Theorem dihlsscpre 41933

Description: Closure of isomorphism H for a lattice 𝐾 when ¬ 𝑋 ≤ 𝑊. (Contributed by NM, 6-Mar-2014.)

Hypotheses

Ref	Expression
dihval.b	⊢ 𝐵 = (Base‘𝐾)
dihval.l	⊢ ≤ = (le‘𝐾)
dihval.j	⊢ ∨ = (join‘𝐾)
dihval.m	⊢ ∧ = (meet‘𝐾)
dihval.a	⊢ 𝐴 = (Atoms‘𝐾)
dihval.h	⊢ 𝐻 = (LHyp‘𝐾)
dihval.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dihval.d	⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)
dihval.c	⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)
dihval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihval.s	⊢ 𝑆 = (LSubSp‘𝑈)
dihval.p	⊢ ⊕ = (LSSum‘𝑈)

Assertion

Ref	Expression
dihlsscpre	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ 𝑆)

Proof of Theorem dihlsscpre

Dummy variables 𝑞 𝑢 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dihval.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		dihval.l	. . 3 ⊢ ≤ = (le‘𝐾)
3		dihval.j	. . 3 ⊢ ∨ = (join‘𝐾)
4		dihval.m	. . 3 ⊢ ∧ = (meet‘𝐾)
5		dihval.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
6		dihval.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
7		dihval.i	. . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
8		dihval.d	. . 3 ⊢ 𝐷 = ((DIsoB‘𝐾)‘𝑊)
9		dihval.c	. . 3 ⊢ 𝐶 = ((DIsoC‘𝐾)‘𝑊)
10		dihval.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11		dihval.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑈)
12		dihval.p	. . 3 ⊢ ⊕ = (LSSum‘𝑈)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	dihvalc 41932	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
14		simp1l 1214	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		simp2l 1216	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → 𝑞 ∈ 𝐴)
16		simp3ll 1261	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → ¬ 𝑞 ≤ 𝑊)
17	15, 16	jca 520	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊))
18		simp2r 1217	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → 𝑟 ∈ 𝐴)
19		simp3rl 1263	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → ¬ 𝑟 ≤ 𝑊)
20	18, 19	jca 520	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊))
21		simp1rl 1255	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → 𝑋 ∈ 𝐵)
22		simp3lr 1262	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
23		simp3rr 1264	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋)
24	22, 23	eqtr4d 2807	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → (𝑞 ∨ (𝑋 ∧ 𝑊)) = (𝑟 ∨ (𝑋 ∧ 𝑊)))
25	1, 2, 3, 4, 5, 6, 8, 9, 10, 12	dihjust 41916	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑋 ∈ 𝐵) ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = (𝑟 ∨ (𝑋 ∧ 𝑊))) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) = ((𝐶‘𝑟) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))
26	14, 17, 20, 21, 24, 25	syl131anc 1408	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) = ((𝐶‘𝑟) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))
27	26	3exp 1135	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ((𝑞 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) → (((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) = ((𝐶‘𝑟) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
28	27	ralrimivv 3212	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) = ((𝐶‘𝑟) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))
29	1, 2, 3, 4, 5, 6	lhpmcvr2 40723	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
30		simpll 778	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	6, 10, 30	dvhlmod 41809	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝑈 ∈ LMod)
32	2, 5, 6, 10, 9, 11	diclss 41892	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐶‘𝑞) ∈ 𝑆)
33	32	adantlr 727	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐶‘𝑞) ∈ 𝑆)
34		hllat 40062	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
35	34	ad3antrrr 742	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝐾 ∈ Lat)
36		simplrl 788	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
37	1, 6	lhpbase 40697	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
38	37	ad3antlr 743	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
39	1, 4	latmcl 18496	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
40	35, 36, 38, 39	syl3anc 1396	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
41	1, 2, 4	latmle2 18521	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
42	35, 36, 38, 41	syl3anc 1396	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝑋 ∧ 𝑊) ≤ 𝑊)
43	1, 2, 6, 10, 8, 11	diblss 41869	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (𝐷‘(𝑋 ∧ 𝑊)) ∈ 𝑆)
44	30, 40, 42, 43	syl12anc 849	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐷‘(𝑋 ∧ 𝑊)) ∈ 𝑆)
45	11, 12	lsmcl 21182	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ LMod ∧ (𝐶‘𝑞) ∈ 𝑆 ∧ (𝐷‘(𝑋 ∧ 𝑊)) ∈ 𝑆) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆)
46	31, 33, 44, 45	syl3anc 1396	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆)
47	46	a1d 26	. . . . . . . . . . 11 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → ((𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋 → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆))
48	47	expr 461	. . . . . . . . . 10 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑞 ∈ 𝐴) → (¬ 𝑞 ≤ 𝑊 → ((𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋 → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆)))
49	48	impd 415	. . . . . . . . 9 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆))
50	49	ancld 559	. . . . . . . 8 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ 𝑞 ∈ 𝐴) → ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆)))
51	50	reximdva 3184	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → ∃𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆)))
52	29, 51	mpd 16	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆))
53		breq1 5114	. . . . . . . . 9 ⊢ (𝑞 = 𝑟 → (𝑞 ≤ 𝑊 ↔ 𝑟 ≤ 𝑊))
54	53	notbid 321	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (¬ 𝑞 ≤ 𝑊 ↔ ¬ 𝑟 ≤ 𝑊))
55		oveq1 7418	. . . . . . . . 9 ⊢ (𝑞 = 𝑟 → (𝑞 ∨ (𝑋 ∧ 𝑊)) = (𝑟 ∨ (𝑋 ∧ 𝑊)))
56	55	eqeq1d 2771	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → ((𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ↔ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
57	54, 56	anbi12d 643	. . . . . . 7 ⊢ (𝑞 = 𝑟 → ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ↔ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋)))
58		fveq2 6882	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝐶‘𝑞) = (𝐶‘𝑟))
59	58	oveq1d 7426	. . . . . . 7 ⊢ (𝑞 = 𝑟 → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) = ((𝐶‘𝑟) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))
60	57, 59	reusv3 5377	. . . . . 6 ⊢ (∃𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆) → (∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) = ((𝐶‘𝑟) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))) ↔ ∃𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
61	52, 60	syl 18	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (∀𝑞 ∈ 𝐴 ∀𝑟 ∈ 𝐴 (((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋)) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) = ((𝐶‘𝑟) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))) ↔ ∃𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
62	28, 61	mpbid 235	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))
63		reusv1 5369	. . . . 5 ⊢ (∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → (∃!𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))) ↔ ∃𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
64	29, 63	syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (∃!𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))) ↔ ∃𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))))
65	62, 64	mpbird 260	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃!𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))))
66		riotacl 7385	. . 3 ⊢ (∃!𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊)))) → (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))) ∈ 𝑆)
67	65, 66	syl 18	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (℩𝑢 ∈ 𝑆 ∀𝑞 ∈ 𝐴 ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) → 𝑢 = ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))) ∈ 𝑆)
68	13, 67	eqeltrd 2869	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ∃!wreu 3373   class class class wbr 5111  ‘cfv 6537  ℩crio 7367  (class class class)co 7411  Basecbs 17269  lecple 17317  joincjn 18367  meetcmee 18368  Latclat 18487  LSSumclsm 19704  LModclmod 20959  LSubSpclss 21030  Atomscatm 39962  HLchlt 40049  LHypclh 40683  DVecHcdvh 41777  DIsoBcdib 41837  DIsoCcdic 41871  DIsoHcdih 41927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-riotaBAD 39652

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  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-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-tpos 8222  df-undef 8269  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-n0 12505  df-z 12592  df-uz 12863  df-fz 13536  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-sca 17326  df-vsca 17327  df-0g 17494  df-proset 18350  df-poset 18369  df-plt 18384  df-lub 18400  df-glb 18401  df-join 18402  df-meet 18403  df-p0 18479  df-p1 18480  df-lat 18488  df-clat 18555  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-submnd 18842  df-grp 19003  df-minusg 19004  df-sbg 19005  df-subg 19189  df-cntz 19387  df-lsm 19706  df-cmn 19852  df-abl 19853  df-mgp 20217  df-rng 20231  df-ur 20264  df-ring 20317  df-oppr 20419  df-dvdsr 20439  df-unit 20440  df-invr 20470  df-dvr 20483  df-drng 20815  df-lmod 20961  df-lss 21031  df-lsp 21071  df-lvec 21202  df-oposet 39875  df-ol 39877  df-oml 39878  df-covers 39965  df-ats 39966  df-atl 39997  df-cvlat 40021  df-hlat 40050  df-llines 40197  df-lplanes 40198  df-lvols 40199  df-lines 40200  df-psubsp 40202  df-pmap 40203  df-padd 40495  df-lhyp 40687  df-laut 40688  df-ldil 40803  df-ltrn 40804  df-trl 40858  df-tendo 41454  df-edring 41456  df-disoa 41728  df-dvech 41778  df-dib 41838  df-dic 41872  df-dih 41928

This theorem is referenced by:  dihvalcqpre  41934  dihlss  41949