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Theorem diblsmopel 41173
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 2737 . . . . 5 (LSubSp‘𝑈) = (LSubSp‘𝑈)
93, 4, 5, 6, 7, 8diblss 41172 . . . 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 41172 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼𝑌) ∈ (LSubSp‘𝑈))
131, 11, 12syl2anc 584 . . 3 (𝜑 → (𝐼𝑌) ∈ (LSubSp‘𝑈))
14 eqid 2737 . . . 4 (+g𝑈) = (+g𝑈)
15 diblsmopel.p . . . 4 = (LSSum‘𝑈)
165, 6, 14, 8, 15dvhopellsm 41119 . . 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 1373 . 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 41147 . . . . . . . . . . . 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 41147 . . . . . . . . . . . 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 632 . . . . . . . . . 10 (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂))))
27 an4 656 . . . . . . . . . . 11 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝑦 = 𝑂𝑤 = 𝑂)))
28 ancom 460 . . . . . . . . . . 11 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝑦 = 𝑂𝑤 = 𝑂)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))))
2927, 28bitri 275 . . . . . . . . . 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 631 . . . . . . . 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 468 . . . . . . . . 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 1089 . . . . . . . . 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 278 . . . . . . . 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 1924 . . . . . 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 6920 . . . . . . . . . 10 𝑇 ∈ V
3837mptex 7243 . . . . . . . . 9 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
3920, 38eqeltri 2837 . . . . . . . 8 𝑂 ∈ V
40 opeq2 4874 . . . . . . . . . . 11 (𝑦 = 𝑂 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑂⟩)
4140oveq1d 7446 . . . . . . . . . 10 (𝑦 = 𝑂 → (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩))
4241eqeq2d 2748 . . . . . . . . 9 (𝑦 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩)))
4342anbi2d 630 . . . . . . . 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 4874 . . . . . . . . . . 11 (𝑤 = 𝑂 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑂⟩)
4544oveq2d 7447 . . . . . . . . . 10 (𝑤 = 𝑂 → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩))
4645eqeq2d 2748 . . . . . . . . 9 (𝑤 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩)))
4746anbi2d 630 . . . . . . . 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 3536 . . . . . . 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 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
502adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑋𝐵𝑋 𝑊))
51 simprl 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥 ∈ (𝐽𝑋))
523, 4, 5, 19, 21diael 41045 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑥 ∈ (𝐽𝑋)) → 𝑥𝑇)
5349, 50, 51, 52syl3anc 1373 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥𝑇)
54 eqid 2737 . . . . . . . . . . . . 13 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
553, 5, 19, 54, 20tendo0cl 40792 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
5649, 55syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
5711adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑌𝐵𝑌 𝑊))
58 simprr 773 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧 ∈ (𝐽𝑌))
593, 4, 5, 19, 21diael 41045 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊) ∧ 𝑧 ∈ (𝐽𝑌)) → 𝑧𝑇)
6049, 57, 58, 59syl3anc 1373 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧𝑇)
61 eqid 2737 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
62 eqid 2737 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
635, 19, 54, 6, 61, 14, 62dvhopvadd 41095 . . . . . . . . . . 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 1381 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
6564eqeq2d 2748 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) ↔ ⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩))
66 vex 3484 . . . . . . . . . . . 12 𝑥 ∈ V
67 vex 3484 . . . . . . . . . . . 12 𝑧 ∈ V
6866, 67coex 7952 . . . . . . . . . . 11 (𝑥𝑧) ∈ V
69 ovex 7464 . . . . . . . . . . 11 (𝑂(+g‘(Scalar‘𝑈))𝑂) ∈ V
7068, 69opth2 5485 . . . . . . . . . 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 2737 . . . . . . . . . . . . . . 15 (+g𝑉) = (+g𝑉)
735, 19, 71, 72dvavadd 41017 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥(+g𝑉)𝑧) = (𝑥𝑧))
7449, 53, 60, 73syl12anc 837 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑥(+g𝑉)𝑧) = (𝑥𝑧))
7574eqeq2d 2748 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐹 = (𝑥(+g𝑉)𝑧) ↔ 𝐹 = (𝑥𝑧)))
7675bicomd 223 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐹 = (𝑥𝑧) ↔ 𝐹 = (𝑥(+g𝑉)𝑧)))
77 eqid 2737 . . . . . . . . . . . . . . . 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 41086 . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . 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 40793 . . . . . . . . . . . . . 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 2777 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂)
8483eqeq2d 2748 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂) ↔ 𝑆 = 𝑂))
8576, 84anbi12d 632 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → ((𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂)) ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8670, 85bitrid 283 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩ ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8765, 86bitrd 279 . . . . . . . 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, 88bitrid 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 279 . . . . 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 1921 . . . 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, 91bitrid 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 1921 . 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 468 . . . . . 6 ((((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
9594bicomi 224 . . . . 5 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
96952exbii 1849 . . . 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 1950 . . . 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 275 . . 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 41028 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑉 ∈ LVec)
100 lveclmod 21105 . . . . . . . . 9 (𝑉 ∈ LVec → 𝑉 ∈ LMod)
101 eqid 2737 . . . . . . . . . 10 (LSubSp‘𝑉) = (LSubSp‘𝑉)
102101lsssssubg 20956 . . . . . . . . 9 (𝑉 ∈ LMod → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1031, 99, 100, 1024syl 19 . . . . . . . 8 (𝜑 → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1043, 4, 5, 71, 21, 101dialss 41048 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐽𝑋) ∈ (LSubSp‘𝑉))
1051, 2, 104syl2anc 584 . . . . . . . 8 (𝜑 → (𝐽𝑋) ∈ (LSubSp‘𝑉))
106103, 105sseldd 3984 . . . . . . 7 (𝜑 → (𝐽𝑋) ∈ (SubGrp‘𝑉))
1073, 4, 5, 71, 21, 101dialss 41048 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐽𝑌) ∈ (LSubSp‘𝑉))
1081, 11, 107syl2anc 584 . . . . . . . 8 (𝜑 → (𝐽𝑌) ∈ (LSubSp‘𝑉))
109103, 108sseldd 3984 . . . . . . 7 (𝜑 → (𝐽𝑌) ∈ (SubGrp‘𝑉))
110 diblsmopel.q . . . . . . . 8 = (LSSum‘𝑉)
11172, 110lsmelval 19667 . . . . . . 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 3196 . . . . . 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 631 . . . 4 (𝜑 → ((𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂) ↔ (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂)))
116115bicomd 223 . . 3 (𝜑 → ((∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
11798, 116bitrid 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 206  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wrex 3070  Vcvv 3480  wss 3951  cop 4632   class class class wbr 5143  cmpt 5225   I cid 5577  cres 5687  ccom 5689  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  +gcplusg 17297  Scalarcsca 17300  lecple 17304  SubGrpcsubg 19138  LSSumclsm 19652  LModclmod 20858  LSubSpclss 20929  LVecclvec 21101  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  TEndoctendo 40754  DVecAcdveca 41004  DIsoAcdia 41030  DVecHcdvh 41080  DIsoBcdib 41140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-riotaBAD 38954
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-tpos 8251  df-undef 8298  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-0g 17486  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-lsm 19654  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-drng 20731  df-lmod 20860  df-lss 20930  df-lvec 21102  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502  df-lines 39503  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161  df-tgrp 40745  df-tendo 40757  df-edring 40759  df-dveca 41005  df-disoa 41031  df-dvech 41081  df-dib 41141
This theorem is referenced by:  dib2dim  41245  dih2dimbALTN  41247
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