Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diblsmopel Structured version   Visualization version   GIF version

Theorem diblsmopel 40771
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 2725 . . . . 5 (LSubSp‘𝑈) = (LSubSp‘𝑈)
93, 4, 5, 6, 7, 8diblss 40770 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ (LSubSp‘𝑈))
101, 2, 9syl2anc 582 . . 3 (𝜑 → (𝐼𝑋) ∈ (LSubSp‘𝑈))
11 diblsmopel.y . . . 4 (𝜑 → (𝑌𝐵𝑌 𝑊))
123, 4, 5, 6, 7, 8diblss 40770 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐼𝑌) ∈ (LSubSp‘𝑈))
131, 11, 12syl2anc 582 . . 3 (𝜑 → (𝐼𝑌) ∈ (LSubSp‘𝑈))
14 eqid 2725 . . . 4 (+g𝑈) = (+g𝑈)
15 diblsmopel.p . . . 4 = (LSSum‘𝑈)
165, 6, 14, 8, 15dvhopellsm 40717 . . 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 1368 . 2 (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ ∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ∧ ⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌)) ∧ ⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩))))
18 excom 2151 . . . 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 40745 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ↔ (𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂)))
231, 2, 22syl2anc 582 . . . . . . . . . . 11 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐼𝑋) ↔ (𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂)))
243, 4, 5, 19, 20, 21, 7dibopelval2 40745 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (⟨𝑧, 𝑤⟩ ∈ (𝐼𝑌) ↔ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)))
251, 11, 24syl2anc 582 . . . . . . . . . . 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 654 . . . . . . . . . . 11 (((𝑥 ∈ (𝐽𝑋) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (𝐽𝑌) ∧ 𝑤 = 𝑂)) ↔ ((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝑦 = 𝑂𝑤 = 𝑂)))
28 ancom 459 . . . . . . . . . . 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 286 . . . . . . . . 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 467 . . . . . . . . 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 1086 . . . . . . . . 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 286 . . . . . . 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 1919 . . . . . 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 6910 . . . . . . . . . 10 𝑇 ∈ V
3837mptex 7235 . . . . . . . . 9 (𝑓𝑇 ↦ ( I ↾ 𝐵)) ∈ V
3920, 38eqeltri 2821 . . . . . . . 8 𝑂 ∈ V
40 opeq2 4876 . . . . . . . . . . 11 (𝑦 = 𝑂 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑂⟩)
4140oveq1d 7434 . . . . . . . . . 10 (𝑦 = 𝑂 → (⟨𝑥, 𝑦⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩))
4241eqeq2d 2736 . . . . . . . . 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 4876 . . . . . . . . . . 11 (𝑤 = 𝑂 → ⟨𝑧, 𝑤⟩ = ⟨𝑧, 𝑂⟩)
4544oveq2d 7435 . . . . . . . . . 10 (𝑤 = 𝑂 → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑤⟩) = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩))
4645eqeq2d 2736 . . . . . . . . 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 3520 . . . . . . 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 479 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
502adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑋𝐵𝑋 𝑊))
51 simprl 769 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥 ∈ (𝐽𝑋))
523, 4, 5, 19, 21diael 40643 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑥 ∈ (𝐽𝑋)) → 𝑥𝑇)
5349, 50, 51, 52syl3anc 1368 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑥𝑇)
54 eqid 2725 . . . . . . . . . . . . 13 ((TEndo‘𝐾)‘𝑊) = ((TEndo‘𝐾)‘𝑊)
553, 5, 19, 54, 20tendo0cl 40390 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
5649, 55syl 17 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑂 ∈ ((TEndo‘𝐾)‘𝑊))
5711adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑌𝐵𝑌 𝑊))
58 simprr 771 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧 ∈ (𝐽𝑌))
593, 4, 5, 19, 21diael 40643 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊) ∧ 𝑧 ∈ (𝐽𝑌)) → 𝑧𝑇)
6049, 57, 58, 59syl3anc 1368 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → 𝑧𝑇)
61 eqid 2725 . . . . . . . . . . . 12 (Scalar‘𝑈) = (Scalar‘𝑈)
62 eqid 2725 . . . . . . . . . . . 12 (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈))
635, 19, 54, 6, 61, 14, 62dvhopvadd 40693 . . . . . . . . . . 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 1376 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩)
6564eqeq2d 2736 . . . . . . . . 9 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (⟨𝐹, 𝑆⟩ = (⟨𝑥, 𝑂⟩(+g𝑈)⟨𝑧, 𝑂⟩) ↔ ⟨𝐹, 𝑆⟩ = ⟨(𝑥𝑧), (𝑂(+g‘(Scalar‘𝑈))𝑂)⟩))
66 vex 3465 . . . . . . . . . . . 12 𝑥 ∈ V
67 vex 3465 . . . . . . . . . . . 12 𝑧 ∈ V
6866, 67coex 7938 . . . . . . . . . . 11 (𝑥𝑧) ∈ V
69 ovex 7452 . . . . . . . . . . 11 (𝑂(+g‘(Scalar‘𝑈))𝑂) ∈ V
7068, 69opth2 5482 . . . . . . . . . 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 2725 . . . . . . . . . . . . . . 15 (+g𝑉) = (+g𝑉)
735, 19, 71, 72dvavadd 40615 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥𝑇𝑧𝑇)) → (𝑥(+g𝑉)𝑧) = (𝑥𝑧))
7449, 53, 60, 73syl12anc 835 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑥(+g𝑉)𝑧) = (𝑥𝑧))
7574eqeq2d 2736 . . . . . . . . . . . 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 2725 . . . . . . . . . . . . . . . 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 40684 . . . . . . . . . . . . . . 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 7436 . . . . . . . . . . . . 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 40391 . . . . . . . . . . . . . 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 582 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(𝑠 ∈ ((TEndo‘𝐾)‘𝑊), 𝑡 ∈ ((TEndo‘𝐾)‘𝑊) ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))𝑂) = 𝑂)
8380, 82eqtrd 2765 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌))) → (𝑂(+g‘(Scalar‘𝑈))𝑂) = 𝑂)
8483eqeq2d 2736 . . . . . . . . . . 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, 85bitrid 282 . . . . . . . . 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 577 . . . . . . 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 282 . . . . . 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 1916 . . . 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 282 . . 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 1916 . 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 467 . . . . . 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 1843 . . . 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 1946 . . . 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 40626 . . . . . . . . 9 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑉 ∈ LVec)
100 lveclmod 21003 . . . . . . . . 9 (𝑉 ∈ LVec → 𝑉 ∈ LMod)
101 eqid 2725 . . . . . . . . . 10 (LSubSp‘𝑉) = (LSubSp‘𝑉)
102101lsssssubg 20854 . . . . . . . . 9 (𝑉 ∈ LMod → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1031, 99, 100, 1024syl 19 . . . . . . . 8 (𝜑 → (LSubSp‘𝑉) ⊆ (SubGrp‘𝑉))
1043, 4, 5, 71, 21, 101dialss 40646 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐽𝑋) ∈ (LSubSp‘𝑉))
1051, 2, 104syl2anc 582 . . . . . . . 8 (𝜑 → (𝐽𝑋) ∈ (LSubSp‘𝑉))
106103, 105sseldd 3977 . . . . . . 7 (𝜑 → (𝐽𝑋) ∈ (SubGrp‘𝑉))
1073, 4, 5, 71, 21, 101dialss 40646 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑌𝐵𝑌 𝑊)) → (𝐽𝑌) ∈ (LSubSp‘𝑉))
1081, 11, 107syl2anc 582 . . . . . . . 8 (𝜑 → (𝐽𝑌) ∈ (LSubSp‘𝑉))
109103, 108sseldd 3977 . . . . . . 7 (𝜑 → (𝐽𝑌) ∈ (SubGrp‘𝑉))
110 diblsmopel.q . . . . . . . 8 = (LSSum‘𝑉)
11172, 110lsmelval 19616 . . . . . . 7 (((𝐽𝑋) ∈ (SubGrp‘𝑉) ∧ (𝐽𝑌) ∈ (SubGrp‘𝑉)) → (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ↔ ∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧)))
112106, 109, 111syl2anc 582 . . . . . 6 (𝜑 → (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ↔ ∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧)))
113 r2ex 3185 . . . . . 6 (∃𝑥 ∈ (𝐽𝑋)∃𝑧 ∈ (𝐽𝑌)𝐹 = (𝑥(+g𝑉)𝑧) ↔ ∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)))
114112, 113bitrdi 286 . . . . 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 222 . . 3 (𝜑 → ((∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ 𝐹 = (𝑥(+g𝑉)𝑧)) ∧ 𝑆 = 𝑂) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
11798, 116bitrid 282 . 2 (𝜑 → (∃𝑥𝑧((𝑥 ∈ (𝐽𝑋) ∧ 𝑧 ∈ (𝐽𝑌)) ∧ (𝐹 = (𝑥(+g𝑉)𝑧) ∧ 𝑆 = 𝑂)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
11817, 93, 1173bitrd 304 1 (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wrex 3059  Vcvv 3461  wss 3944  cop 4636   class class class wbr 5149  cmpt 5232   I cid 5575  cres 5680  ccom 5682  cfv 6549  (class class class)co 7419  cmpo 7421  Basecbs 17183  +gcplusg 17236  Scalarcsca 17239  lecple 17243  SubGrpcsubg 19083  LSSumclsm 19601  LModclmod 20755  LSubSpclss 20827  LVecclvec 20999  HLchlt 38949  LHypclh 39584  LTrncltrn 39701  TEndoctendo 40352  DVecAcdveca 40602  DIsoAcdia 40628  DVecHcdvh 40678  DIsoBcdib 40738
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-riotaBAD 38552
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4910  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-tpos 8232  df-undef 8279  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-1o 8487  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-nn 12246  df-2 12308  df-3 12309  df-4 12310  df-5 12311  df-6 12312  df-n0 12506  df-z 12592  df-uz 12856  df-fz 13520  df-struct 17119  df-sets 17136  df-slot 17154  df-ndx 17166  df-base 17184  df-ress 17213  df-plusg 17249  df-mulr 17250  df-sca 17252  df-vsca 17253  df-0g 17426  df-proset 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-p1 18421  df-lat 18427  df-clat 18494  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-minusg 18902  df-sbg 18903  df-subg 19086  df-lsm 19603  df-cmn 19749  df-abl 19750  df-mgp 20087  df-rng 20105  df-ur 20134  df-ring 20187  df-oppr 20285  df-dvdsr 20308  df-unit 20309  df-invr 20339  df-dvr 20352  df-drng 20638  df-lmod 20757  df-lss 20828  df-lvec 21000  df-oposet 38775  df-ol 38777  df-oml 38778  df-covers 38865  df-ats 38866  df-atl 38897  df-cvlat 38921  df-hlat 38950  df-llines 39098  df-lplanes 39099  df-lvols 39100  df-lines 39101  df-psubsp 39103  df-pmap 39104  df-padd 39396  df-lhyp 39588  df-laut 39589  df-ldil 39704  df-ltrn 39705  df-trl 39759  df-tgrp 40343  df-tendo 40355  df-edring 40357  df-dveca 40603  df-disoa 40629  df-dvech 40679  df-dib 40739
This theorem is referenced by:  dib2dim  40843  dih2dimbALTN  40845
  Copyright terms: Public domain W3C validator