Theorem diblsmopel 39112

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 2738	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
9	3, 4, 5, 6, 7, 8	diblss 39111	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
10	1, 2, 9	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
11		diblsmopel.y	. . . 4 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
12	3, 4, 5, 6, 7, 8	diblss 39111	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
13	1, 11, 12	syl2anc 583	. . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
14		eqid 2738	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
15		diblsmopel.p	. . . 4 ⊢ ✚ = (LSSum‘𝑈)
16	5, 6, 14, 8, 15	dvhopellsm 39058	. . 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 1369	. 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 2164	. . . 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 39086	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ↔ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂)))
23	1, 2, 22	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ↔ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂)))
24	3, 4, 5, 19, 20, 21, 7	dibopelval2 39086	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌) ↔ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)))
25	1, 11, 24	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌) ↔ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)))
26	23, 25	anbi12d 630	. . . . . . . . . 10 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂))))
27		an4 652	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝑦 = 𝑂 ∧ 𝑤 = 𝑂)))
28		ancom 460	. . . . . . . . . . 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 274	. . . . . . . . . 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 286	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ↔ ((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)))))
31	30	anbi1d 629	. . . . . . . 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 468	. . . . . . . . 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 1087	. . . . . . . . 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 277	. . . . . . . 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 286	. . . . . . 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 1928	. . . . . 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 6770	. . . . . . . . . 10 ⊢ 𝑇 ∈ V
38	37	mptex 7081	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
39	20, 38	eqeltri 2835	. . . . . . . 8 ⊢ 𝑂 ∈ V
40		opeq2 4802	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑂 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑂⟩)
41	40	oveq1d 7270	. . . . . . . . . 10 ⊢ (𝑦 = 𝑂 → (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))
42	41	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑦 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)))
43	42	anbi2d 628	. . . . . . . 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 4802	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑂 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑂⟩)
45	44	oveq2d 7271	. . . . . . . . . 10 ⊢ (𝑤 = 𝑂 → (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩))
46	45	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑤 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩)))
47	46	anbi2d 628	. . . . . . . 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 3473	. . . . . . 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 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
50	2	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
51		simprl 767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑥 ∈ (𝐽‘𝑋))
52	3, 4, 5, 19, 21	diael 38984	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑥 ∈ (𝐽‘𝑋)) → 𝑥 ∈ 𝑇)
53	49, 50, 51, 52	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑥 ∈ 𝑇)
54		eqid 2738	. . . . . . . . . . . . 13 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
55	3, 5, 19, 54, 20	tendo0cl 38731	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
56	49, 55	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
57	11	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
58		simprr 769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑧 ∈ (𝐽‘𝑌))
59	3, 4, 5, 19, 21	diael 38984	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊) ∧ 𝑧 ∈ (𝐽‘𝑌)) → 𝑧 ∈ 𝑇)
60	49, 57, 58, 59	syl3anc 1369	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑧 ∈ 𝑇)
61		eqid 2738	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
62		eqid 2738	. . . . . . . . . . . 12 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
63	5, 19, 54, 6, 61, 14, 62	dvhopvadd 39034	. . . . . . . . . . 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 1377	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥 ∘ 𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
65	64	eqeq2d 2749	. . . . . . . . 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 3426	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
67		vex 3426	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
68	66, 67	coex 7751	. . . . . . . . . . 11 ⊢ (𝑥 ∘ 𝑧) ∈ V
69		ovex 7288	. . . . . . . . . . 11 ⊢ (𝑂(+g‘(Scalar‘𝑈))𝑂) ∈ V
70	68, 69	opth2 5389	. . . . . . . . . 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 2738	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝑉) = (+g‘𝑉)
73	5, 19, 71, 72	dvavadd 38956	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (𝑥(+g‘𝑉)𝑧) = (𝑥 ∘ 𝑧))
74	49, 53, 60, 73	syl12anc 833	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑥(+g‘𝑉)𝑧) = (𝑥 ∘ 𝑧))
75	74	eqeq2d 2749	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝐹 = (𝑥(+g‘𝑉)𝑧) ↔ 𝐹 = (𝑥 ∘ 𝑧)))
76	75	bicomd 222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝐹 = (𝑥 ∘ 𝑧) ↔ 𝐹 = (𝑥(+g‘𝑉)𝑧)))
77		eqid 2738	. . . . . . . . . . . . . . . 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 39025	. . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . 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 7272	. . . . . . . . . . . . 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 38732	. . . . . . . . . . . . . 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 583	. . . . . . . . . . . . 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 2778	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂)
84	83	eqeq2d 2749	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂) ↔ 𝑆 = 𝑂))
85	76, 84	anbi12d 630	. . . . . . . . . 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	syl5bb 282	. . . . . . . . 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 278	. . . . . . . 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 578	. . . . . . 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	syl5bb 282	. . . . . 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 278	. . . . 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 1925	. . . 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	syl5bb 282	. . 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 1925	. 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 468	. . . . . 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 223	. . . . 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 1852	. . . 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 1955	. . . 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 274	. . 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 38967	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ LVec)
100		lveclmod 20283	. . . . . . . . 9 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod)
101		eqid 2738	. . . . . . . . . 10 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
102	101	lsssssubg 20135	. . . . . . . . 9 ⊢ (𝑉 ∈ LMod → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
103	1, 99, 100, 102	4syl 19	. . . . . . . 8 ⊢ (𝜑 → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
104	3, 4, 5, 71, 21, 101	dialss 38987	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐽‘𝑋) ∈ (LSubSp‘𝑉))
105	1, 2, 104	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LSubSp‘𝑉))
106	103, 105	sseldd 3918	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (SubGrp‘𝑉))
107	3, 4, 5, 71, 21, 101	dialss 38987	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐽‘𝑌) ∈ (LSubSp‘𝑉))
108	1, 11, 107	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LSubSp‘𝑉))
109	103, 108	sseldd 3918	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (SubGrp‘𝑉))
110		diblsmopel.q	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑉)
111	72, 110	lsmelval 19169	. . . . . . 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 583	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ↔ ∃𝑥 ∈ (𝐽‘𝑋)∃𝑧 ∈ (𝐽‘𝑌)𝐹 = (𝑥(+g‘𝑉)𝑧)))
113		r2ex 3231	. . . . . 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 286	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ↔ ∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧))))
115	114	anbi1d 629	. . . 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 222	. . 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	syl5bb 282	. 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 304	1 ⊢ (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼‘𝑋) ✚ (𝐼‘𝑌)) ↔ (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ∧ 𝑆 = 𝑂)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ⟨cop 4564   class class class wbr 5070   ↦ cmpt 5153   I cid 5479   ↾ cres 5582   ∘ ccom 5584  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257  Basecbs 16840  +_gcplusg 16888  Scalarcsca 16891  lecple 16895  SubGrpcsubg 18664  LSSumclsm 19154  LModclmod 20038  LSubSpclss 20108  LVecclvec 20279  HLchlt 37291  LHypclh 37925  LTrncltrn 38042  TEndoctendo 38693  DVecAcdveca 38943  DIsoAcdia 38969  DVecHcdvh 39019  DIsoBcdib 39079

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-riotaBAD 36894

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-tpos 8013  df-undef 8060  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-sca 16904  df-vsca 16905  df-0g 17069  df-proset 17928  df-poset 17946  df-plt 17963  df-lub 17979  df-glb 17980  df-join 17981  df-meet 17982  df-p0 18058  df-p1 18059  df-lat 18065  df-clat 18132  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-sbg 18497  df-subg 18667  df-lsm 19156  df-cmn 19303  df-abl 19304  df-mgp 19636  df-ur 19653  df-ring 19700  df-oppr 19777  df-dvdsr 19798  df-unit 19799  df-invr 19829  df-dvr 19840  df-drng 19908  df-lmod 20040  df-lss 20109  df-lvec 20280  df-oposet 37117  df-ol 37119  df-oml 37120  df-covers 37207  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292  df-llines 37439  df-lplanes 37440  df-lvols 37441  df-lines 37442  df-psubsp 37444  df-pmap 37445  df-padd 37737  df-lhyp 37929  df-laut 37930  df-ldil 38045  df-ltrn 38046  df-trl 38100  df-tgrp 38684  df-tendo 38696  df-edring 38698  df-dveca 38944  df-disoa 38970  df-dvech 39020  df-dib 39080

This theorem is referenced by:  dib2dim  39184  dih2dimbALTN  39186