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Definition df-lcdual 40458
Description: Dual vector space of functionals with closed kernels. Note: we could also define this directly without mapd by using mapdrn 40520. TODO: see if it makes sense to go back and replace some of the LDual stuff with this. TODO: We could simplify df-mapd 40496 using (Baseβ€˜((LCDualβ€˜πΎ)β€˜π‘Š)). (Contributed by NM, 13-Mar-2015.)
Assertion
Ref Expression
df-lcdual LCDual = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ ((LDualβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ∣ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)})))
Distinct variable group:   𝑓,π‘˜,𝑀

Detailed syntax breakdown of Definition df-lcdual
StepHypRef Expression
1 clcd 40457 . 2 class LCDual
2 vk . . 3 setvar π‘˜
3 cvv 3475 . . 3 class V
4 vw . . . 4 setvar 𝑀
52cv 1541 . . . . 5 class π‘˜
6 clh 38855 . . . . 5 class LHyp
75, 6cfv 6544 . . . 4 class (LHypβ€˜π‘˜)
84cv 1541 . . . . . . 7 class 𝑀
9 cdvh 39949 . . . . . . . 8 class DVecH
105, 9cfv 6544 . . . . . . 7 class (DVecHβ€˜π‘˜)
118, 10cfv 6544 . . . . . 6 class ((DVecHβ€˜π‘˜)β€˜π‘€)
12 cld 37993 . . . . . 6 class LDual
1311, 12cfv 6544 . . . . 5 class (LDualβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))
14 vf . . . . . . . . . . 11 setvar 𝑓
1514cv 1541 . . . . . . . . . 10 class 𝑓
16 clk 37955 . . . . . . . . . . 11 class LKer
1711, 16cfv 6544 . . . . . . . . . 10 class (LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))
1815, 17cfv 6544 . . . . . . . . 9 class ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)
19 coch 40218 . . . . . . . . . . 11 class ocH
205, 19cfv 6544 . . . . . . . . . 10 class (ocHβ€˜π‘˜)
218, 20cfv 6544 . . . . . . . . 9 class ((ocHβ€˜π‘˜)β€˜π‘€)
2218, 21cfv 6544 . . . . . . . 8 class (((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))
2322, 21cfv 6544 . . . . . . 7 class (((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)))
2423, 18wceq 1542 . . . . . 6 wff (((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)
25 clfn 37927 . . . . . . 7 class LFnl
2611, 25cfv 6544 . . . . . 6 class (LFnlβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))
2724, 14, 26crab 3433 . . . . 5 class {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ∣ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)}
28 cress 17173 . . . . 5 class β†Ύs
2913, 27, 28co 7409 . . . 4 class ((LDualβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ∣ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)})
304, 7, 29cmpt 5232 . . 3 class (𝑀 ∈ (LHypβ€˜π‘˜) ↦ ((LDualβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ∣ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)}))
312, 3, 30cmpt 5232 . 2 class (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ ((LDualβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ∣ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)})))
321, 31wceq 1542 1 wff LCDual = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ ((LDualβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) β†Ύs {𝑓 ∈ (LFnlβ€˜((DVecHβ€˜π‘˜)β€˜π‘€)) ∣ (((ocHβ€˜π‘˜)β€˜π‘€)β€˜(((ocHβ€˜π‘˜)β€˜π‘€)β€˜((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“))) = ((LKerβ€˜((DVecHβ€˜π‘˜)β€˜π‘€))β€˜π‘“)})))
Colors of variables: wff setvar class
This definition is referenced by:  lcdfval  40459
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