P. 427 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42601-42700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hdmaprnlem1N 42601	Part of proof of part 12 in [Baer] p. 49 line 10, Gu' ≠ Gs. Our (𝑁‘{𝑣}) is Baer's T. (Contributed by NM, 26-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    ⇒   ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑢)}) ≠ (𝐿‘{𝑠}))
Theorem	hdmaprnlem3N 42602	Part of proof of part 12 in [Baer] p. 49 line 15, T ≠ P. Our (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑣}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
Theorem	hdmaprnlem3uN 42603	Part of proof of part 12 in [Baer] p. 49. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑢}) ≠ (◡𝑀‘(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)})))
Theorem	hdmaprnlem4tN 42604	Lemma for hdmaprnN 42616. TODO: This lemma doesn't quite pay for itself even though used six times. Maybe prove this directly instead. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    ⇒   ⊢ (𝜑 → 𝑡 ∈ 𝑉)
Theorem	hdmaprnlem4N 42605	Part of proof of part 12 in [Baer] p. 49 line 19. (T* =) (Ft)* = Gs. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{𝑠}))
Theorem	hdmaprnlem6N 42606	Part of proof of part 12 in [Baer] p. 49 line 18, G(u'+s) = G(u'+t). (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))    ⇒   ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝐿‘{((𝑆‘𝑢) ✚ (𝑆‘𝑡))}))
Theorem	hdmaprnlem7N 42607	Part of proof of part 12 in [Baer] p. 49 line 19, s-St ∈ G(u'+s) = P*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))    ⇒   ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}))
Theorem	hdmaprnlem8N 42608	Part of proof of part 12 in [Baer] p. 49 line 19, s-St ∈ (Ft)* = T*. (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))    ⇒   ⊢ (𝜑 → (𝑠(-g‘𝐶)(𝑆‘𝑡)) ∈ (𝑀‘(𝑁‘{𝑡})))
Theorem	hdmaprnlem9N 42609	Part of proof of part 12 in [Baer] p. 49 line 21, s=S(t). TODO: we seem to be going back and forth with mapd11 42391 and mapdcnv11N 42411. Use better hypotheses and/or theorems? (Contributed by NM, 27-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ (𝜑 → 𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 }))    &   ⊢ + = (+g‘𝑈)    &   ⊢ (𝜑 → (𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))    ⇒   ⊢ (𝜑 → 𝑠 = (𝑆‘𝑡))
Theorem	hdmaprnlem3eN 42610*	Lemma for hdmaprnN 42616. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ + = (+g‘𝑈)    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ ((𝑁‘{𝑣}) ∖ { 0 })(𝐿‘{((𝑆‘𝑢) ✚ 𝑠)}) = (𝑀‘(𝑁‘{(𝑢 + 𝑡)})))
Theorem	hdmaprnlem10N 42611*	Lemma for hdmaprnN 42616. Show 𝑠 is in the range of 𝑆. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ + = (+g‘𝑈)    ⇒   ⊢ (𝜑 → ∃𝑡 ∈ 𝑉 (𝑆‘𝑡) = 𝑠)
Theorem	hdmaprnlem11N 42612*	Lemma for hdmaprnN 42616. Show 𝑠 is in the range of 𝑆. (Contributed by NM, 29-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ {𝑄}))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    &   ⊢ (𝜑 → 𝑢 ∈ 𝑉)    &   ⊢ (𝜑 → ¬ 𝑢 ∈ (𝑁‘{𝑣}))    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ + = (+g‘𝑈)    ⇒   ⊢ (𝜑 → 𝑠 ∈ ran 𝑆)
Theorem	hdmaprnlem15N 42613*	Lemma for hdmaprnN 42616. Eliminate 𝑢. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 0 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑣 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑣})) = (𝐿‘{𝑠}))    ⇒   ⊢ (𝜑 → 𝑠 ∈ ran 𝑆)
Theorem	hdmaprnlem16N 42614	Lemma for hdmaprnN 42616. Eliminate 𝑣. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 0 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ (𝐷 ∖ { 0 }))    ⇒   ⊢ (𝜑 → 𝑠 ∈ ran 𝑆)
Theorem	hdmaprnlem17N 42615	Lemma for hdmaprnN 42616. Include zero. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 0 = (0g‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑠 ∈ 𝐷)    ⇒   ⊢ (𝜑 → 𝑠 ∈ ran 𝑆)
Theorem	hdmaprnN 42616	Part of proof of part 12 in [Baer] p. 49 line 21, As=B. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ran 𝑆 = 𝐷)
Theorem	hdmapf1oN 42617	Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 42595, this shows the map is an automorphism from the additive group of vectors to the additive group of functionals with closed kernels. (Contributed by NM, 30-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑆:𝑉–1-1-onto→𝐷)
Theorem	hdmap14lem1a 42618	Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐹 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))}))
Theorem	hdmap14lem2a 42619*	Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 0 so it can be used in hdmap14lem10 42629. (Contributed by NM, 31-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
Theorem	hdmap14lem1 42620	Prior to part 14 in [Baer] p. 49, line 25. (Contributed by NM, 31-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑄 = (0g‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))    ⇒   ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑋)}) = (𝐿‘{(𝑆‘(𝐹 · 𝑋))}))
Theorem	hdmap14lem2N 42621*	Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 𝑍 so it can be used in hdmap14lem10 42629. (Contributed by NM, 31-May-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑄 = (0g‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))    ⇒   ⊢ (𝜑 → ∃𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
Theorem	hdmap14lem3 42622*	Prior to part 14 in [Baer] p. 49, line 26. (Contributed by NM, 31-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑄 = (0g‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))    ⇒   ⊢ (𝜑 → ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
Theorem	hdmap14lem4a 42623*	Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 42622 to provide a slightly simpler definition later. (Contributed by NM, 31-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑄 = (0g‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))    ⇒   ⊢ (𝜑 → (∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))))
Theorem	hdmap14lem4 42624*	Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 42622 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 42623 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 42623 into this one. (Contributed by NM, 31-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑄 = (0g‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍}))    ⇒   ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
Theorem	hdmap14lem6 42625*	Case where 𝐹 is zero. (Contributed by NM, 1-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑍 = (0g‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝐿 = (LSpan‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑄 = (0g‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 = 𝑍)    ⇒   ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
Theorem	hdmap14lem7 42626*	Combine cases of 𝐹. TODO: Can this be done at once in hdmap14lem3 42622, in order to get rid of hdmap14lem6 42625? Perhaps modify lspsneu 21228 to become ∃!𝑘 ∈ 𝐾 instead of ∃!𝑘 ∈ (𝐾 ∖ { 0 })? (Contributed by NM, 1-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))
Theorem	hdmap14lem8 42627	Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))    &   ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌)))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → 𝐽 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))))    ⇒   ⊢ (𝜑 → ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
Theorem	hdmap14lem9 42628	Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 1-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))    &   ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌)))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    &   ⊢ (𝜑 → 𝐽 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))))    ⇒   ⊢ (𝜑 → 𝐺 = 𝐼)
Theorem	hdmap14lem10 42629	Part of proof of part 14 in [Baer] p. 49 line 38. (Contributed by NM, 3-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))    &   ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌)))    &   ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))    ⇒   ⊢ (𝜑 → 𝐺 = 𝐼)
Theorem	hdmap14lem11 42630	Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 3-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ✚ = (+g‘𝐶)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐼 ∈ 𝐴)    &   ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))    &   ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌)))    ⇒   ⊢ (𝜑 → 𝐺 = 𝐼)
Theorem	hdmap14lem12 42631*	Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ (𝑉 ∖ { 0 })(𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
Theorem	hdmap14lem13 42632*	Lemma for proof of part 14 in [Baer] p. 50. (Contributed by NM, 6-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐴)    ⇒   ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)) ↔ ∀𝑦 ∈ 𝑉 (𝑆‘(𝐹 · 𝑦)) = (𝐺 ∙ (𝑆‘𝑦))))
Theorem	hdmap14lem14 42633*	Part of proof of part 14 in [Baer] p. 50 line 3. (Contributed by NM, 6-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    ⇒   ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)))
Theorem	hdmap14lem15 42634*	Part of proof of part 14 in [Baer] p. 50 line 3. Convert scalar base of dual to scalar base of vector space. (Contributed by NM, 6-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ∃!𝑔 ∈ 𝐵 ∀𝑥 ∈ 𝑉 (𝑆‘(𝐹 · 𝑥)) = (𝑔 ∙ (𝑆‘𝑥)))
Syntax	chg 42635	Extend class notation with g-map.
class HGMap
Definition	df-hgmap 42636*	Define map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
⊢ HGMap = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑎 ∣ [((DVecH‘𝑘)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝑘)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠 ‘𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝑘)‘𝑤))(𝑚‘𝑣))))}))
Theorem	hgmapffval 42637*	Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑋 → (HGMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑎 ∣ [((DVecH‘𝐾)‘𝑤) / 𝑢][(Base‘(Scalar‘𝑢)) / 𝑏][((HDMap‘𝐾)‘𝑤) / 𝑚]𝑎 ∈ (𝑥 ∈ 𝑏 ↦ (℩𝑦 ∈ 𝑏 ∀𝑣 ∈ (Base‘𝑢)(𝑚‘(𝑥( ·𝑠 ‘𝑢)𝑣)) = (𝑦( ·𝑠 ‘((LCDual‘𝐾)‘𝑤))(𝑚‘𝑣))))}))
Theorem	hgmapfval 42638*	Map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. (Contributed by NM, 25-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐼 = (𝑥 ∈ 𝐵 ↦ (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑥 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣)))))
Theorem	hgmapval 42639*	Value of map from the scalar division ring of the vector space to the scalar division ring of its closed kernel dual. Function sigma of scalar f in part 14 of [Baer] p. 50 line 4. TODO: variable names are inherited from older version. Maybe make more consistent with hdmap14lem15 42634. (Contributed by NM, 25-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑀 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) = (℩𝑦 ∈ 𝐵 ∀𝑣 ∈ 𝑉 (𝑀‘(𝑋 · 𝑣)) = (𝑦 ∙ (𝑀‘𝑣))))
Theorem	hgmapfnN 42640	Functionality of scalar sigma map. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐺 Fn 𝐵)
Theorem	hgmapcl 42641	Closure of scalar sigma map i.e. the map from the vector space scalar base to the dual space scalar base. (Contributed by NM, 6-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐵)
Theorem	hgmapdcl 42642	Closure of the vector space to dual space scalar map, in the scalar sigma map. (Contributed by NM, 6-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝑄 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑄)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘𝐹) ∈ 𝐴)
Theorem	hgmapvs 42643	Part 15 of [Baer] p. 50 line 6. Also line 15 in [Holland95] p. 14. (Contributed by NM, 6-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = ((𝐺‘𝐹) ∙ (𝑆‘𝑋)))
Theorem	hgmapval0 42644	Value of the scalar sigma map at zero. (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → (𝐺‘ 0 ) = 0 )
Theorem	hgmapval1 42645	Value of the scalar sigma map at one. (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → (𝐺‘ 1 ) = 1 )
Theorem	hgmapadd 42646	Part 15 of [Baer] p. 50 line 13. (Contributed by NM, 6-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘(𝑋 + 𝑌)) = ((𝐺‘𝑋) + (𝐺‘𝑌)))
Theorem	hgmapmul 42647	Part 15 of [Baer] p. 50 line 16. The multiplication is reversed after converting to the dual space scalar to the vector space scalar. (Contributed by NM, 7-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘(𝑋 · 𝑌)) = ((𝐺‘𝑌) · (𝐺‘𝑋)))
Theorem	hgmaprnlem1N 42648	Lemma for hgmaprnN 42653. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑧 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑠 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))    &   ⊢ (𝜑 → 𝑘 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑠 = (𝑘 · 𝑡))    ⇒   ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)
Theorem	hgmaprnlem2N 42649	Lemma for hgmaprnN 42653. Part 15 of [Baer] p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero 𝑧 is taken care of automatically. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑧 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑠 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LSpan‘𝐶)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡}))
Theorem	hgmaprnlem3N 42650*	Lemma for hgmaprnN 42653. Eliminate 𝑘. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑧 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑠 ∈ 𝑉)    &   ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))    &   ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐿 = (LSpan‘𝐶)    ⇒   ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)
Theorem	hgmaprnlem4N 42651*	Lemma for hgmaprnN 42653. Eliminate 𝑠. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑧 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)
Theorem	hgmaprnlem5N 42652	Lemma for hgmaprnN 42653. Eliminate 𝑡. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Base‘𝐶)    &   ⊢ 𝑃 = (Scalar‘𝐶)    &   ⊢ 𝐴 = (Base‘𝑃)    &   ⊢ ∙ = ( ·𝑠 ‘𝐶)    &   ⊢ 𝑄 = (0g‘𝐶)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑧 ∈ 𝐴)    ⇒   ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)
Theorem	hgmaprnN 42653	Part of proof of part 16 in [Baer] p. 50 line 23, Fs=G, except that we use the original vector space scalars for the range. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ran 𝐺 = 𝐵)
Theorem	hgmap11 42654	The scalar sigma map is one-to-one. (Contributed by NM, 7-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐺‘𝑋) = (𝐺‘𝑌) ↔ 𝑋 = 𝑌))
Theorem	hgmapf1oN 42655	The scalar sigma map is a one-to-one onto function. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐺:𝐵–1-1-onto→𝐵)
Theorem	hgmapeq0 42656	The scalar sigma map is zero iff its argument is zero. (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐺‘𝑋) = 0 ↔ 𝑋 = 0 ))
Theorem	hdmapipcl 42657	The inner product (Hermitian form) (𝑋, 𝑌) will be defined as ((𝑆‘𝑌)‘𝑋). Show closure. (Contributed by NM, 7-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑌)‘𝑋) ∈ 𝐵)
Theorem	hdmapln1 42658	Linearity property that will be used for inner product. TODO: try to combine hypotheses in hdmap*ln* series. (Contributed by NM, 7-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⨣ = (+g‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑍)‘((𝐴 · 𝑋) + 𝑌)) = ((𝐴 × ((𝑆‘𝑍)‘𝑋)) ⨣ ((𝑆‘𝑍)‘𝑌)))
Theorem	hdmaplna1 42659	Additive property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ ⨣ = (+g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑍)‘(𝑋 + 𝑌)) = (((𝑆‘𝑍)‘𝑋) ⨣ ((𝑆‘𝑍)‘𝑌)))
Theorem	hdmaplns1 42660	Subtraction property of first (inner product) argument. (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝑁 = (-g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑍)‘(𝑋 − 𝑌)) = (((𝑆‘𝑍)‘𝑋)𝑁((𝑆‘𝑍)‘𝑌)))
Theorem	hdmaplnm1 42661	Multiplicative property of first (inner product) argument. (Contributed by NM, 11-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑌)‘(𝐴 · 𝑋)) = (𝐴 × ((𝑆‘𝑌)‘𝑋)))
Theorem	hdmaplna2 42662	Additive property of second (inner product) argument. (Contributed by NM, 10-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ ⨣ = (+g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑆‘(𝑌 + 𝑍))‘𝑋) = (((𝑆‘𝑌)‘𝑋) ⨣ ((𝑆‘𝑍)‘𝑋)))
Theorem	hdmapglnm2 42663	g-linear property of second (inner product) argument. Line 19 in [Holland95] p. 14. (Contributed by NM, 10-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑆‘(𝐴 · 𝑌))‘𝑋) = (((𝑆‘𝑌)‘𝑋) × (𝐺‘𝐴)))
Theorem	hdmapgln2 42664	g-linear property that will be used for inner product. (Contributed by NM, 14-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⨣ = (+g‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑆‘((𝐴 · 𝑌) + 𝑍))‘𝑋) = ((((𝑆‘𝑌)‘𝑋) × (𝐺‘𝐴)) ⨣ ((𝑆‘𝑍)‘𝑋)))
Theorem	hdmaplkr 42665	Kernel of the vector to dual map. Line 16 in [Holland95] p. 14. TODO: eliminate 𝐹 hypothesis. (Contributed by NM, 9-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝐹 = (LFnl‘𝑈)    &   ⊢ 𝑌 = (LKer‘𝑈)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑌‘(𝑆‘𝑋)) = (𝑂‘{𝑋}))
Theorem	hdmapellkr 42666	Membership in the kernel (as shown by hdmaplkr 42665) of the vector to dual map. Line 17 in [Holland95] p. 14. (Contributed by NM, 16-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑌) = 0 ↔ 𝑌 ∈ (𝑂‘{𝑋})))
Theorem	hdmapip0 42667	Zero property that will be used for inner product. (Contributed by NM, 9-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝑍 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑋) = 𝑍 ↔ 𝑋 = 0 ))
Theorem	hdmapip1 42668	Construct a proportional vector 𝑌 whose inner product with the original 𝑋 equals one. (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invr‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ 𝑌 = ((𝑁‘((𝑆‘𝑋)‘𝑋)) · 𝑋)    ⇒   ⊢ (𝜑 → ((𝑆‘𝑋)‘𝑌) = 1 )
Theorem	hdmapip0com 42669	Commutation property of Baer's sigma map (Holland's A map). Line 20 of [Holland95] p. 14. Also part of Lemma 1 of [Baer] p. 110 line 7. (Contributed by NM, 9-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (((𝑆‘𝑋)‘𝑌) = 0 ↔ ((𝑆‘𝑌)‘𝑋) = 0 ))
Theorem	hdmapinvlem1 42670	Line 27 in [Baer] p. 110. We use 𝐶 for Baer's u. Our unit vector 𝐸 has the required properties for his w by hdmapevec2 42588. Our ((𝑆‘𝐸)‘𝐶) means the inner product ⟨𝐶, 𝐸⟩ i.e. his f(u,w) (note argument reversal). (Contributed by NM, 11-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    ⇒   ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐶) = 0 )
Theorem	hdmapinvlem2 42671	Line 28 in [Baer] p. 110, 0 = f(w,u). (Contributed by NM, 11-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    ⇒   ⊢ (𝜑 → ((𝑆‘𝐶)‘𝐸) = 0 )
Theorem	hdmapinvlem3 42672	Line 30 in [Baer] p. 110, f(sw + u, tw - v) = 0. (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))    ⇒   ⊢ (𝜑 → ((𝑆‘((𝐽 · 𝐸) − 𝐷))‘((𝐼 · 𝐸) + 𝐶)) = 0 )
Theorem	hdmapinvlem4 42673	Part 1.1 of Proposition 1 of [Baer] p. 110. We use 𝐶, 𝐷, 𝐼, and 𝐽 for Baer's u, v, s, and t. Our unit vector 𝐸 has the required properties for his w by hdmapevec2 42588. Our ((𝑆‘𝐷)‘𝐶) means his f(u,v) (note argument reversal). (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    &   ⊢ (𝜑 → (𝐼 × (𝐺‘𝐽)) = ((𝑆‘𝐷)‘𝐶))    ⇒   ⊢ (𝜑 → (𝐽 × (𝐺‘𝐼)) = ((𝑆‘𝐶)‘𝐷))
Theorem	hdmapglem5 42674	Part 1.2 in [Baer] p. 110 line 34, f(u,v) alpha = f(v,u). (Contributed by NM, 12-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ − = (-g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐼 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘((𝑆‘𝐷)‘𝐶)) = ((𝑆‘𝐶)‘𝐷))
Theorem	hgmapvvlem1 42675	Involution property of scalar sigma map. Line 10 in [Baer] p. 111, t sigma squared = t. Our 𝐸, 𝐶, 𝐷, 𝑌, 𝑋 correspond to Baer's w, h, k, s, t. (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invr‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → (𝑌 × (𝐺‘𝑋)) = 1 )    ⇒   ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)
Theorem	hgmapvvlem2 42676	Lemma for hgmapvv 42678. Eliminate 𝑌 (Baer's s). (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invr‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝐶 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝐷 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → ((𝑆‘𝐷)‘𝐶) = 1 )    ⇒   ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)
Theorem	hgmapvvlem3 42677	Lemma for hgmapvv 42678. Eliminate ((𝑆‘𝐷)‘𝐶) = 1 (Baer's f(h,k)=1). (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑁 = (invr‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)
Theorem	hgmapvv 42678	Value of a double involution. Part 1.2 of [Baer] p. 110 line 37. (Contributed by NM, 13-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘(𝐺‘𝑋)) = 𝑋)
Theorem	hdmapglem7a 42679*	Lemma for hdmapg 42682. (Contributed by NM, 14-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))
Theorem	hdmapglem7b 42680	Lemma for hdmapg 42682. (Contributed by NM, 14-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ✚ = (+g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸}))    &   ⊢ (𝜑 → 𝑚 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑛 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦)))
Theorem	hdmapglem7 42681	Lemma for hdmapg 42682. Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). In the proof, our 𝐸, (𝑂‘{𝐸}), 𝑋, 𝑌, 𝑘, 𝑢, 𝑙, and 𝑣 correspond respectively to Baer's w, H, x, y, x', x'', y', and y'', and our ((𝑆‘𝑌)‘𝑋) corresponds to Baer's f(x,y). (Contributed by NM, 14-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ × = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ✚ = (+g‘𝑅)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))
Theorem	hdmapg 42682	Apply the scalar sigma function (involution) 𝐺 to an inner product reverses the arguments. The inner product of 𝑋 and 𝑌 is represented by ((𝑆‘𝑌)‘𝑋). Line 15 in [Baer] p. 111, f(x,y) alpha = f(y,x). (Contributed by NM, 14-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝐺‘((𝑆‘𝑌)‘𝑋)) = ((𝑆‘𝑋)‘𝑌))
Theorem	hdmapoc 42683*	Express our constructed orthocomplement (polarity) in terms of the Hilbert space definition of orthocomplement. Lines 24 and 25 in [Holland95] p. 14. (Contributed by NM, 17-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((𝑆‘𝑧)‘𝑦) = 0 })
Syntax	chlh 42684	Extend class notation with the final constructed Hilbert space.
class HLHil
Definition	df-hlhil 42685*	Define our final Hilbert space constructed from a Hilbert lattice. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ HLHil = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ⦋((DVecH‘𝑘)‘𝑤) / 𝑢⦌⦋(Base‘𝑢) / 𝑣⦌({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (+g‘𝑢)⟩, ⟨(Scalar‘ndx), (((EDRing‘𝑘)‘𝑤) sSet ⟨(*𝑟‘ndx), ((HGMap‘𝑘)‘𝑤)⟩)⟩} ∪ {⟨( ·𝑠 ‘ndx), ( ·𝑠 ‘𝑢)⟩, ⟨(·𝑖‘ndx), (𝑥 ∈ 𝑣, 𝑦 ∈ 𝑣 ↦ ((((HDMap‘𝑘)‘𝑤)‘𝑦)‘𝑥))⟩})))
Theorem	hlhilset 42686*	The final Hilbert space constructed from a Hilbert lattice 𝐾 and an arbitrary hyperplane 𝑊 in 𝐾. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐿 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)    &   ⊢ · = ( ·𝑠 ‘𝑈)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐿 = ({⟨(Base‘ndx), 𝑉⟩, ⟨(+g‘ndx), + ⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), · ⟩, ⟨(·𝑖‘ndx), , ⟩}))
Theorem	hlhilsca 42687	The scalar of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (𝐸 sSet ⟨(*𝑟‘ndx), 𝐺⟩)    ⇒   ⊢ (𝜑 → 𝑅 = (Scalar‘𝑈))
Theorem	hlhilbase 42688	The base set of the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑀 = (Base‘𝐿)    ⇒   ⊢ (𝜑 → 𝑀 = (Base‘𝑈))
Theorem	hlhilplus 42689	The vector addition for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝐿)    ⇒   ⊢ (𝜑 → + = (+g‘𝑈))
Theorem	hlhilslem 42690	Lemma for hlhilsbase 42691 etc. (Contributed by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐹 = Slot (𝐹‘ndx)    &   ⊢ (𝐹‘ndx) ≠ (*𝑟‘ndx)    &   ⊢ 𝐶 = (𝐹‘𝐸)    ⇒   ⊢ (𝜑 → 𝐶 = (𝐹‘𝑅))
Theorem	hlhilsbase 42691	The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐶 = (Base‘𝐸)    ⇒   ⊢ (𝜑 → 𝐶 = (Base‘𝑅))
Theorem	hlhilsplus 42692	Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ + = (+g‘𝐸)    ⇒   ⊢ (𝜑 → + = (+g‘𝑅))
Theorem	hlhilsmul 42693	Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.) (Revised by AV, 6-Nov-2024.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ · = (.r‘𝐸)    ⇒   ⊢ (𝜑 → · = (.r‘𝑅))
Theorem	hlhilsbase2 42694	The scalar base set of the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐿)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝐶 = (Base‘𝑆)    ⇒   ⊢ (𝜑 → 𝐶 = (Base‘𝑅))
Theorem	hlhilsplus2 42695	Scalar addition for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐿)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ + = (+g‘𝑆)    ⇒   ⊢ (𝜑 → + = (+g‘𝑅))
Theorem	hlhilsmul2 42696	Scalar multiplication for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐿)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ · = (.r‘𝑆)    ⇒   ⊢ (𝜑 → · = (.r‘𝑅))
Theorem	hlhils0 42697	The scalar ring zero for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐿)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (𝜑 → 0 = (0g‘𝑅))
Theorem	hlhils1N 42698	The scalar ring unity for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐿)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 1 = (1r‘𝑆)    ⇒   ⊢ (𝜑 → 1 = (1r‘𝑅))
Theorem	hlhilvsca 42699	The scalar product for the final constructed Hilbert space. (Contributed by NM, 21-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝐿)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → · = ( ·𝑠 ‘𝑈))
Theorem	hlhilip 42700*	Inner product operation for the final constructed Hilbert space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝐿)    &   ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝑆‘𝑦)‘𝑥))    ⇒   ⊢ (𝜑 → , = (·𝑖‘𝑈))