P. 243 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pcophtb 24201*	The path homotopy equivalence relation on two paths 𝐹, 𝐺 with the same start and end point can be written in terms of the loop 𝐹 − 𝐺 formed by concatenating 𝐹 with the inverse of 𝐺. Thus, all the homotopy information in ≃ph‘𝐽 is available if we restrict our attention to closed loops, as in the definition of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ 𝐻 = (𝑥 ∈ (0[,]1) ↦ (𝐺‘(1 − 𝑥)))    &   ⊢ 𝑃 = ((0[,]1) × {(𝐹‘0)})    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘0) = (𝐺‘0))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐺‘1))    ⇒   ⊢ (𝜑 → ((𝐹(*𝑝‘𝐽)𝐻)( ≃ph‘𝐽)𝑃 ↔ 𝐹( ≃ph‘𝐽)𝐺))
Theorem	om1val 24202*	The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})    &   ⊢ (𝜑 → + = (*𝑝‘𝐽))    &   ⊢ (𝜑 → 𝐾 = (𝐽 ↑ko II))    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → 𝑂 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐾⟩})
Theorem	om1bas 24203*	The base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → 𝐵 = {𝑓 ∈ (II Cn 𝐽) ∣ ((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌)})
Theorem	om1elbas 24204	Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
Theorem	om1addcl 24205	Closure of the group operation of the loop space. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 5-Sep-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑂))    &   ⊢ (𝜑 → 𝐻 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐻(*𝑝‘𝐽)𝐾) ∈ 𝐵)
Theorem	om1plusg 24206	The group operation (which isn't much more than a magma) of the loop space. (Contributed by Mario Carneiro, 11-Feb-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (*𝑝‘𝐽) = (+g‘𝑂))
Theorem	om1tset 24207	The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐽 ↑ko II) = (TopSet‘𝑂))
Theorem	om1opn 24208	The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝐾 = (TopOpen‘𝑂)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → 𝐾 = ((𝐽 ↑ko II) ↾t 𝐵))
Theorem	pi1val 24209	The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    ⇒   ⊢ (𝜑 → 𝐺 = (𝑂 /s ( ≃ph‘𝐽)))
Theorem	pi1bas 24210	The base set of the fundamental group of a topological space at a given base point. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → 𝐾 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → 𝐵 = (𝐾 / ( ≃ph‘𝐽)))
Theorem	pi1blem 24211	Lemma for pi1buni 24212. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → 𝐾 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → ((( ≃ph‘𝐽) “ 𝐾) ⊆ 𝐾 ∧ 𝐾 ⊆ (II Cn 𝐽)))
Theorem	pi1buni 24212	Another way to write the loop space base in terms of the base of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → 𝐾 = (Base‘𝑂))    ⇒   ⊢ (𝜑 → ∪ 𝐵 = 𝐾)
Theorem	pi1bas2 24213	The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    ⇒   ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / ( ≃ph‘𝐽)))
Theorem	pi1eluni 24214	Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    ⇒   ⊢ (𝜑 → (𝐹 ∈ ∪ 𝐵 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌 ∧ (𝐹‘1) = 𝑌)))
Theorem	pi1bas3 24215	The base set of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))    ⇒   ⊢ (𝜑 → 𝐵 = (∪ 𝐵 / 𝑅))
Theorem	pi1cpbl 24216	The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ 𝑅 = (( ≃ph‘𝐽) ∩ (∪ 𝐵 × ∪ 𝐵))    &   ⊢ 𝑂 = (𝐽 Ω1 𝑌)    &   ⊢ + = (+g‘𝑂)    ⇒   ⊢ (𝜑 → ((𝑀𝑅𝑁 ∧ 𝑃𝑅𝑄) → (𝑀 + 𝑃)𝑅(𝑁 + 𝑄)))
Theorem	elpi1 24217*	The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    ⇒   ⊢ (𝜑 → (𝐹 ∈ 𝐵 ↔ ∃𝑓 ∈ (II Cn 𝐽)(((𝑓‘0) = 𝑌 ∧ (𝑓‘1) = 𝑌) ∧ 𝐹 = [𝑓]( ≃ph‘𝐽))))
Theorem	elpi1i 24218	The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘0) = 𝑌)    &   ⊢ (𝜑 → (𝐹‘1) = 𝑌)    ⇒   ⊢ (𝜑 → [𝐹]( ≃ph‘𝐽) ∈ 𝐵)
Theorem	pi1addf 24219	The group operation of π1 is a binary operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝜑 → + :(𝐵 × 𝐵)⟶𝐵)
Theorem	pi1addval 24220	The concatenation of two path-homotopy classes in the fundamental group. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝑀 ∈ ∪ 𝐵)    &   ⊢ (𝜑 → 𝑁 ∈ ∪ 𝐵)    ⇒   ⊢ (𝜑 → ([𝑀]( ≃ph‘𝐽) + [𝑁]( ≃ph‘𝐽)) = [(𝑀(*𝑝‘𝐽)𝑁)]( ≃ph‘𝐽))
Theorem	pi1grplem 24221	Lemma for pi1grp 24222. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Aug-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ 0 = ((0[,]1) × {𝑌})    ⇒   ⊢ (𝜑 → (𝐺 ∈ Grp ∧ [ 0 ]( ≃ph‘𝐽) = (0g‘𝐺)))
Theorem	pi1grp 24222	The fundamental group is a group. Proposition 1.3 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.) (Revised by Mario Carneiro, 10-Aug-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝑋) → 𝐺 ∈ Grp)
Theorem	pi1id 24223	The identity element of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ 0 = ((0[,]1) × {𝑌})    ⇒   ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ∈ 𝑋) → [ 0 ]( ≃ph‘𝐽) = (0g‘𝐺))
Theorem	pi1inv 24224*	An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
⊢ 𝐺 = (𝐽 π1 𝑌)    &   ⊢ 𝑁 = (invg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝑌 ∈ 𝑋)    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘0) = 𝑌)    &   ⊢ (𝜑 → (𝐹‘1) = 𝑌)    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    ⇒   ⊢ (𝜑 → (𝑁‘[𝐹]( ≃ph‘𝐽)) = [𝐼]( ≃ph‘𝐽))
Theorem	pi1xfrf 24225*	Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))    &   ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))    ⇒   ⊢ (𝜑 → 𝐺:𝐵⟶(Base‘𝑄))
Theorem	pi1xfrval 24226*	The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → 𝐼 ∈ (II Cn 𝐽))    &   ⊢ (𝜑 → (𝐹‘1) = (𝐼‘0))    &   ⊢ (𝜑 → (𝐼‘1) = (𝐹‘0))    &   ⊢ (𝜑 → 𝐴 ∈ ∪ 𝐵)    ⇒   ⊢ (𝜑 → (𝐺‘[𝐴]( ≃ph‘𝐽)) = [(𝐼(*𝑝‘𝐽)(𝐴(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽))
Theorem	pi1xfr 24227*	Given a path 𝐹 and its inverse 𝐼 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))
Theorem	pi1xfrcnvlem 24228*	Given a path 𝐹 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   ⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)    ⇒   ⊢ (𝜑 → ◡𝐺 ⊆ 𝐻)
Theorem	pi1xfrcnv 24229*	Given a path 𝐹 between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    &   ⊢ 𝐻 = ran (ℎ ∈ ∪ (Base‘𝑄) ↦ ⟨[ℎ]( ≃ph‘𝐽), [(𝐹(*𝑝‘𝐽)(ℎ(*𝑝‘𝐽)𝐼))]( ≃ph‘𝐽)⟩)    ⇒   ⊢ (𝜑 → (◡𝐺 = 𝐻 ∧ ◡𝐺 ∈ (𝑄 GrpHom 𝑃)))
Theorem	pi1xfrgim 24230*	The mapping 𝐺 between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.)
⊢ 𝑃 = (𝐽 π1 (𝐹‘0))    &   ⊢ 𝑄 = (𝐽 π1 (𝐹‘1))    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝐵 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐼(*𝑝‘𝐽)(𝑔(*𝑝‘𝐽)𝐹))]( ≃ph‘𝐽)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))    &   ⊢ 𝐼 = (𝑥 ∈ (0[,]1) ↦ (𝐹‘(1 − 𝑥)))    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpIso 𝑄))
Theorem	pi1cof 24231*	Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 𝐴)    &   ⊢ 𝑄 = (𝐾 π1 𝐵)    &   ⊢ 𝑉 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝑉 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)    ⇒   ⊢ (𝜑 → 𝐺:𝑉⟶(Base‘𝑄))
Theorem	pi1coval 24232*	The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 10-Aug-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 𝐴)    &   ⊢ 𝑄 = (𝐾 π1 𝐵)    &   ⊢ 𝑉 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝑉 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)    ⇒   ⊢ ((𝜑 ∧ 𝑇 ∈ ∪ 𝑉) → (𝐺‘[𝑇]( ≃ph‘𝐽)) = [(𝐹 ∘ 𝑇)]( ≃ph‘𝐾))
Theorem	pi1coghm 24233*	The mapping 𝐺 between fundamental groups is a group homomorphism. (Contributed by Mario Carneiro, 10-Aug-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
⊢ 𝑃 = (𝐽 π1 𝐴)    &   ⊢ 𝑄 = (𝐾 π1 𝐵)    &   ⊢ 𝑉 = (Base‘𝑃)    &   ⊢ 𝐺 = ran (𝑔 ∈ ∪ 𝑉 ↦ ⟨[𝑔]( ≃ph‘𝐽), [(𝐹 ∘ 𝑔)]( ≃ph‘𝐾)⟩)    &   ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))    &   ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))    &   ⊢ (𝜑 → 𝐴 ∈ 𝑋)    &   ⊢ (𝜑 → (𝐹‘𝐴) = 𝐵)    ⇒   ⊢ (𝜑 → 𝐺 ∈ (𝑃 GrpHom 𝑄))
12.5  Metric subcomplex vector spaces
12.5.1  Subcomplex modules
Syntax	cclm 24234	Syntax for the class of subcomplex modules.
class ℂMod
Definition	df-clm 24235*	Define the class of subcomplex modules, which are left modules over a subring of the field of complex numbers ℂfld, which allows us to use the complex addition, multiplication, etc. in theorems about subcomplex modules. Since the field of complex numbers is commutative and so are its subrings (see subrgcrng 20037), left modules over such subrings are the same as right modules, see rmodislmod 20200. Therefore, we drop the word "left" from "subcomplex left module". (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ ℂMod = {𝑤 ∈ LMod ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂfld ↾s 𝑘) ∧ 𝑘 ∈ (SubRing‘ℂfld))}
Theorem	isclm 24236	A subcomplex module is a left module over a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod ↔ (𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
Theorem	clmsca 24237	The ring of scalars 𝐹 of a subcomplex module is the restriction of the field of complex numbers to the base set of 𝐹. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐹 = (ℂfld ↾s 𝐾))
Theorem	clmsubrg 24238	The base set of the ring of scalars of a subcomplex module is the base set of a subring of the field of complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐾 ∈ (SubRing‘ℂfld))
Theorem	clmlmod 24239	A subcomplex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ LMod)
Theorem	clmgrp 24240	A subcomplex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Grp)
Theorem	clmabl 24241	A subcomplex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ (𝑊 ∈ ℂMod → 𝑊 ∈ Abel)
Theorem	clmring 24242	The scalar ring of a subcomplex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)
Theorem	clmfgrp 24243	The scalar ring of a subcomplex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)
Theorem	clm0 24244	The zero of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 0 = (0g‘𝐹))
Theorem	clm1 24245	The identity of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → 1 = (1r‘𝐹))
Theorem	clmadd 24246	The addition of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → + = (+g‘𝐹))
Theorem	clmmul 24247	The multiplication of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → · = (.r‘𝐹))
Theorem	clmcj 24248	The conjugation of the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → ∗ = (*𝑟‘𝐹))
Theorem	isclmi 24249	Reverse direction of isclm 24236. (Contributed by Mario Carneiro, 30-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ 𝐹 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
Theorem	clmzss 24250	The scalar ring of a subcomplex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → ℤ ⊆ 𝐾)
Theorem	clmsscn 24251	The scalar ring of a subcomplex module is a subset of the complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
Theorem	clmsub 24252	Subtraction in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) → (𝐴 − 𝐵) = (𝐴(-g‘𝐹)𝐵))
Theorem	clmneg 24253	Negation in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → -𝐴 = ((invg‘𝐹)‘𝐴))
Theorem	clmneg1 24254	Minus one is in the scalar ring of a subcomplex module. (Contributed by AV, 28-Sep-2021.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (𝑊 ∈ ℂMod → -1 ∈ 𝐾)
Theorem	clmabs 24255	Norm in the scalar ring of a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝐾) → (abs‘𝐴) = ((norm‘𝐹)‘𝐴))
Theorem	clmacl 24256	Closure of ring addition for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
Theorem	clmmcl 24257	Closure of ring multiplication for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
Theorem	clmsubcl 24258	Closure of ring subtraction for a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾) → (𝑋 − 𝑌) ∈ 𝐾)
Theorem	lmhmclm 24259	The domain of a linear operator is a subcomplex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))
Theorem	clmvscl 24260	Closure of scalar product for a subcomplex module. Analogue of lmodvscl 20149. (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑄 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑄 · 𝑋) ∈ 𝑉)
Theorem	clmvsass 24261	Associative law for scalar product. Analogue of lmodvsass 20157. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 · 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
Theorem	clmvscom 24262	Commutative law for the scalar product. (Contributed by NM, 14-Feb-2008.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → (𝑄 · (𝑅 · 𝑋)) = (𝑅 · (𝑄 · 𝑋)))
Theorem	clmvsdir 24263	Distributive law for scalar product (right-distributivity). (lmodvsdir 20156 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝑄 ∈ 𝐾 ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉)) → ((𝑄 + 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
Theorem	clmvsdi 24264	Distributive law for scalar product (left-distributivity). (lmodvsdi 20155 analog.) (Contributed by NM, 3-Nov-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐴 · (𝑋 + 𝑌)) = ((𝐴 · 𝑋) + (𝐴 · 𝑌)))
Theorem	clmvs1 24265	Scalar product with ring unit. (lmodvs1 20160 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (1 · 𝑋) = 𝑋)
Theorem	clmvs2 24266	A vector plus itself is two times the vector. (Contributed by NM, 1-Feb-2007.) (Revised by AV, 21-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + 𝐴) = (2 · 𝐴))
Theorem	clm0vs 24267	Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 20165 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (0 · 𝑋) = 0 )
Theorem	clmopfne 24268	The (functionalized) operations of addition and multiplication by a scalar of a subcomplex module cannot be identical. (Contributed by NM, 31-May-2008.) (Revised by AV, 3-Oct-2021.)
⊢ · = ( ·sf ‘𝑊)    &   ⊢ + = (+𝑓‘𝑊)    ⇒   ⊢ (𝑊 ∈ ℂMod → + ≠ · )
Theorem	isclmp 24269*	The predicate "is a subcomplex module". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑊 ∈ ℂMod ↔ ((𝑊 ∈ Grp ∧ 𝑆 = (ℂfld ↾s 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))
Theorem	isclmi0 24270*	Properties that determine a subcomplex module. (Contributed by NM, 5-Nov-2006.) (Revised by AV, 4-Oct-2021.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    &   ⊢ 𝑆 = (ℂfld ↾s 𝐾)    &   ⊢ 𝑊 ∈ Grp    &   ⊢ 𝐾 ∈ (SubRing‘ℂfld)    &   ⊢ (𝑥 ∈ 𝑉 → (1 · 𝑥) = 𝑥)    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → (𝑦 · 𝑥) ∈ 𝑉)    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)))    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)))    &   ⊢ ((𝑦 ∈ 𝐾 ∧ 𝑧 ∈ 𝐾 ∧ 𝑥 ∈ 𝑉) → ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))    ⇒   ⊢ 𝑊 ∈ ℂMod
Theorem	clmvneg1 24271	Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 20175 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝑋 ∈ 𝑉) → (-1 · 𝑋) = (𝑁‘𝑋))
Theorem	clmvsneg 24272	Multiplication of a vector by a negated scalar. (lmodvsneg 20176 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐵 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑁 = (invg‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑊 ∈ ℂMod)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑁‘(𝑅 · 𝑋)) = (-𝑅 · 𝑋))
Theorem	clmmulg 24273	The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ ∙ = (.g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑉) → (𝐴 ∙ 𝐵) = (𝐴 · 𝐵))
Theorem	clmsubdir 24274	Scalar multiplication distributive law for subtraction. (lmodsubdir 20190 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ − = (-g‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ ℂMod)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝐴 − 𝐵) · 𝑋) = ((𝐴 · 𝑋) − (𝐵 · 𝑋)))
Theorem	clmpm1dir 24275	Subtractive distributive law for the scalar product of a subcomplex module. (Contributed by NM, 31-Jul-2007.) (Revised by AV, 21-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝐾 = (Base‘(Scalar‘𝑊))    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) · 𝐶) = ((𝐴 · 𝐶) + (-1 · (𝐵 · 𝐶))))
Theorem	clmnegneg 24276	Double negative of a vector. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 21-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (-1 · (-1 · 𝐴)) = 𝐴)
Theorem	clmnegsubdi2 24277	Distribution of negative over vector subtraction. (Contributed by NM, 6-Aug-2007.) (Revised by AV, 29-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (-1 · (𝐴 + (-1 · 𝐵))) = (𝐵 + (-1 · 𝐴)))
Theorem	clmsub4 24278	Rearrangement of 4 terms in a mixed vector addition and subtraction. (Contributed by NM, 5-Aug-2007.) (Revised by AV, 29-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 + 𝐵) + (-1 · (𝐶 + 𝐷))) = ((𝐴 + (-1 · 𝐶)) + (𝐵 + (-1 · 𝐷))))
Theorem	clmvsrinv 24279	A vector minus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → (𝐴 + (-1 · 𝐴)) = 0 )
Theorem	clmvslinv 24280	Minus a vector plus itself. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 28-Sep-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ((-1 · 𝐴) + 𝐴) = 0 )
Theorem	clmvsubval 24281	Value of vector subtraction in terms of addition in a subcomplex module. Analogue of lmodvsubval2 20187. (Contributed by NM, 31-Mar-2014.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = (𝐴 + (-1 · 𝐵)))
Theorem	clmvsubval2 24282	Value of vector subtraction on a subcomplex module. (Contributed by Mario Carneiro, 19-Nov-2013.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴 − 𝐵) = ((-1 · 𝐵) + 𝐴))
Theorem	clmvz 24283	Two ways to express the negative of a vector. (Contributed by NM, 29-Feb-2008.) (Revised by AV, 7-Oct-2021.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ − = (-g‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ 𝑉) → ( 0 − 𝐴) = (-1 · 𝐴))
Theorem	zlmclm 24284	The ℤ-module operation turns an arbitrary abelian group into a subcomplex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    ⇒   ⊢ (𝐺 ∈ Abel ↔ 𝑊 ∈ ℂMod)
Theorem	clmzlmvsca 24285	The scalar product of a subcomplex module matches the scalar product of the derived ℤ-module, which implies, together with zlmbas 20729 and zlmplusg 20731, that any module over ℤ is structure-equivalent to the canonical ℤ-module ℤMod‘𝐺. (Contributed by Mario Carneiro, 30-Oct-2015.)
⊢ 𝑊 = (ℤMod‘𝐺)    &   ⊢ 𝑋 = (Base‘𝐺)    ⇒   ⊢ ((𝐺 ∈ ℂMod ∧ (𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑋)) → (𝐴( ·𝑠 ‘𝐺)𝐵) = (𝐴( ·𝑠 ‘𝑊)𝐵))
Theorem	nmoleub2lem 24286*	Lemma for nmoleub2a 24289 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝜓 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)    &   ⊢ ((((𝜑 ∧ ∀𝑥 ∈ 𝑉 (𝜓 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦 ∈ 𝑉 ∧ 𝑦 ≠ (0g‘𝑆))) → (𝑀‘(𝐹‘𝑦)) ≤ (𝐴 · (𝐿‘𝑦)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → (𝜓 → (𝐿‘𝑥) ≤ 𝑅))    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 (𝜓 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmoleub2lem3 24287*	Lemma for nmoleub2a 24289 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → ℚ ⊆ 𝐾)    &   ⊢ · = ( ·𝑠 ‘𝑆)    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → 𝐵 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 ≠ (0g‘𝑆))    &   ⊢ (𝜑 → ((𝑟 · 𝐵) ∈ 𝑉 → ((𝐿‘(𝑟 · 𝐵)) < 𝑅 → ((𝑀‘(𝐹‘(𝑟 · 𝐵))) / 𝑅) ≤ 𝐴)))    &   ⊢ (𝜑 → ¬ (𝑀‘(𝐹‘𝐵)) ≤ (𝐴 · (𝐿‘𝐵)))    ⇒   ⊢ ¬ 𝜑
Theorem	nmoleub2lem2 24288*	Lemma for nmoleub2a 24289 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → ℚ ⊆ 𝐾)    &   ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥)𝑂𝑅 → (𝐿‘𝑥) ≤ 𝑅))    &   ⊢ (((𝐿‘𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿‘𝑥) < 𝑅 → (𝐿‘𝑥)𝑂𝑅))    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥)𝑂𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmoleub2a 24289*	The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → ℚ ⊆ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) ≤ 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmoleub2b 24290*	The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → ℚ ⊆ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) < 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmoleub3 24291*	The operator norm is the supremum of the value of a linear operator on the unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.) (Proof shortened by AV, 29-Sep-2021.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐾 = (Base‘𝐺)    &   ⊢ (𝜑 → 𝑆 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝑇 ∈ (NrmMod ∩ ℂMod))    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 LMHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ*)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ (𝜑 → ℝ ⊆ 𝐾)    ⇒   ⊢ (𝜑 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 ((𝐿‘𝑥) = 𝑅 → ((𝑀‘(𝐹‘𝑥)) / 𝑅) ≤ 𝐴)))
Theorem	nmhmcn 24292	A linear operator over a normed subcomplex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝐾 = (TopOpen‘𝑇)    &   ⊢ 𝐺 = (Scalar‘𝑆)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
Theorem	cmodscexp 24293	The powers of i belong to the scalar subring of a subcomplex module if i belongs to the scalar subring . (Contributed by AV, 18-Oct-2021.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    ⇒   ⊢ (((𝑊 ∈ ℂMod ∧ i ∈ 𝐾) ∧ 𝑁 ∈ ℕ) → (i↑𝑁) ∈ 𝐾)
Theorem	cmodscmulexp 24294	The scalar product of a vector with powers of i belongs to the base set of a subcomplex module if the scalar subring of th subcomplex module contains i. (Contributed by AV, 18-Oct-2021.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    ⇒   ⊢ ((𝑊 ∈ ℂMod ∧ (i ∈ 𝐾 ∧ 𝐵 ∈ 𝑋 ∧ 𝑁 ∈ ℕ)) → ((i↑𝑁) · 𝐵) ∈ 𝑋)
12.5.2  Subcomplex vector spaces Usually, "complex vector spaces" are vector spaces over the field of the complex numbers, see for example the definition in [Roman] p. 36. In the setting of set.mm, it is convenient to consider collectively vector spaces on subfields of the field of complex numbers. We call these, "subcomplex vector spaces" and collect them in the class ℂVec defined in df-cvs 24296 and characterized in iscvs 24299. These include rational vector spaces (qcvs 24320), real vector spaces (recvs 24318) and complex vector spaces (cncvs 24317). This definition is analogous to the definition of subcomplex modules (and their class ℂMod), which are modules over subrings of the field of complex numbers. Note that ZZ-modules (that are roughly the same thing as Abelian groups, see zlmclm 24284) are subcomplex modules but are not subcomplex vector spaces (see zclmncvs 24321), because the ring ZZ is not a division ring (see zringndrg 20699). Since the field of complex numbers is commutative, so are its subrings, so there is no need to explicitly state "left module" or "left vector space" for subcomplex modules or vector spaces.
Syntax	ccvs 24295	Syntax for the class of subcomplex vector spaces.
class ℂVec
Definition	df-cvs 24296	Define the class of subcomplex vector spaces, which are the subcomplex modules which are also vector spaces. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ ℂVec = (ℂMod ∩ LVec)
Theorem	cvslvec 24297	A subcomplex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ (𝜑 → 𝑊 ∈ ℂVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ LVec)
Theorem	cvsclm 24298	A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019.)
⊢ (𝜑 → 𝑊 ∈ ℂVec)    ⇒   ⊢ (𝜑 → 𝑊 ∈ ℂMod)
Theorem	iscvs 24299	A subcomplex vector space is a subcomplex module over a division ring. For example, the subcomplex modules over the rational or real or complex numbers are subcomplex vector spaces. (Contributed by AV, 4-Oct-2021.)
⊢ (𝑊 ∈ ℂVec ↔ (𝑊 ∈ ℂMod ∧ (Scalar‘𝑊) ∈ DivRing))
Theorem	iscvsp 24300*	The predicate "is a subcomplex vector space". (Contributed by NM, 31-May-2008.) (Revised by AV, 4-Oct-2021.)
⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ + = (+g‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑆 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝑆)    ⇒   ⊢ (𝑊 ∈ ℂVec ↔ ((𝑊 ∈ Grp ∧ (𝑆 ∈ DivRing ∧ 𝑆 = (ℂfld ↾s 𝐾)) ∧ 𝐾 ∈ (SubRing‘ℂfld)) ∧ ∀𝑥 ∈ 𝑉 ((1 · 𝑥) = 𝑥 ∧ ∀𝑦 ∈ 𝐾 ((𝑦 · 𝑥) ∈ 𝑉 ∧ ∀𝑧 ∈ 𝑉 (𝑦 · (𝑥 + 𝑧)) = ((𝑦 · 𝑥) + (𝑦 · 𝑧)) ∧ ∀𝑧 ∈ 𝐾 (((𝑧 + 𝑦) · 𝑥) = ((𝑧 · 𝑥) + (𝑦 · 𝑥)) ∧ ((𝑧 · 𝑦) · 𝑥) = (𝑧 · (𝑦 · 𝑥)))))))