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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | mapdord 40101 | Ordering property of the map defined by df-mapd 40088. Property (b) of [Baer] p. 40. (Contributed by NM, 27-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝑀‘𝑋) ⊆ (𝑀‘𝑌) ↔ 𝑋 ⊆ 𝑌)) | ||
Theorem | mapd11 40102 | The map defined by df-mapd 40088 is one-to-one. Property (c) of [Baer] p. 40. (Contributed by NM, 12-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝑀‘𝑋) = (𝑀‘𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | mapddlssN 40103 | The mapping of a subspace of vector space H to the dual space is a subspace of the dual space. TODO: Make this obsolete, use mapdcl2 40119 instead. (Contributed by NM, 31-Jan-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑇 = (LSubSp‘𝐷) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑅 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘𝑅) ∈ 𝑇) | ||
Theorem | mapdsn 40104* | Value of the map defined by df-mapd 40088 at the span of a singleton. (Contributed by NM, 16-Feb-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = {𝑓 ∈ 𝐹 ∣ (𝑂‘{𝑋}) ⊆ (𝐿‘𝑓)}) | ||
Theorem | mapdsn2 40105* | Value of the map defined by df-mapd 40088 at the span of a singleton. (Contributed by NM, 16-Feb-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → (𝐿‘𝐺) = (𝑂‘{𝑋})) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = {𝑓 ∈ 𝐹 ∣ (𝐿‘𝐺) ⊆ (𝐿‘𝑓)}) | ||
Theorem | mapdsn3 40106 | Value of the map defined by df-mapd 40088 at the span of a singleton. (Contributed by NM, 17-Feb-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑃 = (LSpan‘𝐷) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝐿‘𝐺) = (𝑂‘{𝑋})) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝑃‘{𝐺})) | ||
Theorem | mapd1dim2lem1N 40107* | Value of the map defined by df-mapd 40088 at an atom. (Contributed by NM, 10-Feb-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑀‘𝑄) = {𝑓 ∈ 𝐹 ∣ ∃𝑣 ∈ 𝑄 (𝑂‘{𝑣}) = (𝐿‘𝑓)}) | ||
Theorem | mapdrvallem2 40108* | Lemma for mapdrval 40110. TODO: very long antecedents are dragged through proof in some places - see if it shortens proof to remove unused conjuncts. (Contributed by NM, 2-Feb-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑇 = (LSubSp‘𝐷) & ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑅 ∈ 𝑇) & ⊢ (𝜑 → 𝑅 ⊆ 𝐶) & ⊢ 𝑄 = ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑌 = (0g‘𝐷) ⇒ ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} ⊆ 𝑅) | ||
Theorem | mapdrvallem3 40109* | Lemma for mapdrval 40110. (Contributed by NM, 2-Feb-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑇 = (LSubSp‘𝐷) & ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑅 ∈ 𝑇) & ⊢ (𝜑 → 𝑅 ⊆ 𝐶) & ⊢ 𝑄 = ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑌 = (0g‘𝐷) ⇒ ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} = 𝑅) | ||
Theorem | mapdrval 40110* | Given a dual subspace 𝑅 (of functionals with closed kernels), reconstruct the subspace 𝑄 that maps to it. (Contributed by NM, 12-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑇 = (LSubSp‘𝐷) & ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑅 ∈ 𝑇) & ⊢ (𝜑 → 𝑅 ⊆ 𝐶) & ⊢ 𝑄 = ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)) ⇒ ⊢ (𝜑 → (𝑀‘𝑄) = 𝑅) | ||
Theorem | mapd1o 40111* | The map defined by df-mapd 40088 is one-to-one and onto the set of dual subspaces of functionals with closed kernels. This shows 𝑀 satisfies part of the definition of projectivity of [Baer] p. 40. TODO: change theorems leading to this (lcfr 40048, mapdrval 40110, lclkrs 40002, lclkr 39996,...) to use 𝑇 ∩ 𝒫 𝐶? TODO: maybe get rid of $d's for 𝑔 versus 𝐾𝑈𝑊; propagate to mapdrn 40112 and any others. (Contributed by NM, 12-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑇 = (LSubSp‘𝐷) & ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑀:𝑆–1-1-onto→(𝑇 ∩ 𝒫 𝐶)) | ||
Theorem | mapdrn 40112* | Range of the map defined by df-mapd 40088. (Contributed by NM, 12-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (LDual‘𝑈) & ⊢ 𝑇 = (LSubSp‘𝐷) & ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ran 𝑀 = (𝑇 ∩ 𝒫 𝐶)) | ||
Theorem | mapdunirnN 40113* | Union of the range of the map defined by df-mapd 40088. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)} & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ∪ ran 𝑀 = 𝐶) | ||
Theorem | mapdrn2 40114 | Range of the map defined by df-mapd 40088. TODO: this seems to be needed a lot in hdmaprnlem3eN 40321 etc. Would it be better to change df-mapd 40088 theorems to use LSubSp‘𝐶 instead of ran 𝑀? (Contributed by NM, 13-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝑇 = (LSubSp‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ran 𝑀 = 𝑇) | ||
Theorem | mapdcnvcl 40115 | Closure of the converse of the map defined by df-mapd 40088. (Contributed by NM, 13-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) ⇒ ⊢ (𝜑 → (◡𝑀‘𝑋) ∈ 𝑆) | ||
Theorem | mapdcl 40116 | Closure the value of the map defined by df-mapd 40088. (Contributed by NM, 13-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘𝑋) ∈ ran 𝑀) | ||
Theorem | mapdcnvid1N 40117 | Converse of the value of the map defined by df-mapd 40088. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) ⇒ ⊢ (𝜑 → (◡𝑀‘(𝑀‘𝑋)) = 𝑋) | ||
Theorem | mapdsord 40118 | Strong ordering property of themap defined by df-mapd 40088. (Contributed by NM, 13-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌)) | ||
Theorem | mapdcl2 40119 | The mapping of a subspace of vector space H is a subspace in the dual space of functionals with closed kernels. (Contributed by NM, 31-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝑇 = (LSubSp‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑅 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘𝑅) ∈ 𝑇) | ||
Theorem | mapdcnvid2 40120 | Value of the converse of the map defined by df-mapd 40088. (Contributed by NM, 13-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) ⇒ ⊢ (𝜑 → (𝑀‘(◡𝑀‘𝑋)) = 𝑋) | ||
Theorem | mapdcnvordN 40121 | Ordering property of the converse of the map defined by df-mapd 40088. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) & ⊢ (𝜑 → 𝑌 ∈ ran 𝑀) ⇒ ⊢ (𝜑 → ((◡𝑀‘𝑋) ⊆ (◡𝑀‘𝑌) ↔ 𝑋 ⊆ 𝑌)) | ||
Theorem | mapdcnv11N 40122 | The converse of the map defined by df-mapd 40088 is one-to-one. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) & ⊢ (𝜑 → 𝑌 ∈ ran 𝑀) ⇒ ⊢ (𝜑 → ((◡𝑀‘𝑋) = (◡𝑀‘𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | mapdcv 40123 | Covering property of the converse of the map defined by df-mapd 40088. (Contributed by NM, 14-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝐶 = ( ⋖L ‘𝑈) & ⊢ 𝐷 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐸 = ( ⋖L ‘𝐷) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀‘𝑋)𝐸(𝑀‘𝑌))) | ||
Theorem | mapdincl 40124 | Closure of dual subspace intersection for the map defined by df-mapd 40088. (Contributed by NM, 12-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) & ⊢ (𝜑 → 𝑌 ∈ ran 𝑀) ⇒ ⊢ (𝜑 → (𝑋 ∩ 𝑌) ∈ ran 𝑀) | ||
Theorem | mapdin 40125 | Subspace intersection is preserved by the map defined by df-mapd 40088. Part of property (e) in [Baer] p. 40. (Contributed by NM, 12-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘(𝑋 ∩ 𝑌)) = ((𝑀‘𝑋) ∩ (𝑀‘𝑌))) | ||
Theorem | mapdlsmcl 40126 | Closure of dual subspace sum for the map defined by df-mapd 40088. (Contributed by NM, 13-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) & ⊢ (𝜑 → 𝑌 ∈ ran 𝑀) ⇒ ⊢ (𝜑 → (𝑋 ⊕ 𝑌) ∈ ran 𝑀) | ||
Theorem | mapdlsm 40127 | Subspace sum is preserved by the map defined by df-mapd 40088. Part of property (e) in [Baer] p. 40. (Contributed by NM, 13-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ ✚ = (LSSum‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑀‘(𝑋 ⊕ 𝑌)) = ((𝑀‘𝑋) ✚ (𝑀‘𝑌))) | ||
Theorem | mapd0 40128 | Projectivity map of the zero subspace. Part of property (f) in [Baer] p. 40. TODO: does proof need to be this long for this simple fact? (Contributed by NM, 15-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑂 = (0g‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝑀‘{𝑂}) = { 0 }) | ||
Theorem | mapdcnvatN 40129 | Atoms are preserved by the map defined by df-mapd 40088. (Contributed by NM, 29-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐵 = (LSAtoms‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) ⇒ ⊢ (𝜑 → (◡𝑀‘𝑄) ∈ 𝐴) | ||
Theorem | mapdat 40130 | Atoms are preserved by the map defined by df-mapd 40088. Property (g) in [Baer] p. 41. (Contributed by NM, 14-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐵 = (LSAtoms‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑀‘𝑄) ∈ 𝐵) | ||
Theorem | mapdspex 40131* | The map of a span equals the dual span of some vector (functional). (Contributed by NM, 15-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝑔})) | ||
Theorem | mapdn0 40132 | Transfer nonzero property from domain to range of projectivity mapd. (Contributed by NM, 12-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑍 = (0g‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → 𝐹 ∈ (𝐷 ∖ {𝑍})) | ||
Theorem | mapdncol 40133 | Transfer non-colinearity from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝐽‘{𝐹}) ≠ (𝐽‘{𝐺})) | ||
Theorem | mapdindp 40134 | Transfer (part of) vector independence condition from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝐸 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸})) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐽‘{𝐺, 𝐸})) | ||
Theorem | mapdpglem1 40135 | Lemma for mapdpg 40169. Baer p. 44, last line: "(F(x-y))* <= (Fx)*+(Fy)*." (Contributed by NM, 15-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) | ||
Theorem | mapdpglem2 40136* | Lemma for mapdpg 40169. Baer p. 45, lines 1 and 2: "we have (F(x-y))* = Gt where t necessarily belongs to (Fx)*+(Fy)*." (We scope $d 𝑡𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 15-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) ⇒ ⊢ (𝜑 → ∃𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))(𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) | ||
Theorem | mapdpglem2a 40137* | Lemma for mapdpg 40169. (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) ⇒ ⊢ (𝜑 → 𝑡 ∈ 𝐹) | ||
Theorem | mapdpglem3 40138* | Lemma for mapdpg 40169. Baer p. 45, line 3: "infer ... the existence of a number g in G and of an element z in (Fy)* such that t = gx'-z." (We scope $d 𝑔𝑤𝑧𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 18-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) ⇒ ⊢ (𝜑 → ∃𝑔 ∈ 𝐵 ∃𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) | ||
Theorem | mapdpglem4N 40139* | Lemma for mapdpg 40169. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑋 − 𝑌) ≠ 𝑄) | ||
Theorem | mapdpglem5N 40140* | Lemma for mapdpg 40169. (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) ⇒ ⊢ (𝜑 → 𝑡 ≠ (0g‘𝐶)) | ||
Theorem | mapdpglem6 40141* | Lemma for mapdpg 40169. Baer p. 45, line 4: "If g were 0, then t would be in (Fy)*..." (Contributed by NM, 18-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑔 = 0 ) ⇒ ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌}))) | ||
Theorem | mapdpglem8 40142* | Lemma for mapdpg 40169. Baer p. 45, line 4: "...so that (F(x-y))* <= (Fy)*. This would imply that F(x-y) <= F(y)..." (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑔 = 0 ) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌})) | ||
Theorem | mapdpglem9 40143* | Lemma for mapdpg 40169. Baer p. 45, line 4: "...so that x would consequently belong to Fy." (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑔 = 0 ) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑌})) | ||
Theorem | mapdpglem10 40144* | Lemma for mapdpg 40169. Baer p. 45, line 6: "Hence Fx=Fy, an impossibility." (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑔 = 0 ) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) | ||
Theorem | mapdpglem11 40145* | Lemma for mapdpg 40169. (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) ⇒ ⊢ (𝜑 → 𝑔 ≠ 0 ) | ||
Theorem | mapdpglem12 40146* | Lemma for mapdpg 40169. TODO: Can some commonality with mapdpglem6 40141 through mapdpglem11 40145 be exploited? Also, some consolidation of small lemmas here could be done. (Contributed by NM, 18-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ (𝜑 → 𝑧 = (0g‘𝐶)) ⇒ ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑋}))) | ||
Theorem | mapdpglem13 40147* | Lemma for mapdpg 40169. (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ (𝜑 → 𝑧 = (0g‘𝐶)) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑋})) | ||
Theorem | mapdpglem14 40148* | Lemma for mapdpg 40169. (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ (𝜑 → 𝑧 = (0g‘𝐶)) ⇒ ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋})) | ||
Theorem | mapdpglem15 40149* | Lemma for mapdpg 40169. (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ (𝜑 → 𝑧 = (0g‘𝐶)) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) | ||
Theorem | mapdpglem16 40150* | Lemma for mapdpg 40169. Baer p. 45, line 7: "Likewise we see that z =/= 0." (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) ⇒ ⊢ (𝜑 → 𝑧 ≠ (0g‘𝐶)) | ||
Theorem | mapdpglem17N 40151* | Lemma for mapdpg 40169. Baer p. 45, line 7: "Hence we may form y' = g^-1 z." (Contributed by NM, 20-Mar-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) ⇒ ⊢ (𝜑 → 𝐸 ∈ 𝐹) | ||
Theorem | mapdpglem18 40152* | Lemma for mapdpg 40169. Baer p. 45, line 7: "Then y =/= 0..." (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) ⇒ ⊢ (𝜑 → 𝐸 ≠ (0g‘𝐶)) | ||
Theorem | mapdpglem19 40153* | Lemma for mapdpg 40169. Baer p. 45, line 8: "...is in (Fy)*..." (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝑀‘(𝑁‘{𝑌}))) | ||
Theorem | mapdpglem20 40154* | Lemma for mapdpg 40169. Baer p. 45, line 8: "...so that (Fy)*=Gy'." (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐸})) | ||
Theorem | mapdpglem21 40155* | Lemma for mapdpg 40169. (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) ⇒ ⊢ (𝜑 → (((invr‘𝐴)‘𝑔) · 𝑡) = (𝐺𝑅𝐸)) | ||
Theorem | mapdpglem22 40156* | Lemma for mapdpg 40169. Baer p. 45, line 9: "(F(x-y))* = ... = G(x'-y')." (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝐸)})) | ||
Theorem | mapdpglem23 40157* | Lemma for mapdpg 40169. Baer p. 45, line 10: "and so y' meets all our requirements." Our ℎ is Baer's y'. (Contributed by NM, 20-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ⊕ = (LSSum‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ 𝑄 = (0g‘𝑈) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) & ⊢ 0 = (0g‘𝐴) & ⊢ (𝜑 → 𝑔 ∈ 𝐵) & ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌}))) & ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧)) & ⊢ (𝜑 → 𝑋 ≠ 𝑄) & ⊢ (𝜑 → 𝑌 ≠ 𝑄) & ⊢ 𝐸 = (((invr‘𝐴)‘𝑔) · 𝑧) ⇒ ⊢ (𝜑 → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) | ||
Theorem | mapdpglem30a 40158 | Lemma for mapdpg 40169. (Contributed by NM, 22-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) ⇒ ⊢ (𝜑 → 𝐺 ≠ (0g‘𝐶)) | ||
Theorem | mapdpglem30b 40159 | Lemma for mapdpg 40169. (Contributed by NM, 22-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) & ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ⇒ ⊢ (𝜑 → 𝑖 ≠ (0g‘𝐶)) | ||
Theorem | mapdpglem25 40160 | Lemma for mapdpg 40169. Baer p. 45 line 12: "Then we have Gy' = Gy'' and G(x'-y') = G(x'-y'')." (Contributed by NM, 21-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) & ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) ⇒ ⊢ (𝜑 → ((𝐽‘{ℎ}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))) | ||
Theorem | mapdpglem26 40161* | Lemma for mapdpg 40169. Baer p. 45 line 14: "Consequently there exist numbers u,v in G neither of which is 0 such that y = uy'' and..." (We scope $d 𝑢𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 22-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) & ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑂 = (0g‘𝐴) ⇒ ⊢ (𝜑 → ∃𝑢 ∈ (𝐵 ∖ {𝑂})ℎ = (𝑢 · 𝑖)) | ||
Theorem | mapdpglem27 40162* | Lemma for mapdpg 40169. Baer p. 45 line 16: "v(x'-y'') = x'-y'" (with equality swapped). (Contributed by NM, 22-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) & ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑂 = (0g‘𝐴) ⇒ ⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) | ||
Theorem | mapdpglem29 40163* | Lemma for mapdpg 40169. Baer p. 45 line 16: "But Gx' and Gy'' are distinct points and so x' and y'' are independent elements in B. (Contributed by NM, 22-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) & ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑂 = (0g‘𝐴) & ⊢ (𝜑 → 𝑣 ∈ 𝐵) & ⊢ (𝜑 → ℎ = (𝑢 · 𝑖)) & ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) ⇒ ⊢ (𝜑 → (𝐽‘{𝐺}) ≠ (𝐽‘{𝑖})) | ||
Theorem | mapdpglem28 40164* | Lemma for mapdpg 40169. Baer p. 45 line 18: "vx'-vy'' = x'-uy''". (Contributed by NM, 22-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) & ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑂 = (0g‘𝐴) & ⊢ (𝜑 → 𝑣 ∈ 𝐵) & ⊢ (𝜑 → ℎ = (𝑢 · 𝑖)) & ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) ⇒ ⊢ (𝜑 → ((𝑣 · 𝐺)𝑅(𝑣 · 𝑖)) = (𝐺𝑅(𝑢 · 𝑖))) | ||
Theorem | mapdpglem30 40165* | Lemma for mapdpg 40169. Baer p. 45 line 18: "Hence we deduce (from mapdpglem28 40164, using lvecindp2 20600) that v = 1 and v = u...". TODO: would it be shorter to have only the 𝑣 = (1r‘𝐴) part and use mapdpglem28.u2 in mapdpglem31 40166? (Contributed by NM, 22-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) & ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑂 = (0g‘𝐴) & ⊢ (𝜑 → 𝑣 ∈ 𝐵) & ⊢ (𝜑 → ℎ = (𝑢 · 𝑖)) & ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) & ⊢ (𝜑 → 𝑢 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑣 = (1r‘𝐴) ∧ 𝑣 = 𝑢)) | ||
Theorem | mapdpglem31 40166* | Lemma for mapdpg 40169. Baer p. 45 line 19: "...and we have consequently that y' = y'', as we claimed." (Contributed by NM, 23-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) & ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) & ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) & ⊢ 𝐴 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐴) & ⊢ · = ( ·𝑠 ‘𝐶) & ⊢ 𝑂 = (0g‘𝐴) & ⊢ (𝜑 → 𝑣 ∈ 𝐵) & ⊢ (𝜑 → ℎ = (𝑢 · 𝑖)) & ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) & ⊢ (𝜑 → 𝑢 ∈ 𝐵) ⇒ ⊢ (𝜑 → ℎ = 𝑖) | ||
Theorem | mapdpglem24 40167* | Lemma for mapdpg 40169. Existence part - consolidate hypotheses in mapdpglem23 40157. (Contributed by NM, 21-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) ⇒ ⊢ (𝜑 → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) | ||
Theorem | mapdpglem32 40168* | Lemma for mapdpg 40169. Uniqueness part - consolidate hypotheses in mapdpglem31 40166. (Contributed by NM, 23-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) ⇒ ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ℎ = 𝑖) | ||
Theorem | mapdpg 40169* | Part 1 of proof of the first fundamental theorem of projective geometry. Part (1) in [Baer] p. 44. Our notation corresponds to Baer's as follows: 𝑀 for *, 𝑁‘{} for F(), 𝐽‘{} for G(), 𝑋 for x, 𝐺 for x', 𝑌 for y, ℎ for y'. TODO: Rename variables per mapdhval 40187. (Contributed by NM, 22-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐹 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) ⇒ ⊢ (𝜑 → ∃!ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) | ||
Theorem | baerlem3lem1 40170 | Lemma for baerlem3 40176. (Contributed by NM, 9-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) & ⊢ (𝜑 → 𝑎 ∈ 𝐵) & ⊢ (𝜑 → 𝑏 ∈ 𝐵) & ⊢ (𝜑 → 𝑑 ∈ 𝐵) & ⊢ (𝜑 → 𝑒 ∈ 𝐵) & ⊢ (𝜑 → 𝑗 = ((𝑎 · 𝑌) + (𝑏 · 𝑍))) & ⊢ (𝜑 → 𝑗 = ((𝑑 · (𝑋 − 𝑌)) + (𝑒 · (𝑋 − 𝑍)))) ⇒ ⊢ (𝜑 → 𝑗 = (𝑎 · (𝑌 − 𝑍))) | ||
Theorem | baerlem5alem1 40171 | Lemma for baerlem5a 40177. (Contributed by NM, 13-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) & ⊢ (𝜑 → 𝑎 ∈ 𝐵) & ⊢ (𝜑 → 𝑏 ∈ 𝐵) & ⊢ (𝜑 → 𝑑 ∈ 𝐵) & ⊢ (𝜑 → 𝑒 ∈ 𝐵) & ⊢ (𝜑 → 𝑗 = ((𝑎 · (𝑋 − 𝑌)) + (𝑏 · 𝑍))) & ⊢ (𝜑 → 𝑗 = ((𝑑 · (𝑋 − 𝑍)) + (𝑒 · 𝑌))) ⇒ ⊢ (𝜑 → 𝑗 = (𝑎 · (𝑋 − (𝑌 + 𝑍)))) | ||
Theorem | baerlem5blem1 40172 | Lemma for baerlem5b 40178. (Contributed by NM, 9-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) & ⊢ (𝜑 → 𝑎 ∈ 𝐵) & ⊢ (𝜑 → 𝑏 ∈ 𝐵) & ⊢ (𝜑 → 𝑑 ∈ 𝐵) & ⊢ (𝜑 → 𝑒 ∈ 𝐵) & ⊢ (𝜑 → 𝑗 = ((𝑎 · 𝑌) + (𝑏 · 𝑍))) & ⊢ (𝜑 → 𝑗 = ((𝑑 · (𝑋 − (𝑌 + 𝑍))) + (𝑒 · 𝑋))) ⇒ ⊢ (𝜑 → 𝑗 = ((𝐼‘𝑑) · (𝑌 + 𝑍))) | ||
Theorem | baerlem3lem2 40173 | Lemma for baerlem3 40176. (Contributed by NM, 9-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 − 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{(𝑋 − 𝑍)})))) | ||
Theorem | baerlem5alem2 40174 | Lemma for baerlem5a 40177. (Contributed by NM, 9-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)}) ⊕ (𝑁‘{𝑌})))) | ||
Theorem | baerlem5blem2 40175 | Lemma for baerlem5b 40178. (Contributed by NM, 13-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑅 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ ⨣ = (+g‘𝑅) & ⊢ 𝐿 = (-g‘𝑅) & ⊢ 𝑄 = (0g‘𝑅) & ⊢ 𝐼 = (invg‘𝑅) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))}) ⊕ (𝑁‘{𝑋})))) | ||
Theorem | baerlem3 40176 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Part (3) in [Baer] p. 45. TODO fix ref. (Contributed by NM, 9-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 − 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{(𝑋 − 𝑍)})))) | ||
Theorem | baerlem5a 40177 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. First equation of part (5) in [Baer] p. 46. (Contributed by NM, 10-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)}) ⊕ (𝑁‘{𝑌})))) | ||
Theorem | baerlem5b 40178 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Second equation of part (5) in [Baer] p. 46. (Contributed by NM, 13-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 + 𝑍))}) ⊕ (𝑁‘{𝑋})))) | ||
Theorem | baerlem5amN 40179 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction version of first equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 40181 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 − 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) ⊕ (𝑁‘{𝑌})))) | ||
Theorem | baerlem5bmN 40180 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction version of second equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 40181 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 − 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 − 𝑍))}) ⊕ (𝑁‘{𝑋})))) | ||
Theorem | baerlem5abmN 40181 | An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction versions of first and second equations of part (5) in [Baer] p. 46, conjoined to share commonality in their proofs. TODO: Delete if not needed. (Contributed by NM, 24-May-2015.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑊) ⇒ ⊢ (𝜑 → ((𝑁‘{(𝑋 − (𝑌 − 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) ⊕ (𝑁‘{𝑌}))) ∧ (𝑁‘{(𝑌 − 𝑍)}) = (((𝑁‘{𝑌}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − (𝑌 − 𝑍))}) ⊕ (𝑁‘{𝑋}))))) | ||
Theorem | mapdindp0 40182 | Vector independence lemma. (Contributed by NM, 29-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) & ⊢ (𝜑 → (𝑌 + 𝑍) ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑌 + 𝑍)}) = (𝑁‘{𝑌})) | ||
Theorem | mapdindp1 40183 | Vector independence lemma. (Contributed by NM, 1-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) | ||
Theorem | mapdindp2 40184 | Vector independence lemma. (Contributed by NM, 1-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, (𝑌 + 𝑍)})) | ||
Theorem | mapdindp3 40185 | Vector independence lemma. (Contributed by NM, 29-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)})) | ||
Theorem | mapdindp4 40186 | Vector independence lemma. (Contributed by NM, 29-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LVec) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) ⇒ ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})) | ||
Theorem | mapdhval 40187* | Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐸) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = if(𝑌 = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))) | ||
Theorem | mapdhval0 40188* | Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) | ||
Theorem | mapdhval2 40189* | Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ (𝜑 → 𝑋 ∈ 𝐴) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))) | ||
Theorem | mapdhcl 40190* | Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) | ||
Theorem | mapdheq 40191* | Lemmma for ~? mapdh . The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (Contributed by NM, 4-Apr-2015.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))) | ||
Theorem | mapdheq2 40192* | Lemmma for ~? mapdh . One direction of part (2) in [Baer] p. 45. (Contributed by NM, 4-Apr-2015.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺 → (𝐼‘〈𝑌, 𝐺, 𝑋〉) = 𝐹)) | ||
Theorem | mapdheq2biN 40193* | Lemmma for ~? mapdh . Part (2) in [Baer] p. 45. The bidirectional version of mapdheq2 40192 seems to require an additional hypothesis not mentioned in Baer. TODO fix ref. TODO: We probably don't need this; delete if never used. (Contributed by NM, 4-Apr-2015.) (New usage is discouraged.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝐺 ∈ 𝐷) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) ⇒ ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺 ↔ (𝐼‘〈𝑌, 𝐺, 𝑋〉) = 𝐹)) | ||
Theorem | mapdheq4lem 40194* | Lemma for mapdheq4 40195. Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 − 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)})) | ||
Theorem | mapdheq4 40195* | Lemma for ~? mapdh . Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑍〉) = 𝐸) | ||
Theorem | mapdh6lem1N 40196* | Lemma for mapdh6N 40210. Part (6) in [Baer] p. 47, lines 16-18. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − (𝑌 + 𝑍))})) = (𝐽‘{(𝐹𝑅(𝐺 ✚ 𝐸))})) | ||
Theorem | mapdh6lem2N 40197* | Lemma for mapdh6N 40210. Part (6) in [Baer] p. 47, lines 20-22. (Contributed by NM, 13-Apr-2015.) (New usage is discouraged.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) ⇒ ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 + 𝑍)})) = (𝐽‘{(𝐺 ✚ 𝐸)})) | ||
Theorem | mapdh6aN 40198* | Lemma for mapdh6N 40210. Part (6) in [Baer] p. 47, case 1. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) & ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) & ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) | ||
Theorem | mapdh6b0N 40199* | Lemmma for mapdh6N 40210. (Contributed by NM, 23-Apr-2015.) (New usage is discouraged.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ((𝑁‘{𝑋}) ∩ (𝑁‘{𝑌, 𝑍})) = { 0 }) ⇒ ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | ||
Theorem | mapdh6bN 40200* | Lemmma for mapdh6N 40210. (Contributed by NM, 24-Apr-2015.) (New usage is discouraged.) |
⊢ 𝑄 = (0g‘𝐶) & ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ − = (-g‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) & ⊢ 𝐷 = (Base‘𝐶) & ⊢ 𝑅 = (-g‘𝐶) & ⊢ 𝐽 = (LSpan‘𝐶) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐹 ∈ 𝐷) & ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ + = (+g‘𝑈) & ⊢ ✚ = (+g‘𝐶) & ⊢ (𝜑 → 𝑌 = 0 ) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) ⇒ ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
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