P. 424 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 42301-42400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	hdmaprnlem7N 42301	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 42302	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 42303	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 42085 and mapdcnv11N 42105. 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 42304*	Lemma for hdmaprnN 42310. (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 42305*	Lemma for hdmaprnN 42310. 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 42306*	Lemma for hdmaprnN 42310. 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 42307*	Lemma for hdmaprnN 42310. 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 42308	Lemma for hdmaprnN 42310. 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 42309	Lemma for hdmaprnN 42310. 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 42310	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 42311	Part 12 in [Baer] p. 49. The map from vectors to functionals with closed kernels maps one-to-one onto. Combined with hdmapadd 42289, 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 42312	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 42313*	Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 0 so it can be used in hdmap14lem10 42323. (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 42314	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 42315*	Prior to part 14 in [Baer] p. 49, line 25. TODO: fix to include 𝐹 = 𝑍 so it can be used in hdmap14lem10 42323. (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 42316*	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 42317*	Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 42316 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 42318*	Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 42316 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 42317 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 42317 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 42319*	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 42320*	Combine cases of 𝐹. TODO: Can this be done at once in hdmap14lem3 42316, in order to get rid of hdmap14lem6 42319? Perhaps modify lspsneu 21121 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 42321	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 42322	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 42323	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 42324	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 42325*	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 42326*	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 42327*	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 42328*	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 42329	Extend class notation with g-map.
class HGMap
Definition	df-hgmap 42330*	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 42331*	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 42332*	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 42333*	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 42328. (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 42334	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 42335	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 42336	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 42337	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 42338	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 42339	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 42340	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 42341	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 42342	Lemma for hgmaprnN 42347. (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 42343	Lemma for hgmaprnN 42347. 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 42344*	Lemma for hgmaprnN 42347. 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 42345*	Lemma for hgmaprnN 42347. 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 42346	Lemma for hgmaprnN 42347. 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 42347	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 42348	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 42349	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 42350	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 42351	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 42352	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 42353	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 42354	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 42355	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 42356	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 42357	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 42358	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 42359	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 42360	Membership in the kernel (as shown by hdmaplkr 42359) 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 42361	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 42362	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 42363	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 42364	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 42282. 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 42365	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 42366	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 42367	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 42282. 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 42368	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 42369	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 42370	Lemma for hgmapvv 42372. 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 42371	Lemma for hgmapvv 42372. 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 42372	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 42373*	Lemma for hdmapg 42376. (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 42374	Lemma for hdmapg 42376. (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 42375	Lemma for hdmapg 42376. 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 42376	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 42377*	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 42378	Extend class notation with the final constructed Hilbert space.
class HLHil
Definition	df-hlhil 42379*	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 42380*	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 42381	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 42382	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 42383	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 42384	Lemma for hlhilsbase 42385 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 42385	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 42386	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 42387	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 42388	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 42389	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 42390	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 42391	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 42392	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 42393	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 42394*	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))
Theorem	hlhilipval 42395	Value of 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))    &   ⊢ , = (·𝑖‘𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 , 𝑌) = ((𝑆‘𝑌)‘𝑋))
Theorem	hlhilnvl 42396	The involution operation of the star division ring for the final constructed Hilbert space. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ ∗ = ((HGMap‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ∗ = (*𝑟‘𝑅))
Theorem	hlhillvec 42397	The final constructed Hilbert space is a vector space. (Contributed by NM, 22-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑈 ∈ LVec)
Theorem	hlhildrng 42398	The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 20-Jun-2015.) (Revised by Mario Carneiro, 28-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝑅 = (Scalar‘𝑈)    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	hlhilsrnglem 42399	Lemma for hlhilsrng 42400. (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))    &   ⊢ 𝑅 = (Scalar‘𝑈)    &   ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝐿)    &   ⊢ 𝐵 = (Base‘𝑆)    &   ⊢ + = (+g‘𝑆)    &   ⊢ · = (.r‘𝑆)    &   ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)    ⇒   ⊢ (𝜑 → 𝑅 ∈ *-Ring)
Theorem	hlhilsrng 42400	The star division ring for the final constructed Hilbert space is a division ring. (Contributed by NM, 21-Jun-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ 𝑅 = (Scalar‘𝑈)    ⇒   ⊢ (𝜑 → 𝑅 ∈ *-Ring)