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Theorem diblsmopel 39185
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 2738 . . . . 5 (LSubSp‘𝑈) = (LSubSp‘𝑈)
93, 4, 5, 6, 7, 8diblss 39184 . . . 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 39184 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼𝑌) ∈ (LSubSp‘𝑈))
131, 11, 12syl2anc 584 . . 3 (𝜑 → (𝐼𝑌) ∈ (LSubSp‘𝑈))
14 eqid 2738 . . . 4 (+g𝑈) = (+g𝑈)
15 diblsmopel.p . . . 4 = (LSSum‘𝑈)
165, 6, 14, 8, 15dvhopellsm 39131 . . 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 1370 . 2 (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ ∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
18 excom 2162 . . . 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 39159 . . . . . . . . . . . 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 39159 . . . . . . . . . . . 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 631 . . . . . . . . . 10 (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂))))
27 an4 653 . . . . . . . . . . 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 274 . . . . . . . . . 10 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))))
3026, 29bitrdi 287 . . . . . . . . 9 (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)))))
3130anbi1d 630 . . . . . . . 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 1088 . . . . . . . . 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 277 . . . . . . . 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, 34bitrdi 287 . . . . . . 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 1927 . . . . . 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 6788 . . . . . . . . . 10 𝑇 ∈ V
3837mptex 7099 . . . . . . . . 9 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
3920, 38eqeltri 2835 . . . . . . . 8 𝑂 ∈ V
40 opeq2 4805 . . . . . . . . . . 11 (𝑦 = 𝑂 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑂⟩)
4140oveq1d 7290 . . . . . . . . . 10 (𝑦 = 𝑂 → (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩))
4241eqeq2d 2749 . . . . . . . . 9 (𝑦 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩)))
4342anbi2d 629 . . . . . . . 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 4805 . . . . . . . . . . 11 (𝑤 = 𝑂 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑂⟩)
4544oveq2d 7291 . . . . . . . . . 10 (𝑤 = 𝑂 → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩))
4645eqeq2d 2749 . . . . . . . . 9 (𝑤 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩)))
4746anbi2d 629 . . . . . . . 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 3483 . . . . . . 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 768 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥 ∈ (𝐽𝑋))
523, 4, 5, 19, 21diael 39057 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑥 ∈ (𝐽𝑋)) → 𝑥𝑇)
5349, 50, 51, 52syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥𝑇)
54 eqid 2738 . . . . . . . . . . . . 13 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
553, 5, 19, 54, 20tendo0cl 38804 . . . . . . . . . . . 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 770 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧 ∈ (𝐽𝑌))
593, 4, 5, 19, 21diael 39057 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊) ∧ 𝑧 ∈ (𝐽𝑌)) → 𝑧𝑇)
6049, 57, 58, 59syl3anc 1370 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧𝑇)
61 eqid 2738 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
62 eqid 2738 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
635, 19, 54, 6, 61, 14, 62dvhopvadd 39107 . . . . . . . . . . 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 1378 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
6564eqeq2d 2749 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) ↔ ⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩))
66 vex 3436 . . . . . . . . . . . 12 𝑥 ∈ V
67 vex 3436 . . . . . . . . . . . 12 𝑧 ∈ V
6866, 67coex 7777 . . . . . . . . . . 11 (𝑥𝑧) ∈ V
69 ovex 7308 . . . . . . . . . . 11 (𝑂(+g‘(Scalar‘𝑈))𝑂) ∈ V
7068, 69opth2 5395 . . . . . . . . . 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 2738 . . . . . . . . . . . . . . 15 (+g𝑉) = (+g𝑉)
735, 19, 71, 72dvavadd 39029 . . . . . . . . . . . . . 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 2749 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐹 = (𝑥(+g𝑉)𝑧) ↔ 𝐹 = (𝑥𝑧)))
7675bicomd 222 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐹 = (𝑥𝑧) ↔ 𝐹 = (𝑥(+g𝑉)𝑧)))
77 eqid 2738 . . . . . . . . . . . . . . . 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 39098 . . . . . . . . . . . . . . 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 7292 . . . . . . . . . . . . 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 38805 . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂)
8483eqeq2d 2749 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂) ↔ 𝑆 = 𝑂))
8576, 84anbi12d 631 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → ((𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂)) ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8670, 85syl5bb 283 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩ ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8765, 86bitrd 278 . . . . . . . 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 283 . . . . . 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 278 . . . . 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 1924 . . . 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 283 . . 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 1924 . 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 223 . . . . 5 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
96952exbii 1851 . . . 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 1954 . . . 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 274 . . 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 39040 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑉 ∈ LVec)
100 lveclmod 20368 . . . . . . . . 9 (𝑉 ∈ LVec → 𝑉 ∈ LMod)
101 eqid 2738 . . . . . . . . . 10 (LSubSp‘𝑉) = (LSubSp‘𝑉)
102101lsssssubg 20220 . . . . . . . . 9 (𝑉 ∈ LMod → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1031, 99, 100, 1024syl 19 . . . . . . . 8 (𝜑 → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1043, 4, 5, 71, 21, 101dialss 39060 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐽𝑋) ∈ (LSubSp‘𝑉))
1051, 2, 104syl2anc 584 . . . . . . . 8 (𝜑 → (𝐽𝑋) ∈ (LSubSp‘𝑉))
106103, 105sseldd 3922 . . . . . . 7 (𝜑 → (𝐽𝑋) ∈ (SubGrp‘𝑉))
1073, 4, 5, 71, 21, 101dialss 39060 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐽𝑌) ∈ (LSubSp‘𝑉))
1081, 11, 107syl2anc 584 . . . . . . . 8 (𝜑 → (𝐽𝑌) ∈ (LSubSp‘𝑉))
109103, 108sseldd 3922 . . . . . . 7 (𝜑 → (𝐽𝑌) ∈ (SubGrp‘𝑉))
110 diblsmopel.q . . . . . . . 8 = (LSSum‘𝑉)
11172, 110lsmelval 19254 . . . . . . 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 3232 . . . . . 6 (∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧) ↔ ∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)))
114112, 113bitrdi 287 . . . . 5 (𝜑 → (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ↔ ∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧))))
115114anbi1d 630 . . . 4 (𝜑 → ((𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂) ↔ (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂)))
116115bicomd 222 . . 3 (𝜑 → ((∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
11798, 116syl5bb 283 . 2 (𝜑 → (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
11817, 93, 1173bitrd 305 1 (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  wrex 3065  Vcvv 3432  wss 3887  cop 4567   class class class wbr 5074  cmpt 5157   I cid 5488  cres 5591  ccom 5593  cfv 6433  (class class class)co 7275  cmpo 7277  Basecbs 16912  +gcplusg 16962  Scalarcsca 16965  lecple 16969  SubGrpcsubg 18749  LSSumclsm 19239  LModclmod 20123  LSubSpclss 20193  LVecclvec 20364  HLchlt 37364  LHypclh 37998  LTrncltrn 38115  TEndoctendo 38766  DVecAcdveca 39016  DIsoAcdia 39042  DVecHcdvh 39092  DIsoBcdib 39152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-riotaBAD 36967
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-undef 8089  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-n0 12234  df-z 12320  df-uz 12583  df-fz 13240  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-sca 16978  df-vsca 16979  df-0g 17152  df-proset 18013  df-poset 18031  df-plt 18048  df-lub 18064  df-glb 18065  df-join 18066  df-meet 18067  df-p0 18143  df-p1 18144  df-lat 18150  df-clat 18217  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-lsm 19241  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-drng 19993  df-lmod 20125  df-lss 20194  df-lvec 20365  df-oposet 37190  df-ol 37192  df-oml 37193  df-covers 37280  df-ats 37281  df-atl 37312  df-cvlat 37336  df-hlat 37365  df-llines 37512  df-lplanes 37513  df-lvols 37514  df-lines 37515  df-psubsp 37517  df-pmap 37518  df-padd 37810  df-lhyp 38002  df-laut 38003  df-ldil 38118  df-ltrn 38119  df-trl 38173  df-tgrp 38757  df-tendo 38769  df-edring 38771  df-dveca 39017  df-disoa 39043  df-dvech 39093  df-dib 39153
This theorem is referenced by:  dib2dim  39257  dih2dimbALTN  39259
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