Theorem diblsmopel 38467

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 2798	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
9	3, 4, 5, 6, 7, 8	diblss 38466	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
10	1, 2, 9	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐼‘𝑋) ∈ (LSubSp‘𝑈))
11		diblsmopel.y	. . . 4 ⊢ (𝜑 → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
12	3, 4, 5, 6, 7, 8	diblss 38466	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
13	1, 11, 12	syl2anc 587	. . 3 ⊢ (𝜑 → (𝐼‘𝑌) ∈ (LSubSp‘𝑈))
14		eqid 2798	. . . 4 ⊢ (+g‘𝑈) = (+g‘𝑈)
15		diblsmopel.p	. . . 4 ⊢ ✚ = (LSSum‘𝑈)
16	5, 6, 14, 8, 15	dvhopellsm 38413	. . 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 1368	. 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 2166	. . . 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 38441	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ↔ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂)))
23	1, 2, 22	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ↔ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂)))
24	3, 4, 5, 19, 20, 21, 7	dibopelval2 38441	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌) ↔ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)))
25	1, 11, 24	syl2anc 587	. . . . . . . . . . 11 ⊢ (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌) ↔ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)))
26	23, 25	anbi12d 633	. . . . . . . . . 10 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂))))
27		an4 655	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽‘𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ (𝑦 = 𝑂 ∧ 𝑤 = 𝑂)))
28		ancom 464	. . . . . . . . . . 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	syl6bb 290	. . . . . . . . 9 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼‘𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼‘𝑌)) ↔ ((𝑦 = 𝑂 ∧ 𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)))))
31	30	anbi1d 632	. . . . . . . 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 472	. . . . . . . . 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 1086	. . . . . . . . 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	syl6bb 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 1925	. . . . . 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 6659	. . . . . . . . . 10 ⊢ 𝑇 ∈ V
38	37	mptex 6963	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ∈ V
39	20, 38	eqeltri 2886	. . . . . . . 8 ⊢ 𝑂 ∈ V
40		opeq2 4765	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑂 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑂⟩)
41	40	oveq1d 7150	. . . . . . . . . 10 ⊢ (𝑦 = 𝑂 → (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩))
42	41	eqeq2d 2809	. . . . . . . . 9 ⊢ (𝑦 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩)))
43	42	anbi2d 631	. . . . . . . 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 4765	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑂 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑂⟩)
45	44	oveq2d 7151	. . . . . . . . . 10 ⊢ (𝑤 = 𝑂 → (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩))
46	45	eqeq2d 2809	. . . . . . . . 9 ⊢ (𝑤 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩)))
47	46	anbi2d 631	. . . . . . . 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 3492	. . . . . . 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 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
50	2	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))
51		simprl 770	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑥 ∈ (𝐽‘𝑋))
52	3, 4, 5, 19, 21	diael 38339	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑥 ∈ (𝐽‘𝑋)) → 𝑥 ∈ 𝑇)
53	49, 50, 51, 52	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑥 ∈ 𝑇)
54		eqid 2798	. . . . . . . . . . . . 13 ⊢ ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
55	3, 5, 19, 54, 20	tendo0cl 38086	. . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊))
58		simprr 772	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑧 ∈ (𝐽‘𝑌))
59	3, 4, 5, 19, 21	diael 38339	. . . . . . . . . . . 12 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊) ∧ 𝑧 ∈ (𝐽‘𝑌)) → 𝑧 ∈ 𝑇)
60	49, 57, 58, 59	syl3anc 1368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → 𝑧 ∈ 𝑇)
61		eqid 2798	. . . . . . . . . . . 12 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈)
62		eqid 2798	. . . . . . . . . . . 12 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
63	5, 19, 54, 6, 61, 14, 62	dvhopvadd 38389	. . . . . . . . . . 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 1376	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (⟨𝑥, 𝑂⟩(+g‘𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥 ∘ 𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
65	64	eqeq2d 2809	. . . . . . . . 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 3444	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
67		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
68	66, 67	coex 7617	. . . . . . . . . . 11 ⊢ (𝑥 ∘ 𝑧) ∈ V
69		ovex 7168	. . . . . . . . . . 11 ⊢ (𝑂(+g‘(Scalar‘𝑈))𝑂) ∈ V
70	68, 69	opth2 5337	. . . . . . . . . 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 2798	. . . . . . . . . . . . . . 15 ⊢ (+g‘𝑉) = (+g‘𝑉)
73	5, 19, 71, 72	dvavadd 38311	. . . . . . . . . . . . . 14 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) → (𝑥(+g‘𝑉)𝑧) = (𝑥 ∘ 𝑧))
74	49, 53, 60, 73	syl12anc 835	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑥(+g‘𝑉)𝑧) = (𝑥 ∘ 𝑧))
75	74	eqeq2d 2809	. . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . . 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 38380	. . . . . . . . . . . . . . 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 7152	. . . . . . . . . . . . 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 38087	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 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 2833	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂)
84	83	eqeq2d 2809	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌))) → (𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂) ↔ 𝑆 = 𝑂))
85	76, 84	anbi12d 633	. . . . . . . . . 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 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 582	. . . . . . 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 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 1922	. . . 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 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 1922	. 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 472	. . . . . 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 1850	. . . 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 1951	. . . 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 38322	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑉 ∈ LVec)
100		lveclmod 19871	. . . . . . . . 9 ⊢ (𝑉 ∈ LVec → 𝑉 ∈ LMod)
101		eqid 2798	. . . . . . . . . 10 ⊢ (LSubSp‘𝑉) = (LSubSp‘𝑉)
102	101	lsssssubg 19723	. . . . . . . . 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 38342	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐽‘𝑋) ∈ (LSubSp‘𝑉))
105	1, 2, 104	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (LSubSp‘𝑉))
106	103, 105	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑋) ∈ (SubGrp‘𝑉))
107	3, 4, 5, 71, 21, 101	dialss 38342	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐽‘𝑌) ∈ (LSubSp‘𝑉))
108	1, 11, 107	syl2anc 587	. . . . . . . 8 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (LSubSp‘𝑉))
109	103, 108	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → (𝐽‘𝑌) ∈ (SubGrp‘𝑉))
110		diblsmopel.q	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑉)
111	72, 110	lsmelval 18766	. . . . . . 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 587	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ↔ ∃𝑥 ∈ (𝐽‘𝑋)∃𝑧 ∈ (𝐽‘𝑌)𝐹 = (𝑥(+g‘𝑉)𝑧)))
113		r2ex 3262	. . . . . 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	syl6bb 290	. . . . 5 ⊢ (𝜑 → (𝐹 ∈ ((𝐽‘𝑋) ⊕ (𝐽‘𝑌)) ↔ ∃𝑥∃𝑧((𝑥 ∈ (𝐽‘𝑋) ∧ 𝑧 ∈ (𝐽‘𝑌)) ∧ 𝐹 = (𝑥(+g‘𝑉)𝑧))))
115	114	anbi1d 632	. . . 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	syl5bb 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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ⟨cop 4531   class class class wbr 5030   ↦ cmpt 5110   I cid 5424   ↾ cres 5521   ∘ ccom 5523  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  Basecbs 16475  +_gcplusg 16557  Scalarcsca 16560  lecple 16564  SubGrpcsubg 18265  LSSumclsm 18751  LModclmod 19627  LSubSpclss 19696  LVecclvec 19867  HLchlt 36646  LHypclh 37280  LTrncltrn 37397  TEndoctendo 38048  DVecAcdveca 38298  DIsoAcdia 38324  DVecHcdvh 38374  DIsoBcdib 38434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-riotaBAD 36249

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-tpos 7875  df-undef 7922  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-0g 16707  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-lsm 18753  df-cmn 18900  df-abl 18901  df-mgp 19233  df-ur 19245  df-ring 19292  df-oppr 19369  df-dvdsr 19387  df-unit 19388  df-invr 19418  df-dvr 19429  df-drng 19497  df-lmod 19629  df-lss 19697  df-lvec 19868  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647  df-llines 36794  df-lplanes 36795  df-lvols 36796  df-lines 36797  df-psubsp 36799  df-pmap 36800  df-padd 37092  df-lhyp 37284  df-laut 37285  df-ldil 37400  df-ltrn 37401  df-trl 37455  df-tgrp 38039  df-tendo 38051  df-edring 38053  df-dveca 38299  df-disoa 38325  df-dvech 38375  df-dib 38435

This theorem is referenced by:  dib2dim  38539  dih2dimbALTN  38541