Theorem djhfval 41982

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 41981	. . . 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 6864	. . 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 2808	. 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 6867	. . . . . 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 6865	. . . . . . 7 ⊢ (Base‘𝑈) = (Base‘((DVecH‘𝐾)‘𝑊))
10	7, 9	eqtri 2784	. . . . . 6 ⊢ 𝑉 = (Base‘((DVecH‘𝐾)‘𝑊))
11	6, 10	eqtr4di 2814	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘((DVecH‘𝐾)‘𝑤)) = 𝑉)
12	11	pweqd 4569	. . . 4 ⊢ (𝑤 = 𝑊 → 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) = 𝒫 𝑉)
13		fveq2 6862	. . . . . 6 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ((ocH‘𝐾)‘𝑊))
14		djhval.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
15	13, 14	eqtr4di 2814	. . . . 5 ⊢ (𝑤 = 𝑊 → ((ocH‘𝐾)‘𝑤) = ⊥ )
16	15	fveq1d 6864	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑥) = ( ⊥ ‘𝑥))
17	15	fveq1d 6864	. . . . . 6 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘𝑦) = ( ⊥ ‘𝑦))
18	16, 17	ineq12d 4171	. . . . 5 ⊢ (𝑤 = 𝑊 → ((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)) = (( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))
19	15, 18	fveq12d 6869	. . . 4 ⊢ (𝑤 = 𝑊 → (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))) = ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))
20	12, 12, 19	mpoeq123dv 7466	. . 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 2761	. . 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 6876	. . . . 5 ⊢ 𝑉 ∈ V
23	22	pwex 5334	. . . 4 ⊢ 𝒫 𝑉 ∈ V
24	23, 23	mpoex 8055	. . 3 ⊢ (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))) ∈ V
25	20, 21, 24	fvmpt 6970	. 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 2816	1 ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∩ cin 3901  𝒫 cpw 4552   ↦ cmpt 5178  ‘cfv 6516   ∈ cmpo 7393  Basecbs 17236  LHypclh 40569  DVecHcdvh 41663  ocHcoch 41932  joinHcdjh 41979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-djh 41980

This theorem is referenced by:  djhval  41983