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Theorem diblsmopel 35478
 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 2610 . . . . 5 (LSubSp‘𝑈) = (LSubSp‘𝑈)
93, 4, 5, 6, 7, 8diblss 35477 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ (LSubSp‘𝑈))
101, 2, 9syl2anc 691 . . 3 (𝜑 → (𝐼𝑋) ∈ (LSubSp‘𝑈))
11 diblsmopel.y . . . 4 (𝜑 → (𝑌𝐵𝑌 𝑊))
123, 4, 5, 6, 7, 8diblss 35477 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼𝑌) ∈ (LSubSp‘𝑈))
131, 11, 12syl2anc 691 . . 3 (𝜑 → (𝐼𝑌) ∈ (LSubSp‘𝑈))
14 eqid 2610 . . . 4 (+g𝑈) = (+g𝑈)
15 diblsmopel.p . . . 4 = (LSSum‘𝑈)
165, 6, 14, 8, 15dvhopellsm 35424 . . 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 1318 . 2 (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ ∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
18 excom 2029 . . . 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 35452 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ↔ (𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂)))
231, 2, 22syl2anc 691 . . . . . . . . . . 11 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ↔ (𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂)))
243, 4, 5, 19, 20, 21, 7dibopelval2 35452 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌) ↔ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)))
251, 11, 24syl2anc 691 . . . . . . . . . . 11 (𝜑 → (⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌) ↔ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)))
2623, 25anbi12d 743 . . . . . . . . . 10 (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂))))
27 an4 861 . . . . . . . . . . 11 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝑦 = 𝑂𝑤 = 𝑂)))
28 ancom 465 . . . . . . . . . . 11 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝑦 = 𝑂𝑤 = 𝑂)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))))
2927, 28bitri 263 . . . . . . . . . 10 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))))
3026, 29syl6bb 275 . . . . . . . . 9 (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ↔ ((𝑦 = 𝑂𝑤 = 𝑂) ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)))))
3130anbi1d 737 . . . . . . . 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 679 . . . . . . . . 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 1033 . . . . . . . . 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 266 . . . . . . . 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 275 . . . . . . 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 1839 . . . . . 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⟩)))))
37 fvex 6113 . . . . . . . . . . 11 ((LTrn‘𝐾)‘𝑊) ∈ V
3819, 37eqeltri 2684 . . . . . . . . . 10 𝑇 ∈ V
3938mptex 6390 . . . . . . . . 9 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
4020, 39eqeltri 2684 . . . . . . . 8 𝑂 ∈ V
41 opeq2 4341 . . . . . . . . . . 11 (𝑦 = 𝑂 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑂⟩)
4241oveq1d 6564 . . . . . . . . . 10 (𝑦 = 𝑂 → (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩))
4342eqeq2d 2620 . . . . . . . . 9 (𝑦 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩)))
4443anbi2d 736 . . . . . . . 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⟩))))
45 opeq2 4341 . . . . . . . . . . 11 (𝑤 = 𝑂 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑂⟩)
4645oveq2d 6565 . . . . . . . . . 10 (𝑤 = 𝑂 → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩))
4746eqeq2d 2620 . . . . . . . . 9 (𝑤 = 𝑂 → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩) ↔ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩)))
4847anbi2d 736 . . . . . . . 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⟩))))
4940, 40, 44, 48ceqsex2v 3218 . . . . . . 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⟩)))
501adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
512adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑋𝐵𝑋 𝑊))
52 simprl 790 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥 ∈ (𝐽𝑋))
533, 4, 5, 19, 21diael 35350 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑥 ∈ (𝐽𝑋)) → 𝑥𝑇)
5450, 51, 52, 53syl3anc 1318 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥𝑇)
55 eqid 2610 . . . . . . . . . . . . 13 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
563, 5, 19, 55, 20tendo0cl 35096 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
5750, 56syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
5811adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑌𝐵𝑌 𝑊))
59 simprr 792 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧 ∈ (𝐽𝑌))
603, 4, 5, 19, 21diael 35350 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊) ∧ 𝑧 ∈ (𝐽𝑌)) → 𝑧𝑇)
6150, 58, 59, 60syl3anc 1318 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧𝑇)
62 eqid 2610 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
63 eqid 2610 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
645, 19, 55, 6, 62, 14, 63dvhopvadd 35400 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) ∧ (𝑧𝑇𝑂 ∈ ((TEndo‘𝐾)‘𝑊))) → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
6550, 54, 57, 61, 57, 64syl122anc 1327 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
6665eqeq2d 2620 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) ↔ ⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩))
67 vex 3176 . . . . . . . . . . . 12 𝑥 ∈ V
68 vex 3176 . . . . . . . . . . . 12 𝑧 ∈ V
6967, 68coex 7011 . . . . . . . . . . 11 (𝑥𝑧) ∈ V
70 ovex 6577 . . . . . . . . . . 11 (𝑂(+g‘(Scalar‘𝑈))𝑂) ∈ V
7169, 70opth2 4875 . . . . . . . . . 10 (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩ ↔ (𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂)))
72 diblsmopel.v . . . . . . . . . . . . . . 15 𝑉 = ((DVecA‘𝐾)‘𝑊)
73 eqid 2610 . . . . . . . . . . . . . . 15 (+g𝑉) = (+g𝑉)
745, 19, 72, 73dvavadd 35321 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥(+g𝑉)𝑧) = (𝑥𝑧))
7550, 54, 61, 74syl12anc 1316 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑥(+g𝑉)𝑧) = (𝑥𝑧))
7675eqeq2d 2620 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐹 = (𝑥(+g𝑉)𝑧) ↔ 𝐹 = (𝑥𝑧)))
7776bicomd 212 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐹 = (𝑥𝑧) ↔ 𝐹 = (𝑥(+g𝑉)𝑧)))
78 eqid 2610 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))
795, 19, 55, 6, 62, 78, 63dvhfplusr 35391 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))))
8050, 79syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (+g‘(Scalar‘𝑈)) = (𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))))
8180oveqd 6566 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))𝑂))
823, 5, 19, 55, 20, 78tendo0pl 35097 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑂 ∈ ((TEndo‘𝐾)‘𝑊)) → (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))𝑂) = 𝑂)
8350, 57, 82syl2anc 691 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))𝑂) = 𝑂)
8481, 83eqtrd 2644 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂)
8584eqeq2d 2620 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂) ↔ 𝑆 = 𝑂))
8677, 85anbi12d 743 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → ((𝐹 = (𝑥𝑧) ∧ 𝑆 = (𝑂(+g‘(Scalar‘𝑈))𝑂)) ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8771, 86syl5bb 271 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩ ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8866, 87bitrd 267 . . . . . . . 8 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) ↔ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
8988pm5.32da 671 . . . . . . 7 (𝜑 → (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂))))
9049, 89syl5bb 271 . . . . . 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))))
9136, 90bitrd 267 . . . . 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))))
9291exbidv 1837 . . . 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))))
9318, 92syl5bb 271 . . 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))))
9493exbidv 1837 . 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))))
95 anass 679 . . . . . 6 ((((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)))
9695bicomi 213 . . . . 5 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
97962exbii 1765 . . . 4 (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ ∃𝑥𝑧(((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
98 19.41vv 1902 . . . 4 (∃𝑥𝑧(((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
9997, 98bitri 263 . . 3 (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂))
1005, 72dvalvec 35333 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑉 ∈ LVec)
101 lveclmod 18927 . . . . . . . . 9 (𝑉 ∈ LVec → 𝑉 ∈ LMod)
102 eqid 2610 . . . . . . . . . 10 (LSubSp‘𝑉) = (LSubSp‘𝑉)
103102lsssssubg 18779 . . . . . . . . 9 (𝑉 ∈ LMod → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1041, 100, 101, 1034syl 19 . . . . . . . 8 (𝜑 → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1053, 4, 5, 72, 21, 102dialss 35353 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐽𝑋) ∈ (LSubSp‘𝑉))
1061, 2, 105syl2anc 691 . . . . . . . 8 (𝜑 → (𝐽𝑋) ∈ (LSubSp‘𝑉))
107104, 106sseldd 3569 . . . . . . 7 (𝜑 → (𝐽𝑋) ∈ (SubGrp‘𝑉))
1083, 4, 5, 72, 21, 102dialss 35353 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐽𝑌) ∈ (LSubSp‘𝑉))
1091, 11, 108syl2anc 691 . . . . . . . 8 (𝜑 → (𝐽𝑌) ∈ (LSubSp‘𝑉))
110104, 109sseldd 3569 . . . . . . 7 (𝜑 → (𝐽𝑌) ∈ (SubGrp‘𝑉))
111 diblsmopel.q . . . . . . . 8 = (LSSum‘𝑉)
11273, 111lsmelval 17887 . . . . . . 7 (((𝐽𝑋) ∈ (SubGrp‘𝑉) ∧ (𝐽𝑌) ∈ (SubGrp‘𝑉)) → (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ↔ ∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧)))
113107, 110, 112syl2anc 691 . . . . . 6 (𝜑 → (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ↔ ∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧)))
114 r2ex 3043 . . . . . 6 (∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧) ↔ ∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)))
115113, 114syl6bb 275 . . . . 5 (𝜑 → (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ↔ ∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧))))
116115anbi1d 737 . . . 4 (𝜑 → ((𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂) ↔ (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂)))
117116bicomd 212 . . 3 (𝜑 → ((∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
11899, 117syl5bb 271 . 2 (𝜑 → (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
11917, 94, 1183bitrd 293 1 (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   ↾ cres 5040   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771  lecple 15775  SubGrpcsubg 17411  LSSumclsm 17872  LModclmod 18686  LSubSpclss 18753  LVecclvec 18923  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  TEndoctendo 35058  DVecAcdveca 35308  DIsoAcdia 35335  DVecHcdvh 35385  DIsoBcdib 35445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-riotaBAD 33257 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-undef 7286  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-0g 15925  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-lsm 17874  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-dvr 18506  df-drng 18572  df-lmod 18688  df-lss 18754  df-lvec 18924  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464  df-tgrp 35049  df-tendo 35061  df-edring 35063  df-dveca 35309  df-disoa 35336  df-dvech 35386  df-dib 35446 This theorem is referenced by:  dib2dim  35550  dih2dimbALTN  35552
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