dihlsscpre - Mathbox for Norm Megill

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Theorem dihlsscpre 40762

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 40761	. 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 1194	. . . . . . . 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 1196	. . . . . . . . 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 1241	. . . . . . . . 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 510	. . . . . . . 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 1197	. . . . . . . . 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 1243	. . . . . . . . 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 510	. . . . . . . 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 1235	. . . . . . . 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 1242	. . . . . . . . 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 1244	. . . . . . . . 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 2768	. . . . . . . 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 40745	. . . . . . . 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 1380	. . . . . . 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 1116	. . . . . 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 3189	. . . . 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 39552	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → ∃𝑞 ∈ 𝐴 (¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
30		simpll 765	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	6, 10, 30	dvhlmod 40638	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝑈 ∈ LMod)
32	2, 5, 6, 10, 9, 11	diclss 40721	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐶‘𝑞) ∈ 𝑆)
33	32	adantlr 713	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐶‘𝑞) ∈ 𝑆)
34		hllat 38890	. . . . . . . . . . . . . . . 16 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat)
35	34	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝐾 ∈ Lat)
36		simplrl 775	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝑋 ∈ 𝐵)
37	1, 6	lhpbase 39526	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
38	37	ad3antlr 729	. . . . . . . . . . . . . . 15 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → 𝑊 ∈ 𝐵)
39	1, 4	latmcl 18429	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
40	35, 36, 38, 39	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝑋 ∧ 𝑊) ∈ 𝐵)
41	1, 2, 4	latmle2 18454	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
42	35, 36, 38, 41	syl3anc 1368	. . . . . . . . . . . . . 14 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝑋 ∧ 𝑊) ≤ 𝑊)
43	1, 2, 6, 10, 8, 11	diblss 40698	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊)) → (𝐷‘(𝑋 ∧ 𝑊)) ∈ 𝑆)
44	30, 40, 42, 43	syl12anc 835	. . . . . . . . . . . . 13 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → (𝐷‘(𝑋 ∧ 𝑊)) ∈ 𝑆)
45	11, 12	lsmcl 20970	. . . . . . . . . . . . 13 ⊢ ((𝑈 ∈ LMod ∧ (𝐶‘𝑞) ∈ 𝑆 ∧ (𝐷‘(𝑋 ∧ 𝑊)) ∈ 𝑆) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆)
46	31, 33, 44, 45	syl3anc 1368	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) ∧ (𝑞 ∈ 𝐴 ∧ ¬ 𝑞 ≤ 𝑊)) → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) ∈ 𝑆)
47	46	a1d 25	. . . . . . . . . . 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 455	. . . . . . . . . 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 409	. . . . . . . . 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 549	. . . . . . . 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 3158	. . . . . . 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 15	. . . . . 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 5146	. . . . . . . . 9 ⊢ (𝑞 = 𝑟 → (𝑞 ≤ 𝑊 ↔ 𝑟 ≤ 𝑊))
54	53	notbid 317	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (¬ 𝑞 ≤ 𝑊 ↔ ¬ 𝑟 ≤ 𝑊))
55		oveq1 7422	. . . . . . . . 9 ⊢ (𝑞 = 𝑟 → (𝑞 ∨ (𝑋 ∧ 𝑊)) = (𝑟 ∨ (𝑋 ∧ 𝑊)))
56	55	eqeq1d 2727	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → ((𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋 ↔ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋))
57	54, 56	anbi12d 630	. . . . . . 7 ⊢ (𝑞 = 𝑟 → ((¬ 𝑞 ≤ 𝑊 ∧ (𝑞 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ↔ (¬ 𝑟 ≤ 𝑊 ∧ (𝑟 ∨ (𝑋 ∧ 𝑊)) = 𝑋)))
58		fveq2 6891	. . . . . . . 8 ⊢ (𝑞 = 𝑟 → (𝐶‘𝑞) = (𝐶‘𝑟))
59	58	oveq1d 7430	. . . . . . 7 ⊢ (𝑞 = 𝑟 → ((𝐶‘𝑞) ⊕ (𝐷‘(𝑋 ∧ 𝑊))) = ((𝐶‘𝑟) ⊕ (𝐷‘(𝑋 ∧ 𝑊))))
60	57, 59	reusv3 5399	. . . . . 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 17	. . . . 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 231	. . . 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 5391	. . . . 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 17	. . . 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 256	. . 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 7389	. . 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 17	. 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 2825	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3051 ∃wrex 3060 ∃!wreu 3362 class class class wbr 5143 ‘cfv 6542 ℩crio 7370 (class class class)co 7415 Basecbs 17177 lecple 17237 joincjn 18300 meetcmee 18301 Latclat 18420 LSSumclsm 19591 LModclmod 20745 LSubSpclss 20817 Atomscatm 38790 HLchlt 38877 LHypclh 39512 DVecHcdvh 40606 DIsoBcdib 40666 DIsoCcdic 40700 DIsoHcdih 40756

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-riotaBAD 38480

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-undef 8275 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-sca 17246 df-vsca 17247 df-0g 17420 df-proset 18284 df-poset 18302 df-plt 18319 df-lub 18335 df-glb 18336 df-join 18337 df-meet 18338 df-p0 18414 df-p1 18415 df-lat 18421 df-clat 18488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-submnd 18738 df-grp 18895 df-minusg 18896 df-sbg 18897 df-subg 19080 df-cntz 19270 df-lsm 19593 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-drng 20628 df-lmod 20747 df-lss 20818 df-lsp 20858 df-lvec 20990 df-oposet 38703 df-ol 38705 df-oml 38706 df-covers 38793 df-ats 38794 df-atl 38825 df-cvlat 38849 df-hlat 38878 df-llines 39026 df-lplanes 39027 df-lvols 39028 df-lines 39029 df-psubsp 39031 df-pmap 39032 df-padd 39324 df-lhyp 39516 df-laut 39517 df-ldil 39632 df-ltrn 39633 df-trl 39687 df-tendo 40283 df-edring 40285 df-disoa 40557 df-dvech 40607 df-dib 40667 df-dic 40701 df-dih 40757

This theorem is referenced by: dihvalcqpre 40763 dihlss 40778

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