Theorem djhfval 39390

Description: Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)

Hypotheses

Ref	Expression
djhval.h	⊢ 𝐻 = (LHyp‘𝐾)
djhval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
djhval.v	⊢ 𝑉 = (Base‘𝑈)
djhval.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
djhval.j	⊢ ∨ = ((joinH‘𝐾)‘𝑊)

Assertion

Ref	Expression
djhfval	⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))

Distinct variable groups:   𝑥,𝑦,𝐾   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐻(𝑥,𝑦)   ∨ (𝑥,𝑦)   ⊥ (𝑥,𝑦)   𝑋(𝑥,𝑦)

Proof of Theorem djhfval

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		djhval.j	. . 3 ⊢ ∨ = ((joinH‘𝐾)‘𝑊)
2		djhval.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3	2	djhffval 39389	. . . 4 ⊢ (𝐾 ∈ 𝑋 → (joinH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))))
4	3	fveq1d 6770	. . 3 ⊢ (𝐾 ∈ 𝑋 → ((joinH‘𝐾)‘𝑊) = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))‘𝑊))
5	1, 4	eqtrid 2791	. 2 ⊢ (𝐾 ∈ 𝑋 → ∨ = ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))‘𝑊))
6		2fveq3 6773	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑊)))
7		djhval.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
8		djhval.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
9	8	fveq2i 6771	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘((DVecH‘𝐾)‘𝑊))
10	7, 9	eqtri 2767	. . . . . 6 ⊢ 𝑉 = (Base‘((DVecH‘𝐾)‘𝑊))
11	6, 10	eqtr4di 2797	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
12	11	pweqd 4557	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
13		fveq2 6768	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
14		djhval.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
15	13, 14	eqtr4di 2797	. . . . 5 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ⊥ )
16	15	fveq1d 6770	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑥) = ( ⊥ ‘𝑥))
17	15	fveq1d 6770	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑦) = ( ⊥ ‘𝑦))
18	16, 17	ineq12d 4152	. . . . 5 ⊢ (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)) = (( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))
19	15, 18	fveq12d 6775	. . . 4 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))
20	12, 12, 19	mpoeq123dv 7341	. . 3 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
21		eqid 2739	. . 3 ⊢ (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))
22	7	fvexi 6782	. . . . 5 ⊢ 𝑉 ∈ V
23	22	pwex 5306	. . . 4 ⊢ 𝒫 𝑉 ∈ V
24	23, 23	mpoex 7906	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))) ∈ V
25	20, 21, 24	fvmpt 6869	. 2 ⊢ (𝑊 ∈ 𝐻 → ((𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))‘𝑊) = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
26	5, 25	sylan9eq 2799	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2109   ∩ cin 3890  𝒫 cpw 4538   ↦ cmpt 5161  ‘cfv 6430   ∈ cmpo 7270  Basecbs 16893  LHypclh 37977  DVecHcdvh 39071  ocHcoch 39340  joinHcdjh 39387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-djh 39388

This theorem is referenced by:  djhval  39391