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Theorem diblsmopel 39684
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 2733 . . . . 5 (LSubSpβ€˜π‘ˆ) = (LSubSpβ€˜π‘ˆ)
93, 4, 5, 6, 7, 8diblss 39683 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (πΌβ€˜π‘‹) ∈ (LSubSpβ€˜π‘ˆ))
101, 2, 9syl2anc 585 . . 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 39683 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ (πΌβ€˜π‘Œ) ∈ (LSubSpβ€˜π‘ˆ))
131, 11, 12syl2anc 585 . . 3 (πœ‘ β†’ (πΌβ€˜π‘Œ) ∈ (LSubSpβ€˜π‘ˆ))
14 eqid 2733 . . . 4 (+gβ€˜π‘ˆ) = (+gβ€˜π‘ˆ)
15 diblsmopel.p . . . 4 ✚ = (LSSumβ€˜π‘ˆ)
165, 6, 14, 8, 15dvhopellsm 39630 . . 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 1372 . 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 2163 . . . 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 39658 . . . . . . . . . . . 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 585 . . . . . . . . . . 11 (πœ‘ β†’ (⟨π‘₯, π‘¦βŸ© ∈ (πΌβ€˜π‘‹) ↔ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑦 = 𝑂)))
243, 4, 5, 19, 20, 21, 7dibopelval2 39658 . . . . . . . . . . . 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 585 . . . . . . . . . . 11 (πœ‘ β†’ (βŸ¨π‘§, π‘€βŸ© ∈ (πΌβ€˜π‘Œ) ↔ (𝑧 ∈ (π½β€˜π‘Œ) ∧ 𝑀 = 𝑂)))
2623, 25anbi12d 632 . . . . . . . . . 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 655 . . . . . . . . . . 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 462 . . . . . . . . . . 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 275 . . . . . . . . . 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 287 . . . . . . . . 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 631 . . . . . . . 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 470 . . . . . . . . 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 1090 . . . . . . . . 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 278 . . . . . . . 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 287 . . . . . . 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 1928 . . . . . 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 6860 . . . . . . . . . 10 𝑇 ∈ V
3837mptex 7177 . . . . . . . . 9 (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡)) ∈ V
3920, 38eqeltri 2830 . . . . . . . 8 𝑂 ∈ V
40 opeq2 4835 . . . . . . . . . . 11 (𝑦 = 𝑂 β†’ ⟨π‘₯, π‘¦βŸ© = ⟨π‘₯, π‘‚βŸ©)
4140oveq1d 7376 . . . . . . . . . 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 2744 . . . . . . . . 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 630 . . . . . . . 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 4835 . . . . . . . . . . 11 (𝑀 = 𝑂 β†’ βŸ¨π‘§, π‘€βŸ© = βŸ¨π‘§, π‘‚βŸ©)
4544oveq2d 7377 . . . . . . . . . 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 2744 . . . . . . . . 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 630 . . . . . . . 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 3501 . . . . . . 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 482 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
502adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š))
51 simprl 770 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ π‘₯ ∈ (π½β€˜π‘‹))
523, 4, 5, 19, 21diael 39556 . . . . . . . . . . . 12 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š) ∧ π‘₯ ∈ (π½β€˜π‘‹)) β†’ π‘₯ ∈ 𝑇)
5349, 50, 51, 52syl3anc 1372 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ π‘₯ ∈ 𝑇)
54 eqid 2733 . . . . . . . . . . . . 13 ((TEndoβ€˜πΎ)β€˜π‘Š) = ((TEndoβ€˜πΎ)β€˜π‘Š)
553, 5, 19, 54, 20tendo0cl 39303 . . . . . . . . . . . 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 482 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š))
58 simprr 772 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ 𝑧 ∈ (π½β€˜π‘Œ))
593, 4, 5, 19, 21diael 39556 . . . . . . . . . . . 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 1372 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ 𝑧 ∈ 𝑇)
61 eqid 2733 . . . . . . . . . . . 12 (Scalarβ€˜π‘ˆ) = (Scalarβ€˜π‘ˆ)
62 eqid 2733 . . . . . . . . . . . 12 (+gβ€˜(Scalarβ€˜π‘ˆ)) = (+gβ€˜(Scalarβ€˜π‘ˆ))
635, 19, 54, 6, 61, 14, 62dvhopvadd 39606 . . . . . . . . . . 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 1380 . . . . . . . . . 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 2744 . . . . . . . . 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 3451 . . . . . . . . . . . 12 π‘₯ ∈ V
67 vex 3451 . . . . . . . . . . . 12 𝑧 ∈ V
6866, 67coex 7871 . . . . . . . . . . 11 (π‘₯ ∘ 𝑧) ∈ V
69 ovex 7394 . . . . . . . . . . 11 (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂) ∈ V
7068, 69opth2 5441 . . . . . . . . . 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 2733 . . . . . . . . . . . . . . 15 (+gβ€˜π‘‰) = (+gβ€˜π‘‰)
735, 19, 71, 72dvavadd 39528 . . . . . . . . . . . . . 14 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘₯ ∈ 𝑇 ∧ 𝑧 ∈ 𝑇)) β†’ (π‘₯(+gβ€˜π‘‰)𝑧) = (π‘₯ ∘ 𝑧))
7449, 53, 60, 73syl12anc 836 . . . . . . . . . . . . 13 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (π‘₯(+gβ€˜π‘‰)𝑧) = (π‘₯ ∘ 𝑧))
7574eqeq2d 2744 . . . . . . . . . . . 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 2733 . . . . . . . . . . . . . . . 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 39597 . . . . . . . . . . . . . . 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 7378 . . . . . . . . . . . . 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 39304 . . . . . . . . . . . . . 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 585 . . . . . . . . . . . . 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 2773 . . . . . . . . . . . 12 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂) = 𝑂)
8483eqeq2d 2744 . . . . . . . . . . 11 ((πœ‘ ∧ (π‘₯ ∈ (π½β€˜π‘‹) ∧ 𝑧 ∈ (π½β€˜π‘Œ))) β†’ (𝑆 = (𝑂(+gβ€˜(Scalarβ€˜π‘ˆ))𝑂) ↔ 𝑆 = 𝑂))
8576, 84anbi12d 632 . . . . . . . . . 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 283 . . . . . . . . 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 279 . . . . . . . 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 580 . . . . . . 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 283 . . . . . 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 279 . . . . 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 1925 . . . 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 283 . . 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 1925 . 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 470 . . . . . 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 1852 . . . 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 1955 . . . 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 275 . . 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 39539 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑉 ∈ LVec)
100 lveclmod 20611 . . . . . . . . 9 (𝑉 ∈ LVec β†’ 𝑉 ∈ LMod)
101 eqid 2733 . . . . . . . . . 10 (LSubSpβ€˜π‘‰) = (LSubSpβ€˜π‘‰)
102101lsssssubg 20463 . . . . . . . . 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 39559 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑋 ∈ 𝐡 ∧ 𝑋 ≀ π‘Š)) β†’ (π½β€˜π‘‹) ∈ (LSubSpβ€˜π‘‰))
1051, 2, 104syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (π½β€˜π‘‹) ∈ (LSubSpβ€˜π‘‰))
106103, 105sseldd 3949 . . . . . . 7 (πœ‘ β†’ (π½β€˜π‘‹) ∈ (SubGrpβ€˜π‘‰))
1073, 4, 5, 71, 21, 101dialss 39559 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘Œ ∈ 𝐡 ∧ π‘Œ ≀ π‘Š)) β†’ (π½β€˜π‘Œ) ∈ (LSubSpβ€˜π‘‰))
1081, 11, 107syl2anc 585 . . . . . . . 8 (πœ‘ β†’ (π½β€˜π‘Œ) ∈ (LSubSpβ€˜π‘‰))
109103, 108sseldd 3949 . . . . . . 7 (πœ‘ β†’ (π½β€˜π‘Œ) ∈ (SubGrpβ€˜π‘‰))
110 diblsmopel.q . . . . . . . 8 βŠ• = (LSSumβ€˜π‘‰)
11172, 110lsmelval 19439 . . . . . . 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 585 . . . . . 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 3189 . . . . . 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 287 . . . . 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 631 . . . 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 283 . 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 305 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒwrex 3070  Vcvv 3447   βŠ† wss 3914  βŸ¨cop 4596   class class class wbr 5109   ↦ cmpt 5192   I cid 5534   β†Ύ cres 5639   ∘ ccom 5641  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Basecbs 17091  +gcplusg 17141  Scalarcsca 17144  lecple 17148  SubGrpcsubg 18930  LSSumclsm 19424  LModclmod 20365  LSubSpclss 20436  LVecclvec 20607  HLchlt 37862  LHypclh 38497  LTrncltrn 38614  TEndoctendo 39265  DVecAcdveca 39515  DIsoAcdia 39541  DVecHcdvh 39591  DIsoBcdib 39651
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-riotaBAD 37465
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-tpos 8161  df-undef 8208  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-mulr 17155  df-sca 17157  df-vsca 17158  df-0g 17331  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-p1 18323  df-lat 18329  df-clat 18396  df-mgm 18505  df-sgrp 18554  df-mnd 18565  df-grp 18759  df-minusg 18760  df-sbg 18761  df-subg 18933  df-lsm 19426  df-cmn 19572  df-abl 19573  df-mgp 19905  df-ur 19922  df-ring 19974  df-oppr 20057  df-dvdsr 20078  df-unit 20079  df-invr 20109  df-dvr 20120  df-drng 20221  df-lmod 20367  df-lss 20437  df-lvec 20608  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-llines 38011  df-lplanes 38012  df-lvols 38013  df-lines 38014  df-psubsp 38016  df-pmap 38017  df-padd 38309  df-lhyp 38501  df-laut 38502  df-ldil 38617  df-ltrn 38618  df-trl 38672  df-tgrp 39256  df-tendo 39268  df-edring 39270  df-dveca 39516  df-disoa 39542  df-dvech 39592  df-dib 39652
This theorem is referenced by:  dib2dim  39756  dih2dimbALTN  39758
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