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Theorem diblsmopel 38193
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 = (𝑓𝑇 ↦ ( 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 ∧ 𝑊𝐻))
diblsmopel.x (𝜑 → (𝑋𝐵𝑋 𝑊))
diblsmopel.y (𝜑 → (𝑌𝐵𝑌 𝑊))
Assertion
Ref Expression
diblsmopel (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 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.
StepHypRef Expression
1 diblsmopel.k . . 3 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 diblsmopel.x . . . 4 (𝜑 → (𝑋𝐵𝑋 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 2826 . . . . 5 (LSubSp‘𝑈) = (LSubSp‘𝑈)
93, 4, 5, 6, 7, 8diblss 38192 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ (LSubSp‘𝑈))
101, 2, 9syl2anc 584 . . 3 (𝜑 → (𝐼𝑋) ∈ (LSubSp‘𝑈))
11 diblsmopel.y . . . 4 (𝜑 → (𝑌𝐵𝑌 𝑊))
123, 4, 5, 6, 7, 8diblss 38192 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼𝑌) ∈ (LSubSp‘𝑈))
131, 11, 12syl2anc 584 . . 3 (𝜑 → (𝐼𝑌) ∈ (LSubSp‘𝑈))
14 eqid 2826 . . . 4 (+g𝑈) = (+g𝑈)
15 diblsmopel.p . . . 4 = (LSSum‘𝑈)
165, 6, 14, 8, 15dvhopellsm 38139 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐼𝑋) ∈ (LSubSp‘𝑈) ∧ (𝐼𝑌) ∈ (LSubSp‘𝑈)) → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ ∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
171, 10, 13, 16syl3anc 1365 . 2 (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ ∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
18 excom 2161 . . . 4 (∃𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑧𝑦𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)))
19 diblsmopel.t . . . . . . . . . . . . 13 𝑇 = ((LTrn‘𝐾)‘𝑊)
20 diblsmopel.o . . . . . . . . . . . . 13 𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))
21 diblsmopel.j . . . . . . . . . . . . 13 𝐽 = ((DIsoA‘𝐾)‘𝑊)
223, 4, 5, 19, 20, 21, 7dibopelval2 38167 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ↔ (𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂)))
231, 2, 22syl2anc 584 . . . . . . . . . . 11 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ↔ (𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂)))
243, 4, 5, 19, 20, 21, 7dibopelval2 38167 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌) ↔ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)))
251, 11, 24syl2anc 584 . . . . . . . . . . 11 (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌) ↔ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)))
2623, 25anbi12d 630 . . . . . . . . . 10 (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂))))
27 an4 652 . . . . . . . . . . 11 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝑦 = 𝑂𝑤 = 𝑂)))
28 ancom 461 . . . . . . . . . . 11 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝑦 = 𝑂𝑤 = 𝑂)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))))
2927, 28bitri 276 . . . . . . . . . 10 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))))
3026, 29syl6bb 288 . . . . . . . . 9 (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)))))
3130anbi1d 629 . . . . . . . 8 (𝜑 → (((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ (((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
32 anass 469 . . . . . . . . 9 ((((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
33 df-3an 1083 . . . . . . . . 9 ((𝑦 = 𝑂𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
3432, 33bitr4i 279 . . . . . . . 8 ((((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ (𝑦 = 𝑂𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
3531, 34syl6bb 288 . . . . . . 7 (𝜑 → (((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ (𝑦 = 𝑂𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)))))
36352exbidv 1918 . . . . . 6 (𝜑 → (∃𝑦𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑦𝑤(𝑦 = 𝑂𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)))))
3719fvexi 6683 . . . . . . . . . 10 𝑇 ∈ V
3837mptex 6983 . . . . . . . . 9 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
3920, 38eqeltri 2914 . . . . . . . 8 𝑂 ∈ V
40 opeq2 4803 . . . . . . . . . . 11 (𝑦 = 𝑂 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑂⟩)
4140oveq1d 7165 . . . . . . . . . 10 (𝑦 = 𝑂 → (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩))
4241eqeq2d 2837 . . . . . . . . 9 (𝑦 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩)))
4342anbi2d 628 . . . . . . . 8 (𝑦 = 𝑂 → (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
44 opeq2 4803 . . . . . . . . . . 11 (𝑤 = 𝑂 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑂⟩)
4544oveq2d 7166 . . . . . . . . . 10 (𝑤 = 𝑂 → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩))
4645eqeq2d 2837 . . . . . . . . 9 (𝑤 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩)))
4746anbi2d 628 . . . . . . . 8 (𝑤 = 𝑂 → (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩))))
4839, 39, 43, 47ceqsex2v 3550 . . . . . . 7 (∃𝑦𝑤(𝑦 = 𝑂𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩)))
491adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
502adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑋𝐵𝑋 𝑊))
51 simprl 767 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥 ∈ (𝐽𝑋))
523, 4, 5, 19, 21diael 38065 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑥 ∈ (𝐽𝑋)) → 𝑥𝑇)
5349, 50, 51, 52syl3anc 1365 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥𝑇)
54 eqid 2826 . . . . . . . . . . . . 13 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
553, 5, 19, 54, 20tendo0cl 37812 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
5649, 55syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
5711adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑌𝐵𝑌 𝑊))
58 simprr 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧 ∈ (𝐽𝑌))
593, 4, 5, 19, 21diael 38065 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊) ∧ 𝑧 ∈ (𝐽𝑌)) → 𝑧𝑇)
6049, 57, 58, 59syl3anc 1365 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧𝑇)
61 eqid 2826 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
62 eqid 2826 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
635, 19, 54, 6, 61, 14, 62dvhopvadd 38115 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) ∧ (𝑧𝑇𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
6449, 53, 56, 60, 56, 63syl122anc 1373 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
6564eqeq2d 2837 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) ↔ ⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩))
66 vex 3503 . . . . . . . . . . . 12 𝑥 ∈ V
67 vex 3503 . . . . . . . . . . . 12 𝑧 ∈ V
6866, 67coex 7628 . . . . . . . . . . 11 (𝑥𝑧) ∈ V
69 ovex 7183 . . . . . . . . . . 11 (𝑂(+g‘(Scalar‘𝑈))𝑂) ∈ V
7068, 69opth2 5369 . . . . . . . . . 10 (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩ ↔ (𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂)))
71 diblsmopel.v . . . . . . . . . . . . . . 15 𝑉 = ((DVecA‘𝐾)‘𝑊)
72 eqid 2826 . . . . . . . . . . . . . . 15 (+g𝑉) = (+g𝑉)
735, 19, 71, 72dvavadd 38037 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥(+g𝑉)𝑧) = (𝑥𝑧))
7449, 53, 60, 73syl12anc 834 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑥(+g𝑉)𝑧) = (𝑥𝑧))
7574eqeq2d 2837 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐹 = (𝑥(+g𝑉)𝑧) ↔ 𝐹 = (𝑥𝑧)))
7675bicomd 224 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐹 = (𝑥𝑧) ↔ 𝐹 = (𝑥(+g𝑉)𝑧)))
77 eqid 2826 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
785, 19, 54, 6, 61, 77, 62dvhfplusr 38106 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))))
7949, 78syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))))
8079oveqd 7167 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))𝑂))
813, 5, 19, 54, 20, 77tendo0pl 37813 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))𝑂) = 𝑂)
8249, 56, 81syl2anc 584 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))𝑂) = 𝑂)
8380, 82eqtrd 2861 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂)
8483eqeq2d 2837 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂) ↔ 𝑆 = 𝑂))
8576, 84anbi12d 630 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → ((𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂)) ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8670, 85syl5bb 284 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩ ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8765, 86bitrd 280 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8887pm5.32da 579 . . . . . . 7 (𝜑 → (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂))))
8948, 88syl5bb 284 . . . . . 6 (𝜑 → (∃𝑦𝑤(𝑦 = 𝑂𝑤 = 𝑂 ∧ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂))))
9036, 89bitrd 280 . . . . 5 (𝜑 → (∃𝑦𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂))))
9190exbidv 1915 . . . 4 (𝜑 → (∃𝑧𝑦𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂))))
9218, 91syl5bb 284 . . 3 (𝜑 → (∃𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂))))
9392exbidv 1915 . 2 (𝜑 → (∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩)) ↔ ∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂))))
94 anass 469 . . . . . 6 ((((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
9594bicomi 225 . . . . 5 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
96952exbii 1842 . . . 4 (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ ∃𝑥𝑧(((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
97 19.41vv 1944 . . . 4 (∃𝑥𝑧(((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
9896, 97bitri 276 . . 3 (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
995, 71dvalvec 38048 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑉 ∈ LVec)
100 lveclmod 19814 . . . . . . . . 9 (𝑉 ∈ LVec → 𝑉 ∈ LMod)
101 eqid 2826 . . . . . . . . . 10 (LSubSp‘𝑉) = (LSubSp‘𝑉)
102101lsssssubg 19666 . . . . . . . . 9 (𝑉 ∈ LMod → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1031, 99, 100, 1024syl 19 . . . . . . . 8 (𝜑 → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1043, 4, 5, 71, 21, 101dialss 38068 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐽𝑋) ∈ (LSubSp‘𝑉))
1051, 2, 104syl2anc 584 . . . . . . . 8 (𝜑 → (𝐽𝑋) ∈ (LSubSp‘𝑉))
106103, 105sseldd 3972 . . . . . . 7 (𝜑 → (𝐽𝑋) ∈ (SubGrp‘𝑉))
1073, 4, 5, 71, 21, 101dialss 38068 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐽𝑌) ∈ (LSubSp‘𝑉))
1081, 11, 107syl2anc 584 . . . . . . . 8 (𝜑 → (𝐽𝑌) ∈ (LSubSp‘𝑉))
109103, 108sseldd 3972 . . . . . . 7 (𝜑 → (𝐽𝑌) ∈ (SubGrp‘𝑉))
110 diblsmopel.q . . . . . . . 8 = (LSSum‘𝑉)
11172, 110lsmelval 18710 . . . . . . 7 (((𝐽𝑋) ∈ (SubGrp‘𝑉) ∧ (𝐽𝑌) ∈ (SubGrp‘𝑉)) → (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ↔ ∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧)))
112106, 109, 111syl2anc 584 . . . . . 6 (𝜑 → (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ↔ ∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧)))
113 r2ex 3308 . . . . . 6 (∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧) ↔ ∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)))
114112, 113syl6bb 288 . . . . 5 (𝜑 → (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ↔ ∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧))))
115114anbi1d 629 . . . 4 (𝜑 → ((𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂) ↔ (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂)))
116115bicomd 224 . . 3 (𝜑 → ((∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
11798, 116syl5bb 284 . 2 (𝜑 → (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
11817, 93, 1173bitrd 306 1 (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  wrex 3144  Vcvv 3500  wss 3940  cop 4570   class class class wbr 5063  cmpt 5143   I cid 5458  cres 5556  ccom 5558  cfv 6354  (class class class)co 7150  cmpo 7152  Basecbs 16478  +gcplusg 16560  Scalarcsca 16563  lecple 16567  SubGrpcsubg 18218  LSSumclsm 18695  LModclmod 19570  LSubSpclss 19639  LVecclvec 19810  HLchlt 36372  LHypclh 37006  LTrncltrn 37123  TEndoctendo 37774  DVecAcdveca 38024  DIsoAcdia 38050  DVecHcdvh 38100  DIsoBcdib 38160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608  ax-riotaBAD 35975
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7574  df-1st 7685  df-2nd 7686  df-tpos 7888  df-undef 7935  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8284  df-map 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12888  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-sca 16576  df-vsca 16577  df-0g 16710  df-proset 17533  df-poset 17551  df-plt 17563  df-lub 17579  df-glb 17580  df-join 17581  df-meet 17582  df-p0 17644  df-p1 17645  df-lat 17651  df-clat 17713  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-grp 18051  df-minusg 18052  df-sbg 18053  df-subg 18221  df-lsm 18697  df-cmn 18844  df-abl 18845  df-mgp 19176  df-ur 19188  df-ring 19235  df-oppr 19309  df-dvdsr 19327  df-unit 19328  df-invr 19358  df-dvr 19369  df-drng 19440  df-lmod 19572  df-lss 19640  df-lvec 19811  df-oposet 36198  df-ol 36200  df-oml 36201  df-covers 36288  df-ats 36289  df-atl 36320  df-cvlat 36344  df-hlat 36373  df-llines 36520  df-lplanes 36521  df-lvols 36522  df-lines 36523  df-psubsp 36525  df-pmap 36526  df-padd 36818  df-lhyp 37010  df-laut 37011  df-ldil 37126  df-ltrn 37127  df-trl 37181  df-tgrp 37765  df-tendo 37777  df-edring 37779  df-dveca 38025  df-disoa 38051  df-dvech 38101  df-dib 38161
This theorem is referenced by:  dib2dim  38265  dih2dimbALTN  38267
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