Theorem dihord5b 42061

Description: Part of proof that isomorphism H is order-preserving. TODO: eliminate 3ad2ant1; combine with other way to have one lhpmcvr2 . (Contributed by NM, 7-Mar-2014.)

Hypotheses

Ref	Expression
dihord3.b	⊢ 𝐵 = (Base‘𝐾)
dihord3.l	⊢ ≤ = (le‘𝐾)
dihord3.h	⊢ 𝐻 = (LHyp‘𝐾)
dihord3.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)

Assertion

Ref	Expression
dihord5b	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))

Proof of Theorem dihord5b

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1209	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		simpl3 1211	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊))
3		dihord3.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
4		dihord3.l	. . . 4 ⊢ ≤ = (le‘𝐾)
5		eqid 2762	. . . 4 ⊢ (join‘𝐾) = (join‘𝐾)
6		eqid 2762	. . . 4 ⊢ (meet‘𝐾) = (meet‘𝐾)
7		eqid 2762	. . . 4 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
8		dihord3.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
9	3, 4, 5, 6, 7, 8	lhpmcvr2 40826	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))
10	1, 2, 9	syl2anc 595	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → ∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌))
11		simp1r 1216	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 ≤ 𝑌)
12		simpl2r 1245	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ≤ 𝑊)
13	12	3ad2ant1 1150	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 ≤ 𝑊)
14		simpl1l 1242	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝐾 ∈ HL)
15	14	3ad2ant1 1150	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝐾 ∈ HL)
16	15	hllatd 40166	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝐾 ∈ Lat)
17		simpl2l 1244	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝑋 ∈ 𝐵)
18	17	3ad2ant1 1150	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 ∈ 𝐵)
19		simpl3l 1246	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝑌 ∈ 𝐵)
20	19	3ad2ant1 1150	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑌 ∈ 𝐵)
21		simpl1r 1243	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → 𝑊 ∈ 𝐻)
22	21	3ad2ant1 1150	. . . . . . . . . . . 12 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑊 ∈ 𝐻)
23	3, 8	lhpbase 40800	. . . . . . . . . . . 12 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
24	22, 23	syl 18	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑊 ∈ 𝐵)
25	3, 4, 6	latlem12 18528	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑊) ↔ 𝑋 ≤ (𝑌(meet‘𝐾)𝑊)))
26	16, 18, 20, 24, 25	syl13anc 1398	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑊) ↔ 𝑋 ≤ (𝑌(meet‘𝐾)𝑊)))
27	11, 13, 26	mpbi2and 724	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → 𝑋 ≤ (𝑌(meet‘𝐾)𝑊))
28		simp1l1 1284	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
29		simp1l2 1285	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
30	3, 6	latmcl 18502	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌(meet‘𝐾)𝑊) ∈ 𝐵)
31	16, 20, 24, 30	syl3anc 1397	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑌(meet‘𝐾)𝑊) ∈ 𝐵)
32	3, 4, 6	latmle2 18527	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌(meet‘𝐾)𝑊) ≤ 𝑊)
33	16, 20, 24, 32	syl3anc 1397	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑌(meet‘𝐾)𝑊) ≤ 𝑊)
34		eqid 2762	. . . . . . . . . . 11 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊)
35	3, 4, 8, 34	dibord 41961	. . . . . . . . . 10 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ ((𝑌(meet‘𝐾)𝑊) ∈ 𝐵 ∧ (𝑌(meet‘𝐾)𝑊) ≤ 𝑊)) → ((((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ↔ 𝑋 ≤ (𝑌(meet‘𝐾)𝑊)))
36	28, 29, 31, 33, 35	syl112anc 1400	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → ((((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ↔ 𝑋 ≤ (𝑌(meet‘𝐾)𝑊)))
37	27, 36	mpbird 260	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)))
38		eqid 2762	. . . . . . . . . . . 12 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
39	8, 38, 28	dvhlmod 41912	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → ((DVecH‘𝐾)‘𝑊) ∈ LMod)
40		eqid 2762	. . . . . . . . . . . 12 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
41	40	lsssssubg 21090	. . . . . . . . . . 11 ⊢ (((DVecH‘𝐾)‘𝑊) ∈ LMod → (LSubSp‘((DVecH‘𝐾)‘𝑊)) ⊆ (SubGrp‘((DVecH‘𝐾)‘𝑊)))
42	39, 41	syl 18	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (LSubSp‘((DVecH‘𝐾)‘𝑊)) ⊆ (SubGrp‘((DVecH‘𝐾)‘𝑊)))
43		simp2 1154	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊))
44		eqid 2762	. . . . . . . . . . . 12 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊)
45	4, 7, 8, 38, 44, 40	diclss 41995	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊)) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊)))
46	28, 43, 45	syl2anc 595	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊)))
47	42, 46	sseldd 3937	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊)))
48	3, 4, 8, 38, 34, 40	diblss 41972	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑌(meet‘𝐾)𝑊) ∈ 𝐵 ∧ (𝑌(meet‘𝐾)𝑊) ≤ 𝑊)) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊)))
49	28, 31, 33, 48	syl12anc 849	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊)))
50	42, 49	sseldd 3937	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊)))
51		eqid 2762	. . . . . . . . . 10 ⊢ (LSSum‘((DVecH‘𝐾)‘𝑊)) = (LSSum‘((DVecH‘𝐾)‘𝑊))
52	51	lsmub2 19734	. . . . . . . . 9 ⊢ (((((DIsoC‘𝐾)‘𝑊)‘𝑟) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊)) ∧ (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ∈ (SubGrp‘((DVecH‘𝐾)‘𝑊))) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
53	47, 50, 52	syl2anc 595	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊)) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
54	37, 53	sstrd 3946	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (((DIsoB‘𝐾)‘𝑊)‘𝑋) ⊆ ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
55		dihord3.i	. . . . . . . . 9 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
56	3, 4, 8, 55, 34	dihvalb 42039	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
57	28, 29, 56	syl2anc 595	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑋) = (((DIsoB‘𝐾)‘𝑊)‘𝑋))
58		simp1l3 1286	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊))
59		simp3 1155	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)
60	3, 4, 5, 6, 7, 8, 55, 34, 44, 38, 51	dihvalcq 42038	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ ((𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌)) → (𝐼‘𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
61	28, 58, 43, 59, 60	syl112anc 1400	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑌) = ((((DIsoC‘𝐾)‘𝑊)‘𝑟)(LSSum‘((DVecH‘𝐾)‘𝑊))(((DIsoB‘𝐾)‘𝑊)‘(𝑌(meet‘𝐾)𝑊))))
62	54, 57, 61	3sstr4d 3991	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) ∧ (𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))
63	62	3exp 1136	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → ((𝑟 ∈ (Atoms‘𝐾) ∧ ¬ 𝑟 ≤ 𝑊) → ((𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌 → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))))
64	63	expd 420	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝑟 ∈ (Atoms‘𝐾) → (¬ 𝑟 ≤ 𝑊 → ((𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌 → (𝐼‘𝑋) ⊆ (𝐼‘𝑌)))))
65	64	imp4a 427	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝑟 ∈ (Atoms‘𝐾) → ((¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))))
66	65	rexlimdv 3163	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (∃𝑟 ∈ (Atoms‘𝐾)(¬ 𝑟 ≤ 𝑊 ∧ (𝑟(join‘𝐾)(𝑌(meet‘𝐾)𝑊)) = 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌)))
67	10, 66	mpd 16	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊)) ∧ 𝑋 ≤ 𝑌) → (𝐼‘𝑋) ⊆ (𝐼‘𝑌))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∃wrex 3088   ⊆ wss 3904   class class class wbr 5108  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  lecple 17323  joincjn 18373  meetcmee 18374  Latclat 18493  SubGrpcsubg 19192  LSSumclsm 19710  LModclmod 20992  LSubSpclss 21063  Atomscatm 40065  HLchlt 40152  LHypclh 40786  DVecHcdvh 41880  DIsoBcdib 41940  DIsoCcdic 41974  DIsoHcdih 42030

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-riotaBAD 39755

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-tpos 8220  df-undef 8267  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-er 8692  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-2 12309  df-3 12310  df-4 12311  df-5 12312  df-6 12313  df-n0 12511  df-z 12598  df-uz 12869  df-fz 13542  df-struct 17213  df-sets 17230  df-slot 17248  df-ndx 17260  df-base 17276  df-ress 17297  df-plusg 17329  df-mulr 17330  df-sca 17332  df-vsca 17333  df-0g 17500  df-proset 18356  df-poset 18375  df-plt 18390  df-lub 18406  df-glb 18407  df-join 18408  df-meet 18409  df-p0 18485  df-p1 18486  df-lat 18494  df-clat 18561  df-mgm 18704  df-sgrp 18783  df-mnd 18799  df-submnd 18848  df-grp 19009  df-minusg 19010  df-sbg 19011  df-subg 19195  df-cntz 19393  df-lsm 19712  df-cmn 19858  df-abl 19859  df-mgp 20223  df-rng 20237  df-ur 20270  df-ring 20323  df-oppr 20426  df-dvdsr 20446  df-unit 20447  df-invr 20477  df-dvr 20490  df-drng 20840  df-lmod 20994  df-lss 21064  df-lsp 21104  df-lvec 21235  df-oposet 39978  df-ol 39980  df-oml 39981  df-covers 40068  df-ats 40069  df-atl 40100  df-cvlat 40124  df-hlat 40153  df-llines 40300  df-lplanes 40301  df-lvols 40302  df-lines 40303  df-psubsp 40305  df-pmap 40306  df-padd 40598  df-lhyp 40790  df-laut 40791  df-ldil 40906  df-ltrn 40907  df-trl 40961  df-tendo 41557  df-edring 41559  df-disoa 41831  df-dvech 41881  df-dib 41941  df-dic 41975  df-dih 42031

This theorem is used by:  dihord  42066