Theorem cdlemn8 42085

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 6266	. . 3 ⊢ ((◡𝐹 ∘ 𝐹) ∘ 𝑔) = (◡𝐹 ∘ (𝐹 ∘ 𝑔))
2		simp1 1154	. . . . . . 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 40900	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
8	7	3ad2ant1 1151	. . . . . . . 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 1218	. . . . . . . 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 41460	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇)
13	2, 8, 9, 12	syl3anc 1398	. . . . . . 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 41005	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
16	2, 13, 15	syl2anc 596	. . . . . 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 6851	. . . . . 6 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐵))
18	16, 17	syl 18	. . . . 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 5845	. . . 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 1229	. . . . . 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 41005	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐵–1-1-onto→𝐵)
22	2, 20, 21	syl2anc 596	. . . . 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 6821	. . . . 5 ⊢ (𝑔:𝐵–1-1-onto→𝐵 → 𝑔:𝐵⟶𝐵)
24		fcoi2 6754	. . . . 5 ⊢ (𝑔:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝑔) = 𝑔)
25	22, 23, 24	3syl 19	. . . 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 2798	. . 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 42084	. . . . . 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 500	. . . . 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 501	. . . . . . . 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 6884	. . . . . . 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 7175	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑇 → (( I ↾ 𝑇)‘𝐹) = 𝐹)
37	13, 36	syl 18	. . . . . . 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 2799	. . . . . 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 5845	. . . . 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 2797	. . . 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 5846	. . 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 2823	. 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 41027	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ◡𝐹 ∈ 𝑇)
44	2, 13, 43	syl2anc 596	. . 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 1219	. . . 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 41460	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐺 ∈ 𝑇)
47	2, 8, 45, 46	syl3anc 1398	. . 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 41619	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ◡𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (◡𝐹 ∘ 𝐺) = (𝐺 ∘ ◡𝐹))
49	2, 44, 47, 48	syl3anc 1398	. 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 2797	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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ⟨cop 4593   class class class wbr 5107   ↦ cmpt 5190   I cid 5553  ^◡ccnv 5658   ↾ cres 5661   ∘ ccom 5663  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537  ℩crio 7373  (class class class)co 7417  Basecbs 17307  +_gcplusg 17348  lecple 17355  occoc 17356  Atomscatm 40144  HLchlt 40231  LHypclh 40865  LTrncltrn 40982  TEndoctendo 41633  DVecHcdvh 41959

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205  ax-riotaBAD 39834

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-iun 4956  df-iin 4957  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-undef 8275  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-struct 17245  df-slot 17280  df-ndx 17292  df-base 17308  df-plusg 17361  df-mulr 17362  df-sca 17364  df-vsca 17365  df-proset 18388  df-poset 18407  df-plt 18422  df-lub 18438  df-glb 18439  df-join 18440  df-meet 18441  df-p0 18517  df-p1 18518  df-lat 18526  df-clat 18593  df-oposet 40057  df-ol 40059  df-oml 40060  df-covers 40147  df-ats 40148  df-atl 40179  df-cvlat 40203  df-hlat 40232  df-llines 40379  df-lplanes 40380  df-lvols 40381  df-lines 40382  df-psubsp 40384  df-pmap 40385  df-padd 40677  df-lhyp 40869  df-laut 40870  df-ldil 40985  df-ltrn 40986  df-trl 41040  df-tendo 41636  df-edring 41638  df-dvech 41960

This theorem is used by:  cdlemn9  42086