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 40037
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 𝐡 = (Baseβ€˜πΎ)
diblsmopel.l ≀ = (leβ€˜πΎ)
diblsmopel.h 𝐻 = (LHypβ€˜πΎ)
diblsmopel.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
diblsmopel.o 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
diblsmopel.v 𝑉 = ((DVecAβ€˜πΎ)β€˜π‘Š)
diblsmopel.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
diblsmopel.q βŠ• = (LSSumβ€˜π‘‰)
diblsmopel.p ✚ = (LSSumβ€˜π‘ˆ)
diblsmopel.j 𝐽 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
diblsmopel.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
diblsmopel.k (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
diblsmopel.x (πœ‘ β†’ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
diblsmopel.y (πœ‘ β†’ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š))
Assertion
Ref Expression
diblsmopel (πœ‘ β†’ (⟨𝐹, π‘†βŸ© ∈ ((πΌβ€˜π‘‹) ✚ (πΌβ€˜π‘Œ)) ↔ (𝐹 ∈ ((π½β€˜π‘‹) βŠ• (π½β€˜π‘Œ)) ∧ 𝑆 = 𝑂)))
Distinct variable groups:   𝐡,𝑓   𝑓,𝐻   𝑓,𝐾   𝑇,𝑓   𝑓,π‘Š
Allowed substitution hints:   πœ‘(𝑓)   ✚ (𝑓)   βŠ• (𝑓)   𝑆(𝑓)   π‘ˆ(𝑓)   𝐹(𝑓)   𝐼(𝑓)   𝐽(𝑓)   ≀ (𝑓)   𝑂(𝑓)   𝑉(𝑓)   𝑋(𝑓)   π‘Œ(𝑓)

Proof of Theorem diblsmopel
Dummy variables π‘₯ 𝑀 𝑦 𝑧 𝑠 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 diblsmopel.k . . 3 (πœ‘ β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
2 diblsmopel.x . . . 4 (πœ‘ β†’ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
3 diblsmopel.b . . . . 5 𝐡 = (Baseβ€˜πΎ)
4 diblsmopel.l . . . . 5 ≀ = (leβ€˜πΎ)
5 diblsmopel.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
6 diblsmopel.u . . . . 5 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
7 diblsmopel.i . . . . 5 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
8 eqid 2732 . . . . 5 (LSubSpβ€˜π‘ˆ) = (LSubSpβ€˜π‘ˆ)
93, 4, 5, 6, 7, 8diblss 40036 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) ∈ (LSubSpβ€˜π‘ˆ))
101, 2, 9syl2anc 584 . . 3 (πœ‘ β†’ (πΌβ€˜π‘‹) ∈ (LSubSpβ€˜π‘ˆ))
11 diblsmopel.y . . . 4 (πœ‘ β†’ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š))
123, 4, 5, 6, 7, 8diblss 40036 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ (πΌβ€˜π‘Œ) ∈ (LSubSpβ€˜π‘ˆ))
131, 11, 12syl2anc 584 . . 3 (πœ‘ β†’ (πΌβ€˜π‘Œ) ∈ (LSubSpβ€˜π‘ˆ))
14 eqid 2732 . . . 4 (+gβ€˜π‘ˆ) = (+gβ€˜π‘ˆ)
15 diblsmopel.p . . . 4 ✚ = (LSSumβ€˜π‘ˆ)
165, 6, 14, 8, 15dvhopellsm 39983 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (πΌβ€˜π‘‹) ∈ (LSubSpβ€˜π‘ˆ) ∧ (πΌβ€˜π‘Œ) ∈ (LSubSpβ€˜π‘ˆ)) β†’ (⟨𝐹, π‘†βŸ© ∈ ((πΌβ€˜π‘‹) ✚ (πΌβ€˜π‘Œ)) ↔ βˆƒπ‘₯βˆƒπ‘¦βˆƒπ‘§βˆƒπ‘€((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))))
171, 10, 13, 16syl3anc 1371 . 2 (πœ‘ β†’ (⟨𝐹, π‘†βŸ© ∈ ((πΌβ€˜π‘‹) ✚ (πΌβ€˜π‘Œ)) ↔ βˆƒπ‘₯βˆƒπ‘¦βˆƒπ‘§βˆƒπ‘€((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))))
18 excom 2162 . . . 4 (βˆƒπ‘¦βˆƒπ‘§βˆƒπ‘€((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ βˆƒπ‘§βˆƒπ‘¦βˆƒπ‘€((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)))
19 diblsmopel.t . . . . . . . . . . . . 13 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
20 diblsmopel.o . . . . . . . . . . . . 13 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
21 diblsmopel.j . . . . . . . . . . . . 13 𝐽 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
223, 4, 5, 19, 20, 21, 7dibopelval2 40011 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ↔ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑦 = 𝑂)))
231, 2, 22syl2anc 584 . . . . . . . . . . 11 (πœ‘ β†’ (⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ↔ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑦 = 𝑂)))
243, 4, 5, 19, 20, 21, 7dibopelval2 40011 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ (βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ) ↔ (𝑧 ∈ (π½β€˜π‘Œ) ∧ 𝑀 = 𝑂)))
251, 11, 24syl2anc 584 . . . . . . . . . . 11 (πœ‘ β†’ (βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ) ↔ (𝑧 ∈ (π½β€˜π‘Œ) ∧ 𝑀 = 𝑂)))
2623, 25anbi12d 631 . . . . . . . . . 10 (πœ‘ β†’ ((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ↔ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (π½β€˜π‘Œ) ∧ 𝑀 = 𝑂))))
27 an4 654 . . . . . . . . . . 11 (((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (π½β€˜π‘Œ) ∧ 𝑀 = 𝑂)) ↔ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝑦 = 𝑂 ∧ 𝑀 = 𝑂)))
28 ancom 461 . . . . . . . . . . 11 (((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝑦 = 𝑂 ∧ 𝑀 = 𝑂)) ↔ ((𝑦 = 𝑂 ∧ 𝑀 = 𝑂) ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))))
2927, 28bitri 274 . . . . . . . . . 10 (((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑦 = 𝑂) ∧ (𝑧 ∈ (π½β€˜π‘Œ) ∧ 𝑀 = 𝑂)) ↔ ((𝑦 = 𝑂 ∧ 𝑀 = 𝑂) ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))))
3026, 29bitrdi 286 . . . . . . . . 9 (πœ‘ β†’ ((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ↔ ((𝑦 = 𝑂 ∧ 𝑀 = 𝑂) ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)))))
3130anbi1d 630 . . . . . . . 8 (πœ‘ β†’ (((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ (((𝑦 = 𝑂 ∧ 𝑀 = 𝑂) ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))))
32 anass 469 . . . . . . . . 9 ((((𝑦 = 𝑂 ∧ 𝑀 = 𝑂) ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ ((𝑦 = 𝑂 ∧ 𝑀 = 𝑂) ∧ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))))
33 df-3an 1089 . . . . . . . . 9 ((𝑦 = 𝑂 ∧ 𝑀 = 𝑂 ∧ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))) ↔ ((𝑦 = 𝑂 ∧ 𝑀 = 𝑂) ∧ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))))
3432, 33bitr4i 277 . . . . . . . 8 ((((𝑦 = 𝑂 ∧ 𝑀 = 𝑂) ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ (𝑦 = 𝑂 ∧ 𝑀 = 𝑂 ∧ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))))
3531, 34bitrdi 286 . . . . . . 7 (πœ‘ β†’ (((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ (𝑦 = 𝑂 ∧ 𝑀 = 𝑂 ∧ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)))))
36352exbidv 1927 . . . . . 6 (πœ‘ β†’ (βˆƒπ‘¦βˆƒπ‘€((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ βˆƒπ‘¦βˆƒπ‘€(𝑦 = 𝑂 ∧ 𝑀 = 𝑂 ∧ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)))))
3719fvexi 6905 . . . . . . . . . 10 𝑇 ∈ V
3837mptex 7224 . . . . . . . . 9 (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡)) ∈ V
3920, 38eqeltri 2829 . . . . . . . 8 𝑂 ∈ V
40 opeq2 4874 . . . . . . . . . . 11 (𝑦 = 𝑂 β†’ ⟨π‘₯, π‘¦βŸ© = ⟨π‘₯, π‘‚βŸ©)
4140oveq1d 7423 . . . . . . . . . 10 (𝑦 = 𝑂 β†’ (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©) = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))
4241eqeq2d 2743 . . . . . . . . 9 (𝑦 = 𝑂 β†’ (⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©) ↔ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)))
4342anbi2d 629 . . . . . . . 8 (𝑦 = 𝑂 β†’ (((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))))
44 opeq2 4874 . . . . . . . . . . 11 (𝑀 = 𝑂 β†’ βŸ¨π‘§, π‘€βŸ© = βŸ¨π‘§, π‘‚βŸ©)
4544oveq2d 7424 . . . . . . . . . 10 (𝑀 = 𝑂 β†’ (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©) = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘‚βŸ©))
4645eqeq2d 2743 . . . . . . . . 9 (𝑀 = 𝑂 β†’ (⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©) ↔ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘‚βŸ©)))
4746anbi2d 629 . . . . . . . 8 (𝑀 = 𝑂 β†’ (((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘‚βŸ©))))
4839, 39, 43, 47ceqsex2v 3530 . . . . . . 7 (βˆƒπ‘¦βˆƒπ‘€(𝑦 = 𝑂 ∧ 𝑀 = 𝑂 ∧ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))) ↔ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘‚βŸ©)))
491adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
502adantr 481 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
51 simprl 769 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ π‘₯ ∈ (π½β€˜π‘‹))
523, 4, 5, 19, 21diael 39909 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘₯ ∈ (π½β€˜π‘‹)) β†’ π‘₯ ∈ 𝑇)
5349, 50, 51, 52syl3anc 1371 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ π‘₯ ∈ 𝑇)
54 eqid 2732 . . . . . . . . . . . . 13 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
553, 5, 19, 54, 20tendo0cl 39656 . . . . . . . . . . . 12 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑂 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
5649, 55syl 17 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ 𝑂 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))
5711adantr 481 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š))
58 simprr 771 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ 𝑧 ∈ (π½β€˜π‘Œ))
593, 4, 5, 19, 21diael 39909 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) β†’ 𝑧 ∈ 𝑇)
6049, 57, 58, 59syl3anc 1371 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ 𝑧 ∈ 𝑇)
61 eqid 2732 . . . . . . . . . . . 12 (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘ˆ)
62 eqid 2732 . . . . . . . . . . . 12 (+gβ€˜(Scalarβ€˜π‘ˆ)) = (+gβ€˜(Scalarβ€˜π‘ˆ))
635, 19, 54, 6, 61, 14, 62dvhopvadd 39959 . . . . . . . . . . 11 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘₯ ∈ 𝑇 ∧ 𝑂 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) ∧ (𝑧 ∈ 𝑇 ∧ 𝑂 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š))) β†’ (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘‚βŸ©) = ⟨(π‘₯ ∘ 𝑧), (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂)⟩)
6449, 53, 56, 60, 56, 63syl122anc 1379 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘‚βŸ©) = ⟨(π‘₯ ∘ 𝑧), (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂)⟩)
6564eqeq2d 2743 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘‚βŸ©) ↔ ⟨𝐹, π‘†βŸ© = ⟨(π‘₯ ∘ 𝑧), (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂)⟩))
66 vex 3478 . . . . . . . . . . . 12 π‘₯ ∈ V
67 vex 3478 . . . . . . . . . . . 12 𝑧 ∈ V
6866, 67coex 7920 . . . . . . . . . . 11 (π‘₯ ∘ 𝑧) ∈ V
69 ovex 7441 . . . . . . . . . . 11 (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂) ∈ V
7068, 69opth2 5480 . . . . . . . . . 10 (⟨𝐹, π‘†βŸ© = ⟨(π‘₯ ∘ 𝑧), (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂)⟩ ↔ (𝐹 = (π‘₯ ∘ 𝑧) ∧ 𝑆 = (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂)))
71 diblsmopel.v . . . . . . . . . . . . . . 15 𝑉 = ((DVecAβ€˜πΎ)β€˜π‘Š)
72 eqid 2732 . . . . . . . . . . . . . . 15 (+gβ€˜π‘‰) = (+gβ€˜π‘‰)
735, 19, 71, 72dvavadd 39881 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘₯ ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) β†’ (π‘₯(+gβ€˜π‘‰)𝑧) = (π‘₯ ∘ 𝑧))
7449, 53, 60, 73syl12anc 835 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (π‘₯(+gβ€˜π‘‰)𝑧) = (π‘₯ ∘ 𝑧))
7574eqeq2d 2743 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ↔ 𝐹 = (π‘₯ ∘ 𝑧)))
7675bicomd 222 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝐹 = (π‘₯ ∘ 𝑧) ↔ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)))
77 eqid 2732 . . . . . . . . . . . . . . . 16 (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))) = (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))
785, 19, 54, 6, 61, 77, 62dvhfplusr 39950 . . . . . . . . . . . . . . 15 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (+gβ€˜(Scalarβ€˜π‘ˆ)) = (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))))
7949, 78syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (+gβ€˜(Scalarβ€˜π‘ˆ)) = (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))))
8079oveqd 7425 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂) = (𝑂(𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))𝑂))
813, 5, 19, 54, 20, 77tendo0pl 39657 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑂 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š)) β†’ (𝑂(𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))𝑂) = 𝑂)
8249, 56, 81syl2anc 584 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝑂(𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘Š) ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))𝑂) = 𝑂)
8380, 82eqtrd 2772 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂) = 𝑂)
8483eqeq2d 2743 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝑆 = (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂) ↔ 𝑆 = 𝑂))
8576, 84anbi12d 631 . . . . . . . . . 10 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ ((𝐹 = (π‘₯ ∘ 𝑧) ∧ 𝑆 = (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂)) ↔ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂)))
8670, 85bitrid 282 . . . . . . . . 9 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (⟨𝐹, π‘†βŸ© = ⟨(π‘₯ ∘ 𝑧), (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂)⟩ ↔ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂)))
8765, 86bitrd 278 . . . . . . . 8 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘‚βŸ©) ↔ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂)))
8887pm5.32da 579 . . . . . . 7 (πœ‘ β†’ (((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘‚βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘‚βŸ©)) ↔ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂))))
8948, 88bitrid 282 . . . . . 6 (πœ‘ β†’ (βˆƒπ‘¦βˆƒπ‘€(𝑦 = 𝑂 ∧ 𝑀 = 𝑂 ∧ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©))) ↔ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂))))
9036, 89bitrd 278 . . . . 5 (πœ‘ β†’ (βˆƒπ‘¦βˆƒπ‘€((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂))))
9190exbidv 1924 . . . 4 (πœ‘ β†’ (βˆƒπ‘§βˆƒπ‘¦βˆƒπ‘€((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂))))
9218, 91bitrid 282 . . 3 (πœ‘ β†’ (βˆƒπ‘¦βˆƒπ‘§βˆƒπ‘€((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂))))
9392exbidv 1924 . 2 (πœ‘ β†’ (βˆƒπ‘₯βˆƒπ‘¦βˆƒπ‘§βˆƒπ‘€((⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ∧ βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ)) ∧ ⟨𝐹, π‘†βŸ© = (⟨π‘₯, π‘¦βŸ©(+gβ€˜π‘ˆ)βŸ¨π‘§, π‘€βŸ©)) ↔ βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂))))
94 anass 469 . . . . . 6 ((((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)) ∧ 𝑆 = 𝑂) ↔ ((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂)))
9594bicomi 223 . . . . 5 (((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂)) ↔ (((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)) ∧ 𝑆 = 𝑂))
96952exbii 1851 . . . 4 (βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂)) ↔ βˆƒπ‘₯βˆƒπ‘§(((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)) ∧ 𝑆 = 𝑂))
97 19.41vv 1954 . . . 4 (βˆƒπ‘₯βˆƒπ‘§(((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)) ∧ 𝑆 = 𝑂) ↔ (βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)) ∧ 𝑆 = 𝑂))
9896, 97bitri 274 . . 3 (βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂)) ↔ (βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)) ∧ 𝑆 = 𝑂))
995, 71dvalvec 39892 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑉 ∈ LVec)
100 lveclmod 20716 . . . . . . . . 9 (𝑉 ∈ LVec β†’ 𝑉 ∈ LMod)
101 eqid 2732 . . . . . . . . . 10 (LSubSpβ€˜π‘‰) = (LSubSpβ€˜π‘‰)
102101lsssssubg 20568 . . . . . . . . 9 (𝑉 ∈ LMod β†’ (LSubSpβ€˜π‘‰) βŠ† (SubGrpβ€˜π‘‰))
1031, 99, 100, 1024syl 19 . . . . . . . 8 (πœ‘ β†’ (LSubSpβ€˜π‘‰) βŠ† (SubGrpβ€˜π‘‰))
1043, 4, 5, 71, 21, 101dialss 39912 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (π½β€˜π‘‹) ∈ (LSubSpβ€˜π‘‰))
1051, 2, 104syl2anc 584 . . . . . . . 8 (πœ‘ β†’ (π½β€˜π‘‹) ∈ (LSubSpβ€˜π‘‰))
106103, 105sseldd 3983 . . . . . . 7 (πœ‘ β†’ (π½β€˜π‘‹) ∈ (SubGrpβ€˜π‘‰))
1073, 4, 5, 71, 21, 101dialss 39912 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ (π½β€˜π‘Œ) ∈ (LSubSpβ€˜π‘‰))
1081, 11, 107syl2anc 584 . . . . . . . 8 (πœ‘ β†’ (π½β€˜π‘Œ) ∈ (LSubSpβ€˜π‘‰))
109103, 108sseldd 3983 . . . . . . 7 (πœ‘ β†’ (π½β€˜π‘Œ) ∈ (SubGrpβ€˜π‘‰))
110 diblsmopel.q . . . . . . . 8 βŠ• = (LSSumβ€˜π‘‰)
11172, 110lsmelval 19516 . . . . . . 7 (((π½β€˜π‘‹) ∈ (SubGrpβ€˜π‘‰) ∧ (π½β€˜π‘Œ) ∈ (SubGrpβ€˜π‘‰)) β†’ (𝐹 ∈ ((π½β€˜π‘‹) βŠ• (π½β€˜π‘Œ)) ↔ βˆƒπ‘₯ ∈ (π½β€˜π‘‹)βˆƒπ‘§ ∈ (π½β€˜π‘Œ)𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)))
112106, 109, 111syl2anc 584 . . . . . 6 (πœ‘ β†’ (𝐹 ∈ ((π½β€˜π‘‹) βŠ• (π½β€˜π‘Œ)) ↔ βˆƒπ‘₯ ∈ (π½β€˜π‘‹)βˆƒπ‘§ ∈ (π½β€˜π‘Œ)𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)))
113 r2ex 3195 . . . . . 6 (βˆƒπ‘₯ ∈ (π½β€˜π‘‹)βˆƒπ‘§ ∈ (π½β€˜π‘Œ)𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ↔ βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)))
114112, 113bitrdi 286 . . . . 5 (πœ‘ β†’ (𝐹 ∈ ((π½β€˜π‘‹) βŠ• (π½β€˜π‘Œ)) ↔ βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧))))
115114anbi1d 630 . . . 4 (πœ‘ β†’ ((𝐹 ∈ ((π½β€˜π‘‹) βŠ• (π½β€˜π‘Œ)) ∧ 𝑆 = 𝑂) ↔ (βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)) ∧ 𝑆 = 𝑂)))
116115bicomd 222 . . 3 (πœ‘ β†’ ((βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ 𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧)) ∧ 𝑆 = 𝑂) ↔ (𝐹 ∈ ((π½β€˜π‘‹) βŠ• (π½β€˜π‘Œ)) ∧ 𝑆 = 𝑂)))
11798, 116bitrid 282 . 2 (πœ‘ β†’ (βˆƒπ‘₯βˆƒπ‘§((π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ)) ∧ (𝐹 = (π‘₯(+gβ€˜π‘‰)𝑧) ∧ 𝑆 = 𝑂)) ↔ (𝐹 ∈ ((π½β€˜π‘‹) βŠ• (π½β€˜π‘Œ)) ∧ 𝑆 = 𝑂)))
11817, 93, 1173bitrd 304 1 (πœ‘ β†’ (⟨𝐹, π‘†βŸ© ∈ ((πΌβ€˜π‘‹) ✚ (πΌβ€˜π‘Œ)) ↔ (𝐹 ∈ ((π½β€˜π‘‹) βŠ• (π½β€˜π‘Œ)) ∧ 𝑆 = 𝑂)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474   βŠ† wss 3948  βŸ¨cop 4634   class class class wbr 5148   ↦ cmpt 5231   I cid 5573   β†Ύ cres 5678   ∘ ccom 5680  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  Basecbs 17143  +gcplusg 17196  Scalarcsca 17199  lecple 17203  SubGrpcsubg 18999  LSSumclsm 19501  LModclmod 20470  LSubSpclss 20541  LVecclvec 20712  HLchlt 38215  LHypclh 38850  LTrncltrn 38967  TEndoctendo 39618  DVecAcdveca 39868  DIsoAcdia 39894  DVecHcdvh 39944  DIsoBcdib 40004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-riotaBAD 37818
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-tpos 8210  df-undef 8257  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-sca 17212  df-vsca 17213  df-0g 17386  df-proset 18247  df-poset 18265  df-plt 18282  df-lub 18298  df-glb 18299  df-join 18300  df-meet 18301  df-p0 18377  df-p1 18378  df-lat 18384  df-clat 18451  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-lsm 19503  df-cmn 19649  df-abl 19650  df-mgp 19987  df-ur 20004  df-ring 20057  df-oppr 20149  df-dvdsr 20170  df-unit 20171  df-invr 20201  df-dvr 20214  df-drng 20358  df-lmod 20472  df-lss 20542  df-lvec 20713  df-oposet 38041  df-ol 38043  df-oml 38044  df-covers 38131  df-ats 38132  df-atl 38163  df-cvlat 38187  df-hlat 38216  df-llines 38364  df-lplanes 38365  df-lvols 38366  df-lines 38367  df-psubsp 38369  df-pmap 38370  df-padd 38662  df-lhyp 38854  df-laut 38855  df-ldil 38970  df-ltrn 38971  df-trl 39025  df-tgrp 39609  df-tendo 39621  df-edring 39623  df-dveca 39869  df-disoa 39895  df-dvech 39945  df-dib 40005
This theorem is referenced by:  dib2dim  40109  dih2dimbALTN  40111
  Copyright terms: Public domain W3C validator