝑌)
72	21, 2, 30	latleeqm1 18432	. . . . . . . . 9 ⊢ ((𝐾 ∈ Lat ∧ 𝐶 ∈ (Base‘𝐾) ∧ 𝑌 ∈ (Base‘𝐾)) → (𝐶 ≤ 𝑌 ↔ (𝐶 ∧ 𝑌) = 𝐶))
73	14, 54, 63, 72	syl3anc 1373	. . . . . . . 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 39644	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝑆 ≤ 𝑌)
77	76	3adant3 1132	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝑆 ≤ 𝑌)
78	7	dalem-ddly 39675	. . . . . . . . 9 ⊢ (𝜓 → ¬ 𝑑 ≤ 𝑌)
79	78	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑑 ≤ 𝑌)
80	21, 2, 3, 30, 4	2atjm 39434	. . . . . . . 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 1390	. . . . . . 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 5140	. . . . 5 ⊢ (((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) ∧ 𝑐 = 𝐺) → 𝐶 ≤ 𝑆)
84		hlatl 39348	. . . . . . . . 9 ⊢ (𝐾 ∈ HL → 𝐾 ∈ AtLat)
85	15, 84	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ AtLat)
86	1, 2, 3, 4, 61, 68	dalemcea 39649	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
87	2, 4	atcmp 39299	. . . . . . . 8 ⊢ ((𝐾 ∈ AtLat ∧ 𝐶 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝐶 ≤ 𝑆 ↔ 𝐶 = 𝑆))
88	85, 86, 26, 87	syl3anc 1373	. . . . . . 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 2947	. 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 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2926   class class class wbr 5109  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  lecple 17233  joincjn 18278  meetcmee 18279  Latclat 18396  Atomscatm 39251  AtLatcal 39252  HLchlt 39338  LPlanesclpl 39481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5236  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3756  df-csb 3865  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5110  df-opab 5172  df-mpt 5191  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-riota 7346  df-ov 7392  df-oprab 7393  df-proset 18261  df-poset 18280  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-clat 18464  df-oposet 39164  df-ol 39166  df-oml 39167  df-covers 39254  df-ats 39255  df-atl 39286  df-cvlat 39310  df-hlat 39339  df-llines 39487  df-lplanes 39488

This theorem is referenced by:  dalem28  39689  dalem31N  39692