Theorem mapdfval 38778

Description: Projectivity from vector space H to dual space. (Contributed by NM, 25-Jan-2015.)

Hypotheses

Ref	Expression
mapdval.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdval.s	⊢ 𝑆 = (LSubSp‘𝑈)
mapdval.f	⊢ 𝐹 = (LFnl‘𝑈)
mapdval.l	⊢ 𝐿 = (LKer‘𝑈)
mapdval.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
mapdval.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)

Assertion

Ref	Expression
mapdfval	⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}))

Distinct variable groups:   𝑓,𝑠,𝐾   𝑓,𝐹   𝑆,𝑠   𝑓,𝑊,𝑠

Allowed substitution hints:   𝑆(𝑓)   𝑈(𝑓,𝑠)   𝐹(𝑠)   𝐻(𝑓,𝑠)   𝐿(𝑓,𝑠)   𝑀(𝑓,𝑠)   𝑂(𝑓,𝑠)   𝑋(𝑓,𝑠)

Proof of Theorem mapdfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdval.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
2		mapdval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3	2	mapdffval 38777	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (mapd‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)})))
4	3	fveq1d 6672	. . 3 ⊢ (𝐾 ∈ 𝑋 → ((mapd‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)}))‘𝑊))
5	1, 4	syl5eq 2868	. 2 ⊢ (𝐾 ∈ 𝑋 → 𝑀 = ((𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)}))‘𝑊))
6		fveq2 6670	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = ((DVecH‘𝐾)‘𝑊))
7		mapdval.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8	6, 7	syl6eqr 2874	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((DVecH‘𝐾)‘𝑤) = 𝑈)
9	8	fveq2d 6674	. . . . 5 ⊢ (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = (LSubSp‘𝑈))
10		mapdval.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑈)
11	9, 10	syl6eqr 2874	. . . 4 ⊢ (𝑤 = 𝑊 → (LSubSp‘((DVecH‘𝐾)‘𝑤)) = 𝑆)
12	8	fveq2d 6674	. . . . . 6 ⊢ (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = (LFnl‘𝑈))
13		mapdval.f	. . . . . 6 ⊢ 𝐹 = (LFnl‘𝑈)
14	12, 13	syl6eqr 2874	. . . . 5 ⊢ (𝑤 = 𝑊 → (LFnl‘((DVecH‘𝐾)‘𝑤)) = 𝐹)
15		fveq2 6670	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
16		mapdval.o	. . . . . . . . 9 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
17	15, 16	syl6eqr 2874	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = 𝑂)
18	8	fveq2d 6674	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = (LKer‘𝑈))
19		mapdval.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LKer‘𝑈)
20	18, 19	syl6eqr 2874	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (LKer‘((DVecH‘𝐾)‘𝑤)) = 𝐿)
21	20	fveq1d 6672	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) = (𝐿‘𝑓))
22	17, 21	fveq12d 6677	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) = (𝑂‘(𝐿‘𝑓)))
23	17, 22	fveq12d 6677	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = (𝑂‘(𝑂‘(𝐿‘𝑓))))
24	23, 21	eqeq12d 2837	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ↔ (𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓)))
25	22	sseq1d 3998	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠 ↔ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠))
26	24, 25	anbi12d 632	. . . . 5 ⊢ (𝑤 = 𝑊 → (((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠) ↔ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)))
27	14, 26	rabeqbidv 3485	. . . 4 ⊢ (𝑤 = 𝑊 → {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)} = {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)})
28	11, 27	mpteq12dv 5151	. . 3 ⊢ (𝑤 = 𝑊 → (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)}) = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}))
29		eqid 2821	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)})) = (𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)}))
30	28, 29, 10	mptfvmpt 6990	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑠 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤)) ↦ {𝑓 ∈ (LFnl‘((DVecH‘𝐾)‘𝑤)) ∣ ((((ocH‘𝐾)‘𝑤)‘(((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓))) = ((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓) ∧ (((ocH‘𝐾)‘𝑤)‘((LKer‘((DVecH‘𝐾)‘𝑤))‘𝑓)) ⊆ 𝑠)}))‘𝑊) = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}))
31	5, 30	sylan9eq 2876	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑀 = (𝑠 ∈ 𝑆 ↦ {𝑓 ∈ 𝐹 ∣ ((𝑂‘(𝑂‘(𝐿‘𝑓))) = (𝐿‘𝑓) ∧ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑠)}))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {crab 3142   ⊆ wss 3936   ↦ cmpt 5146  ‘cfv 6355  LSubSpclss 19703  LFnlclfn 36208  LKerclk 36236  LHypclh 37135  DVecHcdvh 38229  ocHcoch 38498  mapdcmpd 38775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-mapd 38776

This theorem is referenced by:  mapdval  38779  mapd1o  38799