Theorem djhfval 40902

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 40901	. . . 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 6904	. . 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 2780	. 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 6907	. . . . . 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 6905	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘((DVecH‘𝐾)‘𝑊))
10	7, 9	eqtri 2756	. . . . . 6 ⊢ 𝑉 = (Base‘((DVecH‘𝐾)‘𝑊))
11	6, 10	eqtr4di 2786	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
12	11	pweqd 4623	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
13		fveq2 6902	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
14		djhval.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
15	13, 14	eqtr4di 2786	. . . . 5 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ⊥ )
16	15	fveq1d 6904	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑥) = ( ⊥ ‘𝑥))
17	15	fveq1d 6904	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑦) = ( ⊥ ‘𝑦))
18	16, 17	ineq12d 4215	. . . . 5 ⊢ (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)) = (( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))
19	15, 18	fveq12d 6909	. . . 4 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))
20	12, 12, 19	mpoeq123dv 7501	. . 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 2728	. . 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 6916	. . . . 5 ⊢ 𝑉 ∈ V
23	22	pwex 5384	. . . 4 ⊢ 𝒫 𝑉 ∈ V
24	23, 23	mpoex 8090	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))) ∈ V
25	20, 21, 24	fvmpt 7010	. 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 2788	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∩ cin 3948  𝒫 cpw 4606   ↦ cmpt 5235  ‘cfv 6553   ∈ cmpo 7428  Basecbs 17187  LHypclh 39489  DVecHcdvh 40583  ocHcoch 40852  joinHcdjh 40899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-djh 40900

This theorem is referenced by:  djhval  40903