Theorem cdlemn8 38841

Description: Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)

Hypotheses

Ref	Expression
cdlemn8.b	⊢ 𝐵 = (Base‘𝐾)
cdlemn8.l	⊢ ≤ = (le‘𝐾)
cdlemn8.a	⊢ 𝐴 = (Atoms‘𝐾)
cdlemn8.h	⊢ 𝐻 = (LHyp‘𝐾)
cdlemn8.p	⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
cdlemn8.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
cdlemn8.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
cdlemn8.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
cdlemn8.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
cdlemn8.s	⊢ + = (+g‘𝑈)
cdlemn8.f	⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
cdlemn8.g	⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)

Assertion

Ref	Expression
cdlemn8	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑔 = (𝐺 ∘ ◡𝐹))

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

Allowed substitution hints:   𝐴(𝑔,𝑠)   𝐵(𝑔,𝑠)   𝑃(𝑔,𝑠)   + (𝑔,ℎ,𝑠)   𝑄(𝑔,𝑠)   𝑅(𝑔,𝑠)   𝑇(𝑔,𝑠)   𝑈(𝑔,ℎ,𝑠)   𝐸(𝑔,ℎ,𝑠)   𝐹(𝑔,ℎ,𝑠)   𝐺(𝑔,ℎ,𝑠)   𝐻(𝑔,𝑠)   𝐾(𝑔,𝑠)   ≤ (𝑔,𝑠)   𝑂(𝑔,ℎ,𝑠)   𝑊(𝑔,𝑠)

Proof of Theorem cdlemn8

Step	Hyp	Ref	Expression
1		coass 6098	. . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝑔) = (◡𝐹 ∘ (𝐹 ∘ 𝑔))
2		simp1 1137	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
3		cdlemn8.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
4		cdlemn8.a	. . . . . . . . . 10 ⊢ 𝐴 = (Atoms‘𝐾)
5		cdlemn8.h	. . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾)
6		cdlemn8.p	. . . . . . . . . 10 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊)
7	3, 4, 5, 6	lhpocnel2 37656	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
8	7	3ad2ant1 1134	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
9		simp2l 1200	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
10		cdlemn8.t	. . . . . . . . 9 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
11		cdlemn8.f	. . . . . . . . 9 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄)
12	3, 4, 5, 10, 11	ltrniotacl 38216	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
13	2, 8, 9, 12	syl3anc 1372	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝐹 ∈ 𝑇)
14		cdlemn8.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
15	14, 5, 10	ltrn1o 37761	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
16	2, 13, 15	syl2anc 587	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝐹:𝐵–1-1-onto→𝐵)
17		f1ococnv1 6646	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐵))
18	16, 17	syl 17	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐵))
19	18	coeq1d 5704	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ((◡𝐹 ∘ 𝐹) ∘ 𝑔) = (( I ↾ 𝐵) ∘ 𝑔))
20		simp32 1211	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑔 ∈ 𝑇)
21	14, 5, 10	ltrn1o 37761	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐵–1-1-onto→𝐵)
22	2, 20, 21	syl2anc 587	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑔:𝐵–1-1-onto→𝐵)
23		f1of 6618	. . . . 5 ⊢ (𝑔:𝐵–1-1-onto→𝐵 → 𝑔:𝐵⟶𝐵)
24		fcoi2 6553	. . . . 5 ⊢ (𝑔:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝑔) = 𝑔)
25	22, 23, 24	3syl 18	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (( I ↾ 𝐵) ∘ 𝑔) = 𝑔)
26	19, 25	eqtr2d 2774	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑔 = ((◡𝐹 ∘ 𝐹) ∘ 𝑔))
27		cdlemn8.o	. . . . . . 7 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
28		cdlemn8.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
29		cdlemn8.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
30		cdlemn8.s	. . . . . . 7 ⊢ + = (+g‘𝑈)
31		cdlemn8.g	. . . . . . 7 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅)
32	14, 3, 4, 5, 6, 27, 10, 28, 29, 30, 11, 31	cdlemn7 38840	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝐺 = ((𝑠‘𝐹) ∘ 𝑔) ∧ ( I ↾ 𝑇) = 𝑠))
33	32	simpld 498	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝐺 = ((𝑠‘𝐹) ∘ 𝑔))
34	32	simprd 499	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ( I ↾ 𝑇) = 𝑠)
35	34	fveq1d 6676	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (( I ↾ 𝑇)‘𝐹) = (𝑠‘𝐹))
36		fvresi 6945	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑇 → (( I ↾ 𝑇)‘𝐹) = 𝐹)
37	13, 36	syl 17	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (( I ↾ 𝑇)‘𝐹) = 𝐹)
38	35, 37	eqtr3d 2775	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑠‘𝐹) = 𝐹)
39	38	coeq1d 5704	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ((𝑠‘𝐹) ∘ 𝑔) = (𝐹 ∘ 𝑔))
40	33, 39	eqtrd 2773	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝐺 = (𝐹 ∘ 𝑔))
41	40	coeq2d 5705	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (◡𝐹 ∘ 𝐺) = (◡𝐹 ∘ (𝐹 ∘ 𝑔)))
42	1, 26, 41	3eqtr4a 2799	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑔 = (◡𝐹 ∘ 𝐺))
43	5, 10	ltrncnv 37783	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
44	2, 13, 43	syl2anc 587	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → ◡𝐹 ∈ 𝑇)
45		simp2r 1201	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊))
46	3, 4, 5, 10, 31	ltrniotacl 38216	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
47	2, 8, 45, 46	syl3anc 1372	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝐺 ∈ 𝑇)
48	5, 10	ltrncom 38375	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ◡𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (◡𝐹 ∘ 𝐺) = (𝐺 ∘ ◡𝐹))
49	2, 44, 47, 48	syl3anc 1372	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → (◡𝐹 ∘ 𝐺) = (𝐺 ∘ ◡𝐹))
50	42, 49	eqtrd 2773	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠‘𝐹), 𝑠⟩ + ⟨𝑔, 𝑂⟩))) → 𝑔 = (𝐺 ∘ ◡𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ⟨cop 4522   class class class wbr 5030   ↦ cmpt 5110   I cid 5428  ^◡ccnv 5524   ↾ cres 5527   ∘ ccom 5529  ⟶wf 6335  –1-1-onto→wf1o 6338  ‘cfv 6339  ℩crio 7126  (class class class)co 7170  Basecbs 16586  +_gcplusg 16668  lecple 16675  occoc 16676  Atomscatm 36900  HLchlt 36987  LHypclh 37621  LTrncltrn 37738  TEndoctendo 38389  DVecHcdvh 38715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-riotaBAD 36590

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-undef 7968  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-n0 11977  df-z 12063  df-uz 12325  df-fz 12982  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-plusg 16681  df-mulr 16682  df-sca 16684  df-vsca 16685  df-proset 17654  df-poset 17672  df-plt 17684  df-lub 17700  df-glb 17701  df-join 17702  df-meet 17703  df-p0 17765  df-p1 17766  df-lat 17772  df-clat 17834  df-oposet 36813  df-ol 36815  df-oml 36816  df-covers 36903  df-ats 36904  df-atl 36935  df-cvlat 36959  df-hlat 36988  df-llines 37135  df-lplanes 37136  df-lvols 37137  df-lines 37138  df-psubsp 37140  df-pmap 37141  df-padd 37433  df-lhyp 37625  df-laut 37626  df-ldil 37741  df-ltrn 37742  df-trl 37796  df-tendo 38392  df-edring 38394  df-dvech 38716

This theorem is referenced by:  cdlemn9  38842