Theorem diblsmopel 42031

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 2762	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
9	3, 4, 5, 6, 7, 8	diblss 42030	. . . 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 42030	. . . 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 2762	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
15		diblsmopel.p	. . . 4 ⊢ ✚ = (LSSum‘𝑈)
16	5, 6, 14, 8, 15	dvhopellsm 41977	. . 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 2199	. . . 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 42005	. . . . . . . . . . . 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 42005	. . . . . . . . . . . 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 1957	. . . . . 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 6896	. . . . . . . . . 10 ⊢ 𝑇 ∈ V
38	37	mptex 7225	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
39	20, 38	eqeltri 2858	. . . . . . . 8 ⊢ 𝑂 ∈ V
40		opeq2 4837	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑂 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑂⟩)
41	40	oveq1d 7431	. . . . . . . . . 10 ⊢ (𝑦 = 𝑂 → (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))
42	41	eqeq2d 2773	. . . . . . . . 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 4837	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑂 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑂⟩)
45	44	oveq2d 7432	. . . . . . . . . 10 ⊢ (𝑤 = 𝑂 → (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩))
46	45	eqeq2d 2773	. . . . . . . . 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 3504	. . . . . . 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 41903	. . . . . . . . . . . 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 2762	. . . . . . . . . . . . 13 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
55	3, 5, 19, 54, 20	tendo0cl 41650	. . . . . . . . . . . 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 41903	. . . . . . . . . . . 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 2762	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
62		eqid 2762	. . . . . . . . . . . 12 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
63	5, 19, 54, 6, 61, 14, 62	dvhopvadd 41953	. . . . . . . . . . 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 2773	. . . . . . . . 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 3457	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
67		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
68	66, 67	coex 7930	. . . . . . . . . . 11 ⊢ (𝑥 ∘ 𝑧) ∈ V
69		ovex 7449	. . . . . . . . . . 11 ⊢ (𝑂(+g‘(Scalar‘𝑈))𝑂) ∈ V
70	68, 69	opth2 5460	. . . . . . . . . 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 2762	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝑉) = (+g‘𝑉)
73	5, 19, 71, 72	dvavadd 41875	. . . . . . . . . . . . . 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 2773	. . . . . . . . . . . 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 2762	. . . . . . . . . . . . . . . 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 41944	. . . . . . . . . . . . . . 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 7433	. . . . . . . . . . . . 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 41651	. . . . . . . . . . . . . 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 2797	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂)
84	83	eqeq2d 2773	. . . . . . . . . . 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 1954	. . . 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 1954	. 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 1882	. . . 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 1983	. . . 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 41886	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ LVec)
100		lveclmod 21291	. . . . . . . . 9 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod)
101		eqid 2762	. . . . . . . . . 10 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
102	101	lsssssubg 21143	. . . . . . . . 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 41906	. . . . . . . . 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 3935	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (SubGrp‘𝑉))
107	3, 4, 5, 71, 21, 101	dialss 41906	. . . . . . . . 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 3935	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (SubGrp‘𝑉))
110		diblsmopel.q	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑉)
111	72, 110	lsmelval 19777	. . . . . . 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 3201	. . . . . 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 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088  Vcvv 3453   ⊆ wss 3902  ⟨cop 4593   class class class wbr 5107   ↦ cmpt 5190   I cid 5553   ↾ cres 5661   ∘ ccom 5663  ‘cfv 6537  (class class class)co 7416   ∈ cmpo 7418  Basecbs 17305  +_gcplusg 17346  Scalarcsca 17349  lecple 17353  SubGrpcsubg 19244  LSSumclsm 19762  LModclmod 21045  LSubSpclss 21116  LVecclvec 21287  HLchlt 40210  LHypclh 40844  LTrncltrn 40961  TEndoctendo 41612  DVecAcdveca 41862  DIsoAcdia 41888  DVecHcdvh 41938  DIsoBcdib 41998

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 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204  ax-riotaBAD 39813

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 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-tpos 8227  df-undef 8274  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-n0 12532  df-z 12619  df-uz 12891  df-fz 13564  df-struct 17243  df-sets 17260  df-slot 17278  df-ndx 17290  df-base 17306  df-ress 17327  df-plusg 17359  df-mulr 17360  df-sca 17362  df-vsca 17363  df-0g 17530  df-proset 18386  df-poset 18405  df-plt 18420  df-lub 18436  df-glb 18437  df-join 18438  df-meet 18439  df-p0 18515  df-p1 18516  df-lat 18524  df-clat 18591  df-mgm 18734  df-sgrp 18823  df-mnd 18839  df-grp 19061  df-minusg 19062  df-sbg 19063  df-subg 19247  df-lsm 19764  df-cmn 19910  df-abl 19911  df-mgp 20275  df-rng 20289  df-ur 20322  df-ring 20375  df-oppr 20479  df-dvdsr 20499  df-unit 20500  df-invr 20530  df-dvr 20543  df-drng 20893  df-lmod 21047  df-lss 21117  df-lvec 21288  df-oposet 40036  df-ol 40038  df-oml 40039  df-covers 40126  df-ats 40127  df-atl 40158  df-cvlat 40182  df-hlat 40211  df-llines 40358  df-lplanes 40359  df-lvols 40360  df-lines 40361  df-psubsp 40363  df-pmap 40364  df-padd 40656  df-lhyp 40848  df-laut 40849  df-ldil 40964  df-ltrn 40965  df-trl 41019  df-tgrp 41603  df-tendo 41615  df-edring 41617  df-dveca 41863  df-disoa 41889  df-dvech 41939  df-dib 41999

This theorem is used by:  dib2dim  42103  dih2dimbALTN  42105