Theorem diblsmopel 42229

Description: Membership in subspace sum for partial isomorphism B. (Contributed by NM, 21-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)

Hypotheses

Ref	Expression
diblsmopel.b	⊢ 𝐵 = (Base‘𝐾)
diblsmopel.l	⊢ ≤ = (le‘𝐾)
diblsmopel.h	⊢ 𝐻 = (LHyp‘𝐾)
diblsmopel.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
diblsmopel.o	⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
diblsmopel.v	⊢ 𝑉 = ((DVecA‘𝐾)‘𝑊)
diblsmopel.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
diblsmopel.q	⊢ ⊕ = (LSSum‘𝑉)
diblsmopel.p	⊢ + = (LSSum‘𝑈)
diblsmopel.j	⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
diblsmopel.i	⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
diblsmopel.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
diblsmopel.x	⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
diblsmopel.y	⊢ (𝜑 → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))

Assertion

Ref	Expression
diblsmopel	⊢ (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) + (𝐼‘𝑌)) ↔ (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ∧ 𝑆 = 𝑂)))

Distinct variable groups:   𝐵,𝑓   𝑓,𝐻   𝑓,𝐾   𝑇,𝑓   𝑓,𝑊

Allowed substitution hints:   𝜑(𝑓)   + (𝑓)   ⊕ (𝑓)   𝑆(𝑓)   𝑈(𝑓)   𝐹(𝑓)   𝐼(𝑓)   𝐽(𝑓)   ≤ (𝑓)   𝑂(𝑓)   𝑉(𝑓)   𝑋(𝑓)   𝑌(𝑓)

Proof of Theorem diblsmopel

Dummy variables 𝑥 𝑤 𝑦 𝑧 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		diblsmopel.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		diblsmopel.x	. . . 4 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
3		diblsmopel.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
4		diblsmopel.l	. . . . 5 ⊢ ≤ = (le‘𝐾)
5		diblsmopel.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
6		diblsmopel.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		diblsmopel.i	. . . . 5 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)
8		eqid 2760	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
9	3, 4, 5, 6, 7, 8	diblss 42228	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
10	1, 2, 9	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
11		diblsmopel.y	. . . 4 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
12	3, 4, 5, 6, 7, 8	diblss 42228	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
13	1, 11, 12	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
14		eqid 2760	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
15		diblsmopel.p	. . . 4 ⊢ + = (LSSum‘𝑈)
16	5, 6, 14, 8, 15	dvhopellsm 42175	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑋) ∈ (LSubSp‘𝑈) ∧ (𝐼‘𝑌) ∈ (LSubSp‘𝑈)) → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) + (𝐼‘𝑌)) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))))
17	1, 10, 13, 16	syl3anc 1398	. 2 ⊢ (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) + (𝐼‘𝑌)) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))))
18		excom 2198	. . . 4 ⊢ (∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑧∃𝑦∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)))
19		diblsmopel.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
20		diblsmopel.o	. . . . . . . . . . . . 13 ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))
21		diblsmopel.j	. . . . . . . . . . . . 13 ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)
22	3, 4, 5, 19, 20, 21, 7	dibopelval2 42203	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ↔ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂)))
23	1, 2, 22	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ↔ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂)))
24	3, 4, 5, 19, 20, 21, 7	dibopelval2 42203	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌) ↔ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)))
25	1, 11, 24	syl2anc 596	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌) ↔ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)))
26	23, 25	anbi12d 644	. . . . . . . . . 10 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂))))
27		an4 669	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝑦 = 𝑂 ∧ 𝑤 = 𝑂)))
28		ancom 466	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝑦 = 𝑂 ∧ 𝑤 = 𝑂)) ↔ ((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))))
29	27, 28	bitri 278	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))))
30	26, 29	bitrdi 290	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ↔ ((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)))))
31	30	anbi1d 643	. . . . . . . 8 ⊢ (𝜑 → (((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ (((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))))
32		anass 474	. . . . . . . . 9 ⊢ ((((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ ((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))))
33		df-3an 1105	. . . . . . . . 9 ⊢ ((𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))) ↔ ((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))))
34	32, 33	bitr4i 281	. . . . . . . 8 ⊢ ((((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ (𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))))
35	31, 34	bitrdi 290	. . . . . . 7 ⊢ (𝜑 → (((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ (𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)))))
36	35	2exbidv 1956	. . . . . 6 ⊢ (𝜑 → (∃𝑦∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑦∃𝑤(𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)))))
37	19	fvexi 6899	. . . . . . . . . 10 ⊢ 𝑇 ∈ V
38	37	mptex 7229	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
39	20, 38	eqeltri 2856	. . . . . . . 8 ⊢ 𝑂 ∈ V
40		opeq2 4833	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑂 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑂⟩)
41	40	oveq1d 7435	. . . . . . . . . 10 ⊢ (𝑦 = 𝑂 → (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))
42	41	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑦 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)))
43	42	anbi2d 642	. . . . . . . 8 ⊢ (𝑦 = 𝑂 → (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))))
44		opeq2 4833	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑂 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑂⟩)
45	44	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑤 = 𝑂 → (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩))
46	45	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑤 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩)))
47	46	anbi2d 642	. . . . . . . 8 ⊢ (𝑤 = 𝑂 → (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩))))
48	39, 39, 43, 47	ceqsex2v 3501	. . . . . . 7 ⊢ (∃𝑦∃𝑤(𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩)))
49	1	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
50	2	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
51		simprl 783	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑥 ∈ (𝐽‘𝑋))
52	3, 4, 5, 19, 21	diael 42101	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑥 ∈ (𝐽‘𝑋)) → 𝑥 ∈ 𝑇)
53	49, 50, 51, 52	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑥 ∈ 𝑇)
54		eqid 2760	. . . . . . . . . . . . 13 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
55	3, 5, 19, 54, 20	tendo0cl 41848	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
56	49, 55	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
57	11	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
58		simprr 785	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑧 ∈ (𝐽‘𝑌))
59	3, 4, 5, 19, 21	diael 42101	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊) ∧ 𝑧 ∈ (𝐽‘𝑌)) → 𝑧 ∈ 𝑇)
60	49, 57, 58, 59	syl3anc 1398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑧 ∈ 𝑇)
61		eqid 2760	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
62		eqid 2760	. . . . . . . . . . . 12 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
63	5, 19, 54, 6, 61, 14, 62	dvhopvadd 42151	. . . . . . . . . . 11 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) ∧ (𝑧 ∈ 𝑇 ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥 ∘ 𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
64	49, 53, 56, 60, 56, 63	syl122anc 1406	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥 ∘ 𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
65	64	eqeq2d 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩) ↔ ⟨𝐹, 𝑆⟩ = ⟨(𝑥 ∘ 𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩))
66		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
67		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
68	66, 67	coex 7942	. . . . . . . . . . 11 ⊢ (𝑥 ∘ 𝑧) ∈ V
69		ovex 7453	. . . . . . . . . . 11 ⊢ (𝑂(+g‘(Scalar‘𝑈))𝑂) ∈ V
70	68, 69	opth2 5448	. . . . . . . . . 10 ⊢ (⟨𝐹, 𝑆⟩ = ⟨(𝑥 ∘ 𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩ ↔ (𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂)))
71		diblsmopel.v	. . . . . . . . . . . . . . 15 ⊢ 𝑉 = ((DVecA‘𝐾)‘𝑊)
72		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝑉) = (+g‘𝑉)
73	5, 19, 71, 72	dvavadd 42073	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (𝑥(+g‘𝑉)𝑧) = (𝑥 ∘ 𝑧))
74	49, 53, 60, 73	syl12anc 850	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑥(+g‘𝑉)𝑧) = (𝑥 ∘ 𝑧))
75	74	eqeq2d 2771	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝐹 = (𝑥(+g‘𝑉)𝑧) ↔ 𝐹 = (𝑥 ∘ 𝑧)))
76	75	bicomd 226	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝐹 = (𝑥 ∘ 𝑧) ↔ 𝐹 = (𝑥(+g‘𝑉)𝑧)))
77		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))
78	5, 19, 54, 6, 61, 77, 62	dvhfplusr 42142	. . . . . . . . . . . . . . 15 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))))
79	49, 78	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))))
80	79	oveqd 7437	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))𝑂))
81	3, 5, 19, 54, 20, 77	tendo0pl 41849	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))𝑂) = 𝑂)
82	49, 56, 81	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))𝑂) = 𝑂)
83	80, 82	eqtrd 2795	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂)
84	83	eqeq2d 2771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂) ↔ 𝑆 = 𝑂))
85	76, 84	anbi12d 644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → ((𝐹 = (𝑥 ∘ 𝑧) ∧ 𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂)) ↔ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂)))
86	70, 85	bitrid 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (⟨𝐹, 𝑆⟩ = ⟨(𝑥 ∘ 𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩ ↔ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂)))
87	65, 86	bitrd 282	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩) ↔ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂)))
88	87	pm5.32da 590	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂))))
89	48, 88	bitrid 286	. . . . . 6 ⊢ (𝜑 → (∃𝑦∃𝑤(𝑦 = 𝑂 ∧ 𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂))))
90	36, 89	bitrd 282	. . . . 5 ⊢ (𝜑 → (∃𝑦∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂))))
91	90	exbidv 1953	. . . 4 ⊢ (𝜑 → (∃𝑧∃𝑦∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂))))
92	18, 91	bitrid 286	. . 3 ⊢ (𝜑 → (∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂))))
93	92	exbidv 1953	. 2 ⊢ (𝜑 → (∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂))))
94		anass 474	. . . . . 6 ⊢ ((((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂)))
95	94	bicomi 227	. . . . 5 ⊢ (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧)) ∧ 𝑆 = 𝑂))
96	95	2exbii 1881	. . . 4 ⊢ (∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ ∃𝑥∃𝑧(((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧)) ∧ 𝑆 = 𝑂))
97		19.41vv 1982	. . . 4 ⊢ (∃𝑥∃𝑧(((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ (∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧)) ∧ 𝑆 = 𝑂))
98	96, 97	bitri 278	. . 3 ⊢ (∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧)) ∧ 𝑆 = 𝑂))
99	5, 71	dvalvec 42084	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ LVec)
100		lveclmod 21381	. . . . . . . . 9 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod)
101		eqid 2760	. . . . . . . . . 10 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
102	101	lsssssubg 21233	. . . . . . . . 9 ⊢ (𝑉 ∈ LMod → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
103	1, 99, 100, 102	4syl 20	. . . . . . . 8 ⊢ (𝜑 → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
104	3, 4, 5, 71, 21, 101	dialss 42104	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐽‘𝑋) ∈ (LSubSp‘𝑉))
105	1, 2, 104	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LSubSp‘𝑉))
106	103, 105	sseldd 3909	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (SubGrp‘𝑉))
107	3, 4, 5, 71, 21, 101	dialss 42104	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐽‘𝑌) ∈ (LSubSp‘𝑉))
108	1, 11, 107	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LSubSp‘𝑉))
109	103, 108	sseldd 3909	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (SubGrp‘𝑉))
110		diblsmopel.q	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑉)
111	72, 110	lsmelval 19863	. . . . . . 7 ⊢ (((𝐽‘𝑋) ∈ (SubGrp‘𝑉) ∧ (𝐽‘𝑌) ∈ (SubGrp‘𝑉)) → (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ↔ ∃𝑥 ∈ (𝐽‘𝑋)∃𝑧 ∈ (𝐽‘𝑌)𝐹 = (𝑥(+g‘𝑉)𝑧)))
112	106, 109, 111	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ↔ ∃𝑥 ∈ (𝐽‘𝑋)∃𝑧 ∈ (𝐽‘𝑌)𝐹 = (𝑥(+g‘𝑉)𝑧)))
113		r2ex 3199	. . . . . 6 ⊢ (∃𝑥 ∈ (𝐽‘𝑋)∃𝑧 ∈ (𝐽‘𝑌)𝐹 = (𝑥(+g‘𝑉)𝑧) ↔ ∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧)))
114	112, 113	bitrdi 290	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ↔ ∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧))))
115	114	anbi1d 643	. . . 4 ⊢ (𝜑 → ((𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ∧ 𝑆 = 𝑂) ↔ (∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧)) ∧ 𝑆 = 𝑂)))
116	115	bicomd 226	. . 3 ⊢ (𝜑 → ((∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ∧ 𝑆 = 𝑂)))
117	98, 116	bitrid 286	. 2 ⊢ (𝜑 → (∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝐹 = (𝑥(+g‘𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ∧ 𝑆 = 𝑂)))
118	17, 93, 117	3bitrd 308	1 ⊢ (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) + (𝐼‘𝑌)) ↔ (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ∧ 𝑆 = 𝑂)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ⟨cop 4589   class class class wbr 5102   ↦ cmpt 5185   I cid 5544   ↾ cres 5652   ∘ ccom 5654  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  Basecbs 17387  +_gcplusg 17428  Scalarcsca 17431  lecple 17435  SubGrpcsubg 19330  LSSumclsm 19848  LModclmod 21135  LSubSpclss 21206  LVecclvec 21377  HLchlt 40408  LHypclh 41042  LTrncltrn 41159  TEndoctendo 41810  DVecAcdveca 42060  DIsoAcdia 42086  DVecHcdvh 42136  DIsoBcdib 42196

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-riotaBAD 40011

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-tpos 8243  df-undef 8290  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-n0 12607  df-z 12694  df-uz 12966  df-fz 13640  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-mulr 17442  df-sca 17444  df-vsca 17445  df-0g 17612  df-proset 18468  df-poset 18487  df-plt 18502  df-lub 18518  df-glb 18519  df-join 18520  df-meet 18521  df-p0 18597  df-p1 18598  df-lat 18606  df-clat 18673  df-mgm 18816  df-sgrp 18908  df-mnd 18924  df-grp 19147  df-minusg 19148  df-sbg 19149  df-subg 19333  df-lsm 19850  df-cmn 19996  df-abl 19997  df-mgp 20361  df-rng 20375  df-ur 20408  df-ring 20461  df-oppr 20567  df-dvdsr 20587  df-unit 20588  df-invr 20618  df-dvr 20631  df-drng 20982  df-lmod 21137  df-lss 21207  df-lvec 21378  df-oposet 40234  df-ol 40236  df-oml 40237  df-covers 40324  df-ats 40325  df-atl 40356  df-cvlat 40380  df-hlat 40409  df-llines 40556  df-lplanes 40557  df-lvols 40558  df-lines 40559  df-psubsp 40561  df-pmap 40562  df-padd 40854  df-lhyp 41046  df-laut 41047  df-ldil 41162  df-ltrn 41163  df-trl 41217  df-tgrp 41801  df-tendo 41813  df-edring 41815  df-dveca 42061  df-disoa 42087  df-dvech 42137  df-dib 42197

This theorem is used by:  dib2dim  42301  dih2dimbALTN  42303