Theorem dihord2pre 42027

Description: Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)

Hypotheses

Ref	Expression
dihjust.b	⊢ 𝐵 = (Base‘𝐾)
dihjust.l	⊢ ≤ = (le‘𝐾)
dihjust.j	⊢ ∨ = (join‘𝐾)
dihjust.m	⊢ ∧ = (meet‘𝐾)
dihjust.a	⊢ 𝐴 = (Atoms‘𝐾)
dihjust.h	⊢ 𝐻 = (LHyp‘𝐾)
dihjust.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
dihjust.J	⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊)
dihjust.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dihjust.s	⊢ ⊕ = (LSSum‘𝑈)
dihord2c.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dihord2c.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dihord2c.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
dihord2.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
dihord2.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dihord2.d	⊢ + = (+g‘𝑈)
dihord2.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁)

Assertion

Ref	Expression
dihord2pre	⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊))

Distinct variable groups:   𝐴,ℎ   𝑃,ℎ   𝐵,ℎ   ℎ,𝐻   ℎ,𝐾   ≤ ,ℎ   ℎ,𝑁   𝑇,ℎ   ℎ,𝑊

Allowed substitution hints:   + (ℎ)   ⊕ (ℎ)   𝑄(ℎ)   𝑅(ℎ)   𝑈(ℎ)   𝐸(ℎ)   𝐺(ℎ)   𝐼(ℎ)   𝐽(ℎ)   ∨ (ℎ)   ∧ (ℎ)   𝑂(ℎ)   𝑋(ℎ)   𝑌(ℎ)

Proof of Theorem dihord2pre

Dummy variables 𝑓 𝑔 𝑠 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 1210	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)))
2		simpl2l 1245	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑋 ∈ 𝐵)
3		simpl2r 1246	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑌 ∈ 𝐵)
4		simpl3 1212	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊))))
5		simprl 782	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑓 ∈ 𝑇)
6		simprr 784	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))
7		dihjust.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
8		dihjust.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
9		dihjust.j	. . . . . . 7 ⊢ ∨ = (join‘𝐾)
10		dihjust.m	. . . . . . 7 ⊢ ∧ = (meet‘𝐾)
11		dihjust.a	. . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾)
12		dihjust.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
13		dihjust.i	. . . . . . 7 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
14		dihjust.J	. . . . . . 7 ⊢ 𝐽 = ((DIsoC‘𝐾)‘𝑊)
15		dihjust.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
16		dihjust.s	. . . . . . 7 ⊢ ⊕ = (LSSum‘𝑈)
17		dihord2c.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
18		dihord2c.r	. . . . . . 7 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
19		dihord2c.o	. . . . . . 7 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
20		dihord2.p	. . . . . . 7 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
21		dihord2.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
22		dihord2.d	. . . . . . 7 ⊢ + = (+g‘𝑈)
23		dihord2.g	. . . . . . 7 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑁)
24	7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	dihord11c 42026	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊))) ∧ 𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ∃𝑦 ∈ (𝐽‘𝑁)∃𝑧 ∈ (𝐼‘(𝑌 ∧ 𝑊))⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧))
25	1, 2, 3, 4, 5, 6, 24	syl123anc 1414	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ∃𝑦 ∈ (𝐽‘𝑁)∃𝑧 ∈ (𝐼‘(𝑌 ∧ 𝑊))⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧))
26		simpl11 1267	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
27		simpl13 1269	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊))
28	8, 11, 12, 20, 17, 21, 14, 23	dicelval3 41982	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) → (𝑦 ∈ (𝐽‘𝑁) ↔ ∃𝑠 ∈ 𝐸 𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩))
29	26, 27, 28	syl2anc 595	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑦 ∈ (𝐽‘𝑁) ↔ ∃𝑠 ∈ 𝐸 𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩))
30		simp11l 1303	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → 𝐾 ∈ HL)
31	30	adantr 485	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝐾 ∈ HL)
32	31	hllatd 40166	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝐾 ∈ Lat)
33		simp11r 1304	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → 𝑊 ∈ 𝐻)
34	33	adantr 485	. . . . . . . . . . 11 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑊 ∈ 𝐻)
35	7, 12	lhpbase 40800	. . . . . . . . . . 11 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵)
36	34, 35	syl 18	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → 𝑊 ∈ 𝐵)
37	7, 10	latmcl 18500	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ∈ 𝐵)
38	32, 3, 36, 37	syl3anc 1398	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑌 ∧ 𝑊) ∈ 𝐵)
39	7, 8, 10	latmle2 18525	. . . . . . . . . 10 ⊢ ((𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 ∧ 𝑊) ≤ 𝑊)
40	32, 3, 36, 39	syl3anc 1398	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑌 ∧ 𝑊) ≤ 𝑊)
41	7, 8, 12, 17, 18, 19, 13	dibelval3 41949	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑌 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ≤ 𝑊)) → (𝑧 ∈ (𝐼‘(𝑌 ∧ 𝑊)) ↔ ∃𝑔 ∈ 𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊))))
42	26, 38, 40, 41	syl12anc 849	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑧 ∈ (𝐼‘(𝑌 ∧ 𝑊)) ↔ ∃𝑔 ∈ 𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊))))
43	29, 42	anbi12d 643	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ((𝑦 ∈ (𝐽‘𝑁) ∧ 𝑧 ∈ (𝐼‘(𝑌 ∧ 𝑊))) ↔ (∃𝑠 ∈ 𝐸 𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ ∃𝑔 ∈ 𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊)))))
44		reeanv 3237	. . . . . . . 8 ⊢ (∃𝑠 ∈ 𝐸 ∃𝑔 ∈ 𝑇 (𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊))) ↔ (∃𝑠 ∈ 𝐸 𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ ∃𝑔 ∈ 𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊))))
45		simpll1 1231	. . . . . . . . . . . 12 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) ∧ ((𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)))
46		simplr 780	. . . . . . . . . . . 12 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) ∧ ((𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)))
47		simpr 489	. . . . . . . . . . . 12 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) ∧ ((𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ((𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩)))
48	7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	dihord10 42025	. . . . . . . . . . . 12 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊)) ∧ ((𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))
49	45, 46, 47, 48	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) ∧ ((𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))
50	49	3exp2 1373	. . . . . . . . . 10 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ((𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → ((𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) → (⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊)))))
51		oveq12 7419	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → (𝑦 + 𝑧) = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩))
52	51	eqeq2d 2774	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) ↔ ⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩)))
53	52	imbi1d 344	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → ((⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊)) ↔ (⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))))
54	53	imbi2d 343	. . . . . . . . . . . . . 14 ⊢ ((𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → (((𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))) ↔ ((𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) → (⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊)))))
55	54	biimprd 251	. . . . . . . . . . . . 13 ⊢ ((𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → (((𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) → (⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))) → ((𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊)))))
56	55	com23 87	. . . . . . . . . . . 12 ⊢ ((𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ 𝑧 = ⟨𝑔, 𝑂⟩) → ((𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) → (((𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) → (⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊)))))
57	56	impr 459	. . . . . . . . . . 11 ⊢ ((𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊))) → (((𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) → (⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))))
58	57	com12 33	. . . . . . . . . 10 ⊢ (((𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊) → (⟨𝑓, 𝑂⟩ = (⟨(𝑠‘𝐺), 𝑠⟩ + ⟨𝑔, 𝑂⟩) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))) → ((𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊))) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))))
59	50, 58	syl6 36	. . . . . . . . 9 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ((𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇) → ((𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊))) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊)))))
60	59	rexlimdvv 3221	. . . . . . . 8 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (∃𝑠 ∈ 𝐸 ∃𝑔 ∈ 𝑇 (𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊))) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))))
61	44, 60	biimtrrid 246	. . . . . . 7 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ((∃𝑠 ∈ 𝐸 𝑦 = ⟨(𝑠‘𝐺), 𝑠⟩ ∧ ∃𝑔 ∈ 𝑇 (𝑧 = ⟨𝑔, 𝑂⟩ ∧ (𝑅‘𝑔) ≤ (𝑌 ∧ 𝑊))) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))))
62	43, 61	sylbid 243	. . . . . 6 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → ((𝑦 ∈ (𝐽‘𝑁) ∧ 𝑧 ∈ (𝐼‘(𝑌 ∧ 𝑊))) → (⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))))
63	62	rexlimdvv 3221	. . . . 5 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (∃𝑦 ∈ (𝐽‘𝑁)∃𝑧 ∈ (𝐼‘(𝑌 ∧ 𝑊))⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊)))
64	25, 63	mpd 16	. . . 4 ⊢ (((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) ∧ (𝑓 ∈ 𝑇 ∧ (𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊))) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))
65	64	exp32 425	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → (𝑓 ∈ 𝑇 → ((𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))))
66	65	ralrimiv 3156	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → ∀𝑓 ∈ 𝑇 ((𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊)))
67		simp11 1222	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
68	30	hllatd 40166	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → 𝐾 ∈ Lat)
69		simp2l 1218	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → 𝑋 ∈ 𝐵)
70	33, 35	syl 18	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → 𝑊 ∈ 𝐵)
71	7, 10	latmcl 18500	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ∈ 𝐵)
72	68, 69, 70, 71	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → (𝑋 ∧ 𝑊) ∈ 𝐵)
73	7, 8, 10	latmle2 18525	. . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑋 ∧ 𝑊) ≤ 𝑊)
74	68, 69, 70, 73	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → (𝑋 ∧ 𝑊) ≤ 𝑊)
75		simp2r 1219	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → 𝑌 ∈ 𝐵)
76	68, 75, 70, 37	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → (𝑌 ∧ 𝑊) ∈ 𝐵)
77	68, 75, 70, 39	syl3anc 1398	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → (𝑌 ∧ 𝑊) ≤ 𝑊)
78	7, 8, 11, 12, 17, 18	trlord 41371	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑋 ∧ 𝑊) ∈ 𝐵 ∧ (𝑋 ∧ 𝑊) ≤ 𝑊) ∧ ((𝑌 ∧ 𝑊) ∈ 𝐵 ∧ (𝑌 ∧ 𝑊) ≤ 𝑊)) → ((𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊) ↔ ∀𝑓 ∈ 𝑇 ((𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))))
79	67, 72, 74, 76, 77, 78	syl122anc 1406	. 2 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → ((𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊) ↔ ∀𝑓 ∈ 𝑇 ((𝑅‘𝑓) ≤ (𝑋 ∧ 𝑊) → (𝑅‘𝑓) ≤ (𝑌 ∧ 𝑊))))
80	66, 79	mpbird 260	1 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊)) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐽‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑊))) ⊆ ((𝐽‘𝑁) ⊕ (𝐼‘(𝑌 ∧ 𝑊)))) → (𝑋 ∧ 𝑊) ≤ (𝑌 ∧ 𝑊))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3905  ⟨cop 4595   class class class wbr 5109   ↦ cmpt 5192   I cid 5555   ↾ cres 5663  ‘cfv 6536  ℩crio 7366  (class class class)co 7410  Basecbs 17273  +_gcplusg 17314  lecple 17321  occoc 17322  joincjn 18371  meetcmee 18372  Latclat 18491  LSSumclsm 19708  Atomscatm 40065  HLchlt 40152  LHypclh 40786  LTrncltrn 40903  trLctrl 40960  TEndoctendo 41554  DVecHcdvh 41880  DIsoBcdib 41940  DIsoCcdic 41974

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11160  ax-resscn 11161  ax-1cn 11162  ax-icn 11163  ax-addcl 11164  ax-addrcl 11165  ax-mulcl 11166  ax-mulrcl 11167  ax-mulcom 11168  ax-addass 11169  ax-mulass 11170  ax-distr 11171  ax-i2m1 11172  ax-1ne0 11173  ax-1rid 11174  ax-rnegex 11175  ax-rrecex 11176  ax-cnre 11177  ax-pre-lttri 11178  ax-pre-lttrn 11179  ax-pre-ltadd 11180  ax-pre-mulgt0 11181  ax-riotaBAD 39755

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-iin 4959  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  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 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-tpos 8218  df-undef 8265  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11447  df-neg 11448  df-nn 12238  df-2 12307  df-3 12308  df-4 12309  df-5 12310  df-6 12311  df-n0 12509  df-z 12596  df-uz 12867  df-fz 13540  df-struct 17211  df-sets 17228  df-slot 17246  df-ndx 17258  df-base 17274  df-ress 17295  df-plusg 17327  df-mulr 17328  df-sca 17330  df-vsca 17331  df-0g 17498  df-proset 18354  df-poset 18373  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18492  df-clat 18559  df-mgm 18702  df-sgrp 18781  df-mnd 18797  df-submnd 18846  df-grp 19007  df-minusg 19008  df-sbg 19009  df-subg 19193  df-lsm 19710  df-cmn 19856  df-abl 19857  df-mgp 20221  df-rng 20235  df-ur 20268  df-ring 20321  df-oppr 20424  df-dvdsr 20444  df-unit 20445  df-invr 20475  df-dvr 20488  df-drng 20838  df-lmod 20992  df-lss 21062  df-lvec 21233  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

This theorem is used by:  dihord2pre2  42028